diff --git a/examples/hsgp.py b/examples/hsgp.py
new file mode 100644
index 000000000..6f308a3f2
--- /dev/null
+++ b/examples/hsgp.py
@@ -0,0 +1,348 @@
+# ---
+# jupyter:
+#   jupytext:
+#     cell_metadata_filter: -all
+#     custom_cell_magics: kql
+#     text_representation:
+#       extension: .py
+#       format_name: percent
+#       format_version: '1.3'
+#       jupytext_version: 1.11.2
+#   kernelspec:
+#     display_name: .venv
+#     language: python
+#     name: python3
+# ---
+
+# %% [markdown]
+# # Hilbert Space Gaussian Processes
+#
+# Standard GP inference requires inverting an $n \times n$ Gram matrix at
+# $\mathcal{O}(n^3)$ cost, which limits practical use to a few thousand points.
+# The Hilbert Space Gaussian Process (HSGP) approximation
+# ([Solin and Sarkka, 2020](https://link.springer.com/article/10.1007/s11222-019-09886-w))
+# projects the GP onto a truncated basis of Laplacian eigenfunctions, reducing
+# the cost to $\mathcal{O}(nm^2)$ for setup and $\mathcal{O}(m^3)$ per
+# optimisation step, where $m \ll n$ is the number of basis functions. See
+# also the PyMC tutorial by
+# [Orduz and Capretto](https://www.pymc.io/projects/examples/en/latest/gaussian_processes/HSGP-Basic.html)
+# and the standalone introduction by
+# [Orduz (2022)](https://juanitorduz.github.io/hsgp_intro/).
+
+# %%
+from jax import config
+
+config.update("jax_enable_x64", True)
+
+from examples.utils import use_mpl_style
+import jax.numpy as jnp
+import jax.random as jr
+from jaxtyping import install_import_hook
+import matplotlib as mpl
+import matplotlib.pyplot as plt
+
+with install_import_hook("gpjax", "beartype.beartype"):
+    import gpjax as gpx
+
+key = jr.key(42)
+use_mpl_style()
+cols = mpl.rcParams["axes.prop_cycle"].by_key()["color"]
+
+# %% [markdown]
+# ## The HSGP approximation
+#
+# For a stationary kernel $k$ with spectral density $S$, the HSGP approximates
+# the covariance on a bounded domain $[-L, L]$:
+#
+# $$k(x, x') \approx \sum_{j=1}^{m} S\!\left(\sqrt{\lambda_j}\right) \phi_j(x)\,\phi_j(x').$$
+#
+# The eigenfunctions $\phi_j(x) = L^{-1/2}\sin\!\bigl(j\pi(x + L) / 2L\bigr)$
+# are sinusoids of increasing frequency, and the eigenvalues
+# $\lambda_j = (j\pi / 2L)^2$ grow quadratically. The spectral density $S$
+# weights each eigenfunction's contribution. In matrix form the approximate
+# Gram matrix is $\tilde{K} = \Phi\,\Lambda\,\Phi^\top$ where
+# $\Lambda = \mathrm{diag}\!\bigl(S(\sqrt{\lambda_1}), \ldots, S(\sqrt{\lambda_m})\bigr)$.
+#
+# Two parameters govern approximation quality:
+#
+# - **$m$** (number of basis functions): more terms capture shorter
+#   lengthscales, at higher computational cost.
+# - **$L$** (domain half-width): must enclose all inputs after centering,
+#   but setting it much larger than necessary wastes basis functions.
+
+# %% [markdown]
+# ## Eigenfunctions and spectral weights
+#
+# The HSGP has two ingredients: Laplacian eigenfunctions (which depend only on
+# the domain, not the kernel) and spectral weights (which encode the kernel's
+# properties). We visualise both below.
+
+# %%
+x_grid = jnp.linspace(-3.0, 3.0, 300)[:, None]
+
+base_rbf = gpx.kernels.RBF(n_dims=1)
+hsgp_rbf = gpx.kernels.HSGP(
+    base_kernel=base_rbf, num_basis_fns=20, domain_half_width=4.0, center=0.0
+)
+
+phi = hsgp_rbf.eigenfunctions(x_grid)
+
+fig, ax = plt.subplots(figsize=(7.5, 2.5))
+for j in range(6):
+    ax.plot(x_grid, phi[:, j], label=f"$\\phi_{{{j + 1}}}$", color=cols[j % len(cols)])
+ax.set_xlabel("$x$")
+ax.set_ylabel("$\\phi_j(x)$")
+ax.legend(loc="center left", bbox_to_anchor=(1.0, 0.5), fontsize=8)
+ax.set_title("Laplacian eigenfunctions")
+
+# %% [markdown]
+# The eigenfunctions are sinusoids of increasing frequency, forming an
+# orthonormal basis on $[-L, L]$. Because they are independent of the kernel
+# hyperparameters, they need only be evaluated once for a given domain.
+#
+# The spectral weights $S(\sqrt{\lambda_j})$ determine each eigenfunction's
+# contribution. We compare weights for the RBF kernel (Gaussian spectral
+# density, rapid decay) and the Matern-5/2 (heavier-tailed, slower decay).
+
+# %%
+base_m52 = gpx.kernels.Matern52(n_dims=1)
+hsgp_m52 = gpx.kernels.HSGP(
+    base_kernel=base_m52, num_basis_fns=20, domain_half_width=4.0, center=0.0
+)
+
+weights_rbf = hsgp_rbf.spectral_weights()
+weights_m52 = hsgp_m52.spectral_weights()
+
+fig, axes = plt.subplots(1, 2, figsize=(7.5, 2.5), sharey=True)
+basis_indices = jnp.arange(1, 21)
+
+axes[0].stem(basis_indices, weights_rbf, linefmt=cols[0], markerfmt="o", basefmt=" ")
+axes[0].set_title("RBF")
+axes[0].set_xlabel("Basis index $j$")
+axes[0].set_ylabel("$S(\\sqrt{\\lambda_j})$")
+
+axes[1].stem(basis_indices, weights_m52, linefmt=cols[1], markerfmt="o", basefmt=" ")
+axes[1].set_title("Matern-5/2")
+axes[1].set_xlabel("Basis index $j$")
+
+# %% [markdown]
+# The RBF weights decay rapidly, so few basis functions suffice. The
+# Matern-5/2 retains appreciable weight at higher frequencies and needs more
+# terms for the same approximation quality.
+
+# %% [markdown]
+# ## Approximation quality
+#
+# How faithfully does the HSGP Gram matrix reproduce the exact kernel? We
+# compare the *approximation error* $|K - \tilde{K}|$ at $m = 10$, $25$, and
+# $50$ for both the RBF and Matern-5/2 kernels. Plotting the error on a
+# shared log scale makes the convergence difference clearly visible.
+
+# %%
+from matplotlib.colors import LogNorm
+
+x_small = jnp.linspace(-3.0, 3.0, 80)[:, None]
+m_values = [10, 25, 50]
+
+
+def gram_errors(base_kernel, x, m_values):
+    """Return absolute error matrices |K_exact - K_hsgp| for several values of m."""
+    gram_exact = base_kernel.gram(x).to_dense()
+    errors = []
+    for num_basis in m_values:
+        hsgp = gpx.kernels.HSGP(
+            base_kernel=base_kernel,
+            num_basis_fns=num_basis,
+            domain_half_width=4.0,
+            center=0.0,
+        )
+        errors.append(jnp.abs(gram_exact - hsgp.gram(x).to_dense()))
+    return gram_exact, errors
+
+
+gram_exact_rbf, errors_rbf = gram_errors(base_rbf, x_small, m_values)
+gram_exact_m52, errors_m52 = gram_errors(base_m52, x_small, m_values)
+
+log_norm = LogNorm(vmin=1e-6, vmax=1.0)
+fig, axes = plt.subplots(2, 3, figsize=(8, 5), sharex=True, sharey=True)
+
+for j, m in enumerate(m_values):
+    im = axes[0, j].imshow(errors_rbf[j], norm=log_norm, cmap="inferno")
+    axes[0, j].set_title(f"$m = {m}$", fontsize=9)
+    axes[0, j].set_xticks([])
+    axes[0, j].set_yticks([])
+
+    axes[1, j].imshow(errors_m52[j], norm=log_norm, cmap="inferno")
+    axes[1, j].set_xticks([])
+    axes[1, j].set_yticks([])
+
+axes[0, 0].set_ylabel("RBF", fontsize=10)
+axes[1, 0].set_ylabel("Matern-5/2", fontsize=10)
+fig.colorbar(im, ax=axes, label="$|K - \\tilde{K}|$", shrink=0.8)
+fig.suptitle("HSGP approximation error", fontsize=11, y=1.0)
+
+# %% [markdown]
+# The RBF error shrinks rapidly toward machine precision in the interior,
+# reflecting the Gaussian spectral density's fast decay. The Matern-5/2
+# retains appreciable error at $m = 10$ across a wider region, consistent
+# with its heavier spectral tail. By $m = 50$ both kernels are well
+# approximated in the interior; the residual error along the edges is the
+# unavoidable boundary artefact of the Dirichlet eigenbasis.
+
+# %% [markdown]
+# ## Regression with real data
+#
+# We apply the HSGP to daily nitrogen dioxide (NO$_2$) measurements from
+# Marylebone Road in central London, one of the UK's most heavily trafficked
+# roadside sites. The data come from the
+# [Automatic Urban and Rural Network (AURN)](https://uk-air.defra.gov.uk/networks/site-info?site_id=MY1).
+# We aggregate to daily means over approximately ten years (2016--2025),
+# giving several thousand data points where exact GP inference is impractical.
+
+# %%
+from pathlib import Path
+import tempfile
+from urllib.request import urlretrieve
+import warnings
+
+import pandas as pd
+import rdata
+
+SITE = "MY1"
+YEARS = range(2016, 2026)
+
+frames = []
+for year in YEARS:
+    url = f"https://uk-air.defra.gov.uk/openair/R_data/{SITE}_{year}.RData"
+    tmp = Path(tempfile.gettempdir()) / f"{SITE}_{year}.RData"
+    urlretrieve(url, tmp)
+    with warnings.catch_warnings():
+        warnings.simplefilter("ignore")
+        parsed = rdata.read_rda(str(tmp))
+    frames.append(parsed[next(iter(parsed.keys()))])
+
+hourly = pd.concat(frames, ignore_index=True)
+hourly["date"] = pd.to_datetime(hourly["date"], unit="s")
+daily = hourly[["date", "NO2"]].set_index("date").resample("D").mean().dropna()
+
+# %%
+fig, ax = plt.subplots(figsize=(7.5, 2.5))
+ax.plot(daily.index, daily["NO2"], linewidth=0.5, color=cols[0])
+ax.set_xlabel("Date")
+ax.set_ylabel("NO$_2$ ($\\mu$g m$^{-3}$)")
+ax.set_title("Daily mean NO$_2$ at Marylebone Road, London")
+
+# %% [markdown]
+# Clear features include an annual cycle from heating-season emissions, an
+# abrupt dip during the 2020 COVID-19 lockdowns, and a gradual downward trend
+# from tightening vehicle emission standards.
+
+# %%
+# Convert dates to a numeric index and standardise for numerical stability.
+t = (daily.index - daily.index[0]).days.values.astype(float)
+y_obs = daily["NO2"].values
+
+t_mean, t_std = t.mean(), t.std()
+y_mean, y_std = y_obs.mean(), y_obs.std()
+
+t_norm = ((t - t_mean) / t_std)[:, None]
+y_norm = ((y_obs - y_mean) / y_std)[:, None]
+
+D = gpx.Dataset(X=t_norm, y=y_norm)
+
+# %% [markdown]
+# We build a GP prior with an HSGP kernel using a Matern-3/2 base kernel. We
+# use $m = 30$ basis functions and set $L$ to 1.2 times the half-range of
+# the normalised inputs, providing a modest buffer beyond the data boundary.
+
+# %%
+data_half_range = float((t_norm.max() - t_norm.min()) / 2.0)
+L = 1.2 * data_half_range
+
+base_kernel = gpx.kernels.Matern32(n_dims=1)
+hsgp_kernel = gpx.kernels.HSGP(
+    base_kernel=base_kernel,
+    num_basis_fns=30,
+    domain_half_width=L,
+    center=float(t_norm.mean()),
+)
+
+prior = gpx.gps.Prior(mean_function=gpx.mean_functions.Zero(), kernel=hsgp_kernel)
+likelihood = gpx.likelihoods.Gaussian(num_datapoints=D.n)
+posterior = prior * likelihood
+
+# %% [markdown]
+# We optimise the negative conjugate marginal log-likelihood using L-BFGS-B,
+# learning the kernel lengthscale, variance, and observation noise.
+
+# %%
+opt_posterior, history = gpx.fit_scipy(
+    model=posterior,
+    objective=lambda p, d: -gpx.objectives.conjugate_mll(p, d),
+    train_data=D,
+    trainable=gpx.parameters.Parameter,
+)
+
+print(
+    f"Learned lengthscale: {opt_posterior.prior.kernel.base_kernel.lengthscale[...]:.4f}"
+)
+print(
+    f"Learned variance:    {opt_posterior.prior.kernel.base_kernel.variance[...]:.4f}"
+)
+print(f"Learned obs. noise:  {opt_posterior.likelihood.obs_stddev[...] ** 2:.4f}")
+
+# %% [markdown]
+# With optimised hyperparameters, we compute the posterior predictive on a
+# fine grid spanning the training domain.
+
+# %%
+t_test = jnp.linspace(t_norm.min() - 0.05, t_norm.max() + 0.05, 500)[:, None]
+
+latent_dist = opt_posterior.predict(
+    t_test, train_data=D, return_covariance_type="diagonal"
+)
+predictive_dist = opt_posterior.likelihood(latent_dist)
+
+pred_mean = predictive_dist.mean * y_std + y_mean
+pred_std = jnp.sqrt(predictive_dist.variance) * y_std
+
+# Convert test grid back to dates for plotting.
+t_test_days = (t_test.squeeze() * t_std + t_mean).astype(int)
+dates_test = daily.index[0] + pd.to_timedelta(t_test_days, unit="D")
+
+# %%
+fig, ax = plt.subplots(figsize=(7.5, 3.0))
+ax.scatter(daily.index, y_obs, s=1, alpha=0.3, color=cols[0], label="Observations")
+ax.plot(dates_test, pred_mean, color=cols[1], linewidth=1.5, label="Predictive mean")
+ax.fill_between(
+    dates_test,
+    pred_mean - 2 * pred_std,
+    pred_mean + 2 * pred_std,
+    alpha=0.2,
+    color=cols[1],
+    label="Two sigma",
+)
+ax.set_xlabel("Date")
+ax.set_ylabel("NO$_2$ ($\\mu$g m$^{-3}$)")
+ax.legend(loc="upper right", fontsize=8)
+ax.set_title("HSGP posterior for daily NO$_2$ at Marylebone Road")
+
+# %% [markdown]
+# The HSGP posterior captures seasonal structure and the long-term downward
+# trend, with credible intervals that widen where data are sparser. The
+# computation completed in seconds on a single CPU; an exact GP over the same
+# dataset would require forming and decomposing a dense Gram matrix with over
+# ten million entries.
+
+# %% [markdown]
+# ## References
+#
+# - Solin, A. and Sarkka, S. (2020).
+#   [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://link.springer.com/article/10.1007/s11222-019-09886-w).
+#   *Statistics and Computing*.
+# - Orduz, J. and Martin, O. A.
+#   [PyMC HSGP tutorial](https://www.pymc.io/projects/examples/en/latest/gaussian_processes/HSGP-Basic.html).
+# - Orduz, J. (2022).
+#   [Introduction to the HSGP](https://juanitorduz.github.io/hsgp_intro/).
+# - UK [Automatic Urban and Rural Network](https://uk-air.defra.gov.uk/networks/site-info?site_id=MY1)
+#   air quality data via the openair data service.
diff --git a/gpjax/gps.py b/gpjax/gps.py
index ae25c8777..bd813aee5 100644
--- a/gpjax/gps.py
+++ b/gpjax/gps.py
@@ -46,7 +46,9 @@
 from gpjax.linalg.operations import (
     lower_cholesky,
 )
+from gpjax.linalg.operators import LowRank
 from gpjax.linalg.utils import add_jitter
+from gpjax.linalg.woodbury import woodbury_solve
 from gpjax.mean_functions import AbstractMeanFunction
 from gpjax.parameters import (
     Parameter,
@@ -380,7 +382,9 @@ def sample_fn(test_inputs: Float[Array, "N D"]) -> Float[Array, "N B"]:
 
 #######################
 # GP Posteriors
-#######################from gpjax.linalg.operators import LinearOperator
+#######################
+
+
 class AbstractPosterior(nnx.Module, tp.Generic[P, L]):
     r"""Abstract Gaussian process posterior.
 
@@ -587,60 +591,151 @@ def predict(
 
         kernel = self.prior.kernel
         x, y = train_data.X, train_data.y
-        P = self.likelihood.num_outputs
+        num_outputs = self.likelihood.num_outputs
 
-        # Prepare targets via likelihood protocol (identity for single-output,
-        # output-major reshape for multi-output)
         mx = self.prior.mean_function(x)
         y_flat, mx_flat = self.likelihood.prepare_targets(y, mx)
         noise = self.likelihood.noise_vector(train_data.n)
 
         Kxx = kernel.gram(x)
-        Kxx_dense = add_jitter(Kxx.to_dense(), self.jitter)
-        Sigma_dense = Kxx_dense + jnp.diag(noise)
-        Sigma = psd(Dense(Sigma_dense))
-        L_sigma = lower_cholesky(Sigma)
-
         Kxt = kernel.cross_covariance(x, test_inputs)
-        L_inv_Kxt = solve(L_sigma, Kxt)
-        L_inv_y_diff = solve(L_sigma, y_flat - mx_flat)
 
-        mean_t_raw = self.prior.mean_function(test_inputs)
-        mean_t = jnp.tile(mean_t_raw, (P, 1)) if P > 1 else mean_t_raw
-        mean = mean_t + jnp.matmul(L_inv_Kxt.T, L_inv_y_diff)
+        # Posterior test-point mean: m(x*) + Kxt^T Sigma^{-1} (y - m(x))
+        mean_test_raw = self.prior.mean_function(test_inputs)
+        mean_test = (
+            jnp.tile(mean_test_raw, (num_outputs, 1))
+            if num_outputs > 1
+            else mean_test_raw
+        )
 
-        # Diagonal covariance not yet supported for multi-output
-        if return_covariance_type == "diagonal" and P > 1:
+        if return_covariance_type == "diagonal" and num_outputs > 1:
             warnings.warn(
-                "Diagonal covariance is not yet supported for multi-output GPs. "
-                "Returning full covariance.",
+                "Diagonal covariance is not yet supported for "
+                "multi-output GPs. Returning full covariance.",
                 stacklevel=2,
             )
             return_covariance_type = "dense"
 
-        def _return_full_covariance(L_inv_Kxt, t):
-            Ktt = kernel.gram(t)
-            covariance = Ktt.to_dense() - jnp.matmul(L_inv_Kxt.T, L_inv_Kxt)
-            covariance = add_jitter(covariance, self.prior.jitter)
-            covariance = psd(Dense(covariance))
-            return covariance
+        if isinstance(Kxx, LowRank):
+            mean, cov = self._predict_lowrank(
+                kernel,
+                Kxx,
+                Kxt,
+                y_flat,
+                mx_flat,
+                noise,
+                mean_test,
+                test_inputs,
+                return_covariance_type,
+            )
+        else:
+            mean, cov = self._predict_dense(
+                kernel,
+                Kxx,
+                Kxt,
+                y_flat,
+                mx_flat,
+                noise,
+                mean_test,
+                test_inputs,
+                return_covariance_type,
+            )
 
-        def _return_diagonal_covariance(L_inv_Kxt, t):
-            Ktt = kernel.diagonal(t).diagonal
-            covariance = Ktt - jnp.einsum("ij, ji->i", L_inv_Kxt.T, L_inv_Kxt)
+        return GaussianDistribution(loc=jnp.atleast_1d(mean.squeeze()), scale=cov)
+
+    def _predict_lowrank(
+        self,
+        kernel,
+        Kxx,
+        Kxt,
+        y_flat,
+        mx_flat,
+        noise,
+        mean_test,
+        test_inputs,
+        return_covariance_type,
+    ):
+        r"""Posterior prediction using the Woodbury identity.
+
+        When K_xx = W W^T (low-rank), the marginal covariance is
+        Sigma = W W^T + D with D = diag(noise + jitter).  The Woodbury
+        identity gives Sigma^{-1} in O(N m^2) instead of O(N^3).
+        """
+        W = Kxx.factor
+        noise_with_jitter = noise + self.jitter
+
+        # alpha = Sigma^{-1} (y - m(x))
+        alpha = woodbury_solve(W, noise_with_jitter, (y_flat - mx_flat).squeeze())
+        mean = mean_test + jnp.matmul(Kxt.T, alpha[:, None])
+
+        # Sigma^{-1} Kxt  for the covariance update
+        Sigma_inv_Kxt = woodbury_solve(W, noise_with_jitter, Kxt)
+
+        def _full_covariance(Sigma_inv_Kxt, test_inputs):
+            Ktt = kernel.gram(test_inputs).to_dense()
+            covariance = Ktt - jnp.matmul(Kxt.T, Sigma_inv_Kxt)
+            return psd(Dense(add_jitter(covariance, self.prior.jitter)))
+
+        def _diagonal_covariance(Sigma_inv_Kxt, test_inputs):
+            Ktt_diag = kernel.diagonal(test_inputs).diagonal
+            covariance = Ktt_diag - jnp.einsum("ij,ij->j", Kxt, Sigma_inv_Kxt)
             covariance += self.prior.jitter
-            covariance = psd(Dense(jnp.diag(jnp.atleast_1d(covariance.squeeze()))))
-            return covariance
+            return psd(Dense(jnp.diag(jnp.atleast_1d(covariance.squeeze()))))
 
         cov = jax.lax.cond(
             return_covariance_type == "dense",
-            _return_full_covariance,
-            _return_diagonal_covariance,
+            _full_covariance,
+            _diagonal_covariance,
+            Sigma_inv_Kxt,
+            test_inputs,
+        )
+
+        return mean, cov
+
+    def _predict_dense(
+        self,
+        kernel,
+        Kxx,
+        Kxt,
+        y_flat,
+        mx_flat,
+        noise,
+        mean_test,
+        test_inputs,
+        return_covariance_type,
+    ):
+        r"""Posterior prediction via the standard Cholesky decomposition.
+
+        Sigma = K_xx + D,  L L^T = Sigma,  then solves use L^{-1}.
+        """
+        Sigma_dense = add_jitter(Kxx.to_dense(), self.jitter) + jnp.diag(noise)
+        L_sigma = lower_cholesky(psd(Dense(Sigma_dense)))
+
+        L_inv_Kxt = solve(L_sigma, Kxt)
+        L_inv_diff = solve(L_sigma, y_flat - mx_flat)
+
+        mean = mean_test + jnp.matmul(L_inv_Kxt.T, L_inv_diff)
+
+        def _full_covariance(L_inv_Kxt, test_inputs):
+            Ktt = kernel.gram(test_inputs).to_dense()
+            covariance = Ktt - jnp.matmul(L_inv_Kxt.T, L_inv_Kxt)
+            return psd(Dense(add_jitter(covariance, self.prior.jitter)))
+
+        def _diagonal_covariance(L_inv_Kxt, test_inputs):
+            Ktt_diag = kernel.diagonal(test_inputs).diagonal
+            covariance = Ktt_diag - jnp.einsum("ij, ji->i", L_inv_Kxt.T, L_inv_Kxt)
+            covariance += self.prior.jitter
+            return psd(Dense(jnp.diag(jnp.atleast_1d(covariance.squeeze()))))
+
+        cov = jax.lax.cond(
+            return_covariance_type == "dense",
+            _full_covariance,
+            _diagonal_covariance,
             L_inv_Kxt,
             test_inputs,
         )
 
-        return GaussianDistribution(loc=jnp.atleast_1d(mean.squeeze()), scale=cov)
+        return mean, cov
 
     def sample_approx(
         self,
diff --git a/gpjax/kernels/__init__.py b/gpjax/kernels/__init__.py
index 5ad342b86..cf8b06042 100644
--- a/gpjax/kernels/__init__.py
+++ b/gpjax/kernels/__init__.py
@@ -19,7 +19,7 @@
 from gpjax.kernels.additive import (
     OrthogonalAdditiveKernel,
 )
-from gpjax.kernels.approximations import RFF
+from gpjax.kernels.approximations import HSGP, RFF
 from gpjax.kernels.base import (
     AbstractKernel,
     Constant,
@@ -32,6 +32,7 @@
     DenseKernelComputation,
     DiagonalKernelComputation,
     EigenKernelComputation,
+    HSGPComputation,
 )
 from gpjax.kernels.multioutput import (
     ICMKernel,
@@ -57,6 +58,7 @@
 )
 
 __all__ = [
+    "HSGP",
     "RBF",
     "RFF",
     "AbstractKernel",
@@ -68,6 +70,7 @@
     "DiagonalKernelComputation",
     "EigenKernelComputation",
     "GraphKernel",
+    "HSGPComputation",
     "ICMKernel",
     "LCMKernel",
     "Linear",
diff --git a/gpjax/kernels/approximations/__init__.py b/gpjax/kernels/approximations/__init__.py
index 9b08299e5..1f6a804e9 100644
--- a/gpjax/kernels/approximations/__init__.py
+++ b/gpjax/kernels/approximations/__init__.py
@@ -1,3 +1,4 @@
+from gpjax.kernels.approximations.hsgp import HSGP
 from gpjax.kernels.approximations.rff import RFF
 
-__all__ = ["RFF"]
+__all__ = ["HSGP", "RFF"]
diff --git a/gpjax/kernels/approximations/hsgp.py b/gpjax/kernels/approximations/hsgp.py
new file mode 100644
index 000000000..3c3b977bf
--- /dev/null
+++ b/gpjax/kernels/approximations/hsgp.py
@@ -0,0 +1,175 @@
+"""Hilbert Space Gaussian Process (HSGP) kernel approximation.
+
+Reference:
+    Solin & Sarkka (2019). "Hilbert Space Methods for Reduced-Rank
+    Gaussian Process Regression." Statistics and Computing.
+"""
+
+import beartype.typing as tp
+import jax.numpy as jnp
+from jaxtyping import Float
+
+from gpjax.kernels.base import AbstractKernel
+from gpjax.kernels.computations.hsgp import HSGPComputation
+from gpjax.kernels.stationary.base import StationaryKernel
+from gpjax.typing import Array
+
+
+class HSGP(AbstractKernel):
+    r"""Hilbert Space Gaussian Process approximation (1D).
+
+    Approximates a stationary kernel by projecting onto the eigenbasis of the
+    Laplacian operator with Dirichlet boundary conditions on :math:`[-L, L]`.
+    The approximate covariance function is:
+
+    .. math::
+        \tilde{k}(x, x') = \sum_{j=1}^{m} S(\sqrt{\lambda_j})\,
+        \phi_j(x)\,\phi_j(x')
+
+    where :math:`\lambda_j = (j\pi / 2L)^2` are the eigenvalues,
+    :math:`\phi_j(x) = L^{-1/2}\sin(j\pi(x + L) / 2L)` are the
+    eigenfunctions, and :math:`S` is the spectral density of the base
+    kernel.
+
+    The linearised form decomposes the GP function as:
+
+    .. math::
+        f(x) \approx \Phi(x)\,\mathrm{diag}(\sqrt{S})\,\beta,
+        \quad \beta \sim \mathcal{N}(0, I_m)
+
+    which is a Bayesian linear model with :math:`m` basis functions.
+
+    Args:
+        base_kernel: Stationary kernel to approximate.  Must provide a
+            ``spectral_density`` property returning a
+            :class:`~gpjax.kernels.stationary.utils.SpectralDensity`.
+        num_basis_fns: Number of basis functions :math:`m`.
+        domain_half_width: Half-width :math:`L` of the approximation domain.
+            Inputs should lie well inside :math:`[-L, L]` (after centering).
+        center: Center of the data domain.  If ``None``, it is set
+            automatically on the first call to :meth:`eigenfunctions` as the
+            midpoint of the observed input range.
+        compute_engine: Computation engine (default
+            :class:`~gpjax.kernels.computations.hsgp.HSGPComputation`).
+
+    Example:
+        >>> import gpjax as gpx
+        >>> import jax.numpy as jnp
+        >>> X = jnp.linspace(-1, 1, 50)[:, None]
+        >>> base = gpx.kernels.Matern52(n_dims=1)
+        >>> hsgp = gpx.kernels.HSGP(base, num_basis_fns=20, domain_half_width=5.0)
+        >>> K = hsgp.gram(X)  # approximate Gram matrix
+    """
+
+    compute_engine: HSGPComputation
+
+    def __init__(
+        self,
+        base_kernel: StationaryKernel,
+        num_basis_fns: int,
+        domain_half_width: float,
+        center: tp.Union[float, None] = None,
+        compute_engine: HSGPComputation = HSGPComputation(),
+    ):
+        if not isinstance(base_kernel, StationaryKernel):
+            raise TypeError(
+                "HSGP can only approximate stationary kernels. "
+                f"Got {type(base_kernel).__name__}."
+            )
+        _ = base_kernel.spectral_density
+
+        super().__init__(
+            active_dims=base_kernel.active_dims,
+            n_dims=base_kernel.n_dims,
+            compute_engine=compute_engine,
+        )
+        self.base_kernel = base_kernel
+        self.num_basis_fns = num_basis_fns
+        self.domain_half_width = domain_half_width
+        self._center = center
+        self.name = f"{self.base_kernel.name} (HSGP)"
+
+    def eigenvalues(self) -> Float[Array, " m"]:
+        r"""Square roots of the Laplacian eigenvalues.
+
+        .. math::
+            \sqrt{\lambda_j} = \frac{j\pi}{2L}, \quad j = 1, \ldots, m
+
+        Returns:
+            Array of shape ``(m,)``.
+        """
+        indices = jnp.arange(1, self.num_basis_fns + 1)
+        return indices * jnp.pi / (2.0 * self.domain_half_width)
+
+    def eigenfunctions(self, x: Float[Array, "N 1"]) -> Float[Array, "N m"]:
+        r"""Laplacian eigenfunctions evaluated at *x*.
+
+        .. math::
+            \phi_j(x) = \frac{1}{\sqrt{L}}
+            \sin\!\Bigl(\frac{j\pi\,(x + L)}{2L}\Bigr)
+
+        where *x* is first shifted by the stored center.
+
+        Args:
+            x: Input locations of shape ``(N, 1)``.
+
+        Returns:
+            Basis matrix :math:`\Phi` of shape ``(N, m)``.
+        """
+        if self._center is None:
+            self._center = float((x.max() + x.min()) / 2.0)
+
+        half_width = self.domain_half_width
+        x_centered = x - self._center
+        sqrt_eigenvalues = self.eigenvalues()
+        return jnp.sin((x_centered + half_width) * sqrt_eigenvalues) / jnp.sqrt(
+            half_width
+        )
+
+    def spectral_weights(self) -> Float[Array, " m"]:
+        r"""Spectral density evaluated at the eigenvalue square roots.
+
+        .. math::
+            S(\sqrt{\lambda_j})
+
+        Returns:
+            Array of shape ``(m,)`` with the spectral weights.
+        """
+        omega = self.eigenvalues()
+        return self.base_kernel.spectral_density(
+            omega,
+            self.base_kernel.variance[...],
+            self.base_kernel.lengthscale[...],
+        )
+
+    def compute_basis(
+        self, x: Float[Array, "N 1"]
+    ) -> tuple[Float[Array, "N m"], Float[Array, " m"]]:
+        r"""Linearised HSGP decomposition.
+
+        Returns :math:`(\Phi, \sqrt{S})` such that
+
+        .. math::
+            f(x) \approx \Phi\,\mathrm{diag}(\sqrt{S})\,\beta,
+            \quad \beta \sim \mathcal{N}(0, I_m).
+
+        Args:
+            x: Input locations of shape ``(N, 1)``.
+
+        Returns:
+            Tuple ``(phi, sqrt_spectral_weights)`` where ``phi`` has shape
+            ``(N, m)`` and ``sqrt_spectral_weights`` has shape ``(m,)``.
+        """
+        phi = self.eigenfunctions(x)
+        sqrt_spectral_weights = jnp.sqrt(self.spectral_weights())
+        return phi, sqrt_spectral_weights
+
+    def __call__(
+        self,
+        x: Float[Array, " D"],
+        y: Float[Array, " D"],
+    ) -> None:
+        raise RuntimeError(
+            "HSGP does not support pointwise kernel evaluation. "
+            "Use gram(), cross_covariance(), or compute_basis() instead."
+        )
diff --git a/gpjax/kernels/computations/__init__.py b/gpjax/kernels/computations/__init__.py
index 60eb5a1b3..1de8b5f5f 100644
--- a/gpjax/kernels/computations/__init__.py
+++ b/gpjax/kernels/computations/__init__.py
@@ -21,6 +21,7 @@
 from gpjax.kernels.computations.dense import DenseKernelComputation
 from gpjax.kernels.computations.diagonal import DiagonalKernelComputation
 from gpjax.kernels.computations.eigen import EigenKernelComputation
+from gpjax.kernels.computations.hsgp import HSGPComputation
 
 __all__ = [
     "AbstractKernelComputation",
@@ -29,4 +30,5 @@
     "DenseKernelComputation",
     "DiagonalKernelComputation",
     "EigenKernelComputation",
+    "HSGPComputation",
 ]
diff --git a/gpjax/kernels/computations/basis_functions.py b/gpjax/kernels/computations/basis_functions.py
index d4a06c2a2..07c8818d8 100644
--- a/gpjax/kernels/computations/basis_functions.py
+++ b/gpjax/kernels/computations/basis_functions.py
@@ -7,67 +7,69 @@
 from gpjax.kernels.computations.base import AbstractKernelComputation
 from gpjax.linalg import (
     Diagonal,
+    LowRank,
 )
+from gpjax.linalg.utils import psd
 from gpjax.typing import Array
 
 K = tp.TypeVar("K", bound="gpjax.kernels.approximations.RFF")
 
-# TODO: Use low rank linear operator!
-
 
 class BasisFunctionComputation(AbstractKernelComputation):
-    r"""Compute engine class for finite basis function approximations to a kernel."""
+    r"""Compute engine for finite basis function approximations (RFF)."""
+
+    def gram(self, kernel: K, x: Float[Array, "N D"]) -> LowRank:
+        features = self.compute_features(kernel, x)
+        weighted_features = features * jnp.sqrt(self.scaling(kernel))
+        return psd(LowRank(weighted_features))
 
     def _cross_covariance(
         self, kernel: K, x: Float[Array, "N D"], y: Float[Array, "M D"]
     ) -> Float[Array, "N M"]:
-        z1 = self.compute_features(kernel, x)
-        z2 = self.compute_features(kernel, y)
-        return self.scaling(kernel) * jnp.matmul(z1, z2.T)
+        features_x = self.compute_features(kernel, x)
+        features_y = self.compute_features(kernel, y)
+        return self.scaling(kernel) * jnp.matmul(features_x, features_y.T)
 
     def _gram(self, kernel: K, inputs: Float[Array, "N D"]) -> Float[Array, "N N"]:
-        z1 = self.compute_features(kernel, inputs)
-        return self.scaling(kernel) * jnp.matmul(z1, z1.T)
+        features = self.compute_features(kernel, inputs)
+        return self.scaling(kernel) * jnp.matmul(features, features.T)
 
     def diagonal(self, kernel: K, inputs: Float[Array, "N D"]) -> Diagonal:
-        r"""For a given kernel, compute the elementwise diagonal of the
-        NxN gram matrix on an input matrix of shape NxD.
+        r"""Diagonal of the approximate Gram matrix.
 
         Args:
-            kernel (AbstractKernel): the kernel function.
-            inputs (Float[Array, "N D"]): The input matrix.
+            kernel: The RFF kernel.
+            inputs: Input matrix of shape ``(N, D)``.
 
-        Returns
-        -------
-            Diagonal: The computed diagonal variance entries.
+        Returns:
+            Diagonal variance entries.
         """
         return super().diagonal(kernel.base_kernel, inputs)
 
     def compute_features(
         self, kernel: K, x: Float[Array, "N D"]
     ) -> Float[Array, "N L"]:
-        r"""Compute the features for the inputs.
+        r"""Compute random Fourier features :math:`[\cos(z), \sin(z)]`.
 
         Args:
-            kernel: the kernel function.
-            x: the inputs to the kernel function of shape `(N, D)`.
+            kernel: The RFF kernel.
+            x: Inputs of shape ``(N, D)``.
 
         Returns:
-            A matrix of shape $N \times L$ representing the random fourier features where $L = 2M$.
+            Feature matrix of shape ``(N, 2M)`` where ``M = num_basis_fns``.
         """
         frequencies = kernel.frequencies
-        scaling_factor = kernel.base_kernel.lengthscale[...]
-        z = jnp.matmul(x, (frequencies / scaling_factor).T)
-        z = jnp.concatenate([jnp.cos(z), jnp.sin(z)], axis=-1)
-        return z
+        lengthscale = kernel.base_kernel.lengthscale[...]
+        projected = jnp.matmul(x, (frequencies / lengthscale).T)
+        return jnp.concatenate([jnp.cos(projected), jnp.sin(projected)], axis=-1)
 
     def scaling(self, kernel: K) -> Float[Array, ""]:
-        r"""Compute the scaling factor for the covariance matrix.
+        r"""Variance scaling factor: :math:`\sigma^2 / M`.
 
         Args:
-            kernel: the kernel function.
+            kernel: The RFF kernel.
 
         Returns:
-            A scalar array representing the scaling factor.
+            Scalar scaling factor.
         """
         return kernel.base_kernel.variance[...] / kernel.num_basis_fns
diff --git a/gpjax/kernels/computations/hsgp.py b/gpjax/kernels/computations/hsgp.py
new file mode 100644
index 000000000..8baadb90f
--- /dev/null
+++ b/gpjax/kernels/computations/hsgp.py
@@ -0,0 +1,52 @@
+"""Compute engine for the Hilbert Space GP approximation."""
+
+import typing as tp
+
+import jax.numpy as jnp
+from jaxtyping import Float
+
+import gpjax
+from gpjax.kernels.computations.base import AbstractKernelComputation
+from gpjax.linalg import Diagonal, LowRank
+from gpjax.linalg.utils import psd
+from gpjax.typing import Array
+
+K = tp.TypeVar("K", bound="gpjax.kernels.approximations.HSGP")
+
+
+class HSGPComputation(AbstractKernelComputation):
+    r"""Compute engine for the HSGP kernel approximation.
+
+    Builds the Gram matrix from the low-rank decomposition
+    :math:`\tilde{K} = \Phi \Lambda \Phi^\top` where
+    :math:`\Phi` contains Laplacian eigenfunctions and
+    :math:`\Lambda = \mathrm{diag}(S(\sqrt{\lambda_1}), \ldots,
+    S(\sqrt{\lambda_m}))`.
+    """
+
+    def _weighted_basis(
+        self, kernel: K, x: Float[Array, "N D"]
+    ) -> tuple[Float[Array, "N m"], Float[Array, "N m"]]:
+        r"""Return ``(phi, weighted_phi)`` where ``weighted_phi = phi * sqrt(S)``."""
+        phi, sqrt_spectral_weights = kernel.compute_basis(x)
+        weighted_phi = phi * sqrt_spectral_weights[None, :]
+        return phi, weighted_phi
+
+    def gram(self, kernel: K, x: Float[Array, "N D"]) -> LowRank:
+        _, weighted_phi = self._weighted_basis(kernel, x)
+        return psd(LowRank(weighted_phi))
+
+    def _gram(self, kernel: K, x: Float[Array, "N D"]) -> Float[Array, "N N"]:
+        _, weighted_phi = self._weighted_basis(kernel, x)
+        return weighted_phi @ weighted_phi.T
+
+    def _cross_covariance(
+        self, kernel: K, x: Float[Array, "N D"], y: Float[Array, "M D"]
+    ) -> Float[Array, "N M"]:
+        _, weighted_phi_x = self._weighted_basis(kernel, x)
+        _, weighted_phi_y = self._weighted_basis(kernel, y)
+        return weighted_phi_x @ weighted_phi_y.T
+
+    def diagonal(self, kernel: K, x: Float[Array, "N D"]) -> Diagonal:
+        _, weighted_phi = self._weighted_basis(kernel, x)
+        return psd(Diagonal(jnp.sum(weighted_phi**2, axis=1)))
diff --git a/gpjax/kernels/stationary/base.py b/gpjax/kernels/stationary/base.py
index 5de7572e2..9108abc56 100644
--- a/gpjax/kernels/stationary/base.py
+++ b/gpjax/kernels/stationary/base.py
@@ -18,13 +18,13 @@
 from flax import nnx
 import jax.numpy as jnp
 from jaxtyping import Float
-import numpyro.distributions as npd
 
 from gpjax.kernels.base import AbstractKernel
 from gpjax.kernels.computations import (
     AbstractKernelComputation,
     DenseKernelComputation,
 )
+from gpjax.kernels.stationary.utils import SpectralDensity
 from gpjax.parameters import (
     NonNegativeReal,
     PositiveReal,
@@ -95,11 +95,13 @@ def __init__(
                 self.variance = tp.cast(NonNegativeReal[ScalarFloat], self.variance)
 
     @property
-    def spectral_density(self) -> npd.Normal | npd.StudentT:
-        r"""The spectral density of the kernel.
+    def spectral_density(self) -> SpectralDensity:
+        r"""Spectral density :math:`S(\omega)` of this kernel.
 
-        Returns:
-            Callable[[Float[Array, "D"]], Float[Array, "D"]]: The spectral density function.
+        Subclasses override this to return a
+        :class:`~gpjax.kernels.stationary.utils.SpectralDensity` that
+        supports ``sample()`` (for RFF) and
+        ``__call__(omega, variance, lengthscale)`` (for HSGP).
         """
         raise NotImplementedError(
             f"Kernel {self.name} does not have a spectral density."
diff --git a/gpjax/kernels/stationary/matern12.py b/gpjax/kernels/stationary/matern12.py
index 66f992ae1..e452f8144 100644
--- a/gpjax/kernels/stationary/matern12.py
+++ b/gpjax/kernels/stationary/matern12.py
@@ -15,10 +15,10 @@
 
 import jax.numpy as jnp
 from jaxtyping import Float
-import numpyro.distributions as npd
 
 from gpjax.kernels.stationary.base import StationaryKernel
 from gpjax.kernels.stationary.utils import (
+    SpectralDensity,
     build_student_t_distribution,
     euclidean_distance,
 )
@@ -48,5 +48,14 @@ def __call__(self, x: Float[Array, " D"], y: Float[Array, " D"]) -> ScalarFloat:
         return K.squeeze()
 
     @property
-    def spectral_density(self) -> npd.StudentT:
-        return build_student_t_distribution(nu=1)
+    def spectral_density(self) -> SpectralDensity:
+        r"""Matern-1/2 spectral density.
+
+        .. math::
+            S(\omega) = \sigma^2 \,\frac{2/\ell}{1/\ell^2 + \omega^2}
+        """
+
+        def _evaluate(omega, variance, lengthscale):
+            return variance * (2.0 / lengthscale) / (1.0 / lengthscale**2 + omega**2)
+
+        return SpectralDensity(build_student_t_distribution(nu=1), _evaluate)
diff --git a/gpjax/kernels/stationary/matern32.py b/gpjax/kernels/stationary/matern32.py
index e47e031d9..a288f3ac4 100644
--- a/gpjax/kernels/stationary/matern32.py
+++ b/gpjax/kernels/stationary/matern32.py
@@ -15,10 +15,10 @@
 
 import jax.numpy as jnp
 from jaxtyping import Float
-import numpyro.distributions as npd
 
 from gpjax.kernels.stationary.base import StationaryKernel
 from gpjax.kernels.stationary.utils import (
+    SpectralDensity,
     build_student_t_distribution,
     euclidean_distance,
 )
@@ -54,5 +54,17 @@ def __call__(
         return K.squeeze()
 
     @property
-    def spectral_density(self) -> npd.StudentT:
-        return build_student_t_distribution(nu=3)
+    def spectral_density(self) -> SpectralDensity:
+        r"""Matern-3/2 spectral density.
+
+        .. math::
+            S(\omega) = \sigma^2 \,
+            \frac{4\,\alpha^3}{(3/\ell^2 + \omega^2)^2},
+            \quad \alpha = \sqrt{3}/\ell
+        """
+
+        def _evaluate(omega, variance, lengthscale):
+            alpha = jnp.sqrt(3.0) / lengthscale
+            return variance * 4.0 * alpha**3 / (3.0 / lengthscale**2 + omega**2) ** 2
+
+        return SpectralDensity(build_student_t_distribution(nu=3), _evaluate)
diff --git a/gpjax/kernels/stationary/matern52.py b/gpjax/kernels/stationary/matern52.py
index 84ca61069..1c1e7c943 100644
--- a/gpjax/kernels/stationary/matern52.py
+++ b/gpjax/kernels/stationary/matern52.py
@@ -15,10 +15,10 @@
 
 import jax.numpy as jnp
 from jaxtyping import Float
-import numpyro.distributions as npd
 
 from gpjax.kernels.stationary.base import StationaryKernel
 from gpjax.kernels.stationary.utils import (
+    SpectralDensity,
     build_student_t_distribution,
     euclidean_distance,
 )
@@ -53,5 +53,22 @@ def __call__(
         return K.squeeze()
 
     @property
-    def spectral_density(self) -> npd.StudentT:
-        return build_student_t_distribution(nu=5)
+    def spectral_density(self) -> SpectralDensity:
+        r"""Matern-5/2 spectral density.
+
+        .. math::
+            S(\omega) = \sigma^2 \,
+            \frac{(16/3)\,\alpha^5}{(5/\ell^2 + \omega^2)^3},
+            \quad \alpha = \sqrt{5}/\ell
+        """
+
+        def _evaluate(omega, variance, lengthscale):
+            alpha = jnp.sqrt(5.0) / lengthscale
+            return (
+                variance
+                * (16.0 / 3.0)
+                * alpha**5
+                / (5.0 / lengthscale**2 + omega**2) ** 3
+            )
+
+        return SpectralDensity(build_student_t_distribution(nu=5), _evaluate)
diff --git a/gpjax/kernels/stationary/rbf.py b/gpjax/kernels/stationary/rbf.py
index 44ea74d0e..e1da294dc 100644
--- a/gpjax/kernels/stationary/rbf.py
+++ b/gpjax/kernels/stationary/rbf.py
@@ -18,7 +18,7 @@
 import numpyro.distributions as npd
 
 from gpjax.kernels.stationary.base import StationaryKernel
-from gpjax.kernels.stationary.utils import squared_distance
+from gpjax.kernels.stationary.utils import SpectralDensity, squared_distance
 from gpjax.typing import (
     Array,
     ScalarFloat,
@@ -44,5 +44,20 @@ def __call__(self, x: Float[Array, " D"], y: Float[Array, " D"]) -> ScalarFloat:
         return K.squeeze()
 
     @property
-    def spectral_density(self) -> npd.Normal:
-        return npd.Normal(0.0, 1.0)
+    def spectral_density(self) -> SpectralDensity:
+        r"""RBF spectral density.
+
+        .. math::
+            S(\omega) = \sigma^2 \sqrt{2\pi}\,\ell\,
+            \exp\!\bigl(-\tfrac{1}{2}\ell^2 \omega^2\bigr)
+        """
+
+        def _evaluate(omega, variance, lengthscale):
+            return (
+                variance
+                * jnp.sqrt(2.0 * jnp.pi)
+                * lengthscale
+                * jnp.exp(-0.5 * lengthscale**2 * omega**2)
+            )
+
+        return SpectralDensity(npd.Normal(0.0, 1.0), _evaluate)
diff --git a/gpjax/kernels/stationary/utils.py b/gpjax/kernels/stationary/utils.py
index bbe7b0a7d..e0e3261b1 100644
--- a/gpjax/kernels/stationary/utils.py
+++ b/gpjax/kernels/stationary/utils.py
@@ -12,6 +12,7 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 # ==============================================================================
+import beartype.typing as tp
 import jax.numpy as jnp
 from jaxtyping import Float
 import numpyro.distributions as npd
@@ -23,45 +24,97 @@
 
 
 def build_student_t_distribution(nu: int) -> npd.StudentT:
-    r"""Build a Student's t distribution with a fixed smoothness parameter.
-
-    For a fixed half-integer smoothness parameter, compute the spectral density of a
-    Matérn kernel; a Student's t distribution.
+    r"""Student's t distribution for Matern spectral densities.
 
     Args:
-        nu (int): The smoothness parameter of the Matérn kernel.
+        nu: Degrees of freedom (equals the Matern smoothness parameter
+            :math:`\nu` mapped to the nearest integer: 1, 3, or 5).
+
+    Returns:
+        A standard Student's t distribution with ``df=nu``.
+    """
+    return npd.StudentT(df=nu, loc=0.0, scale=1.0)
+
 
-    Returns
-    -------
-        tfp.Distribution: A Student's t distribution with the same smoothness parameter.
+class SpectralDensity:
+    r"""Spectral density :math:`S(\omega)` of a stationary kernel.
+
+    This class serves two roles:
+
+    1. **Sampling** (for RFF): delegates to a wrapped NumPyro distribution
+       via :meth:`sample`, drawing frequency samples :math:`\omega`.
+    2. **Evaluation** (for HSGP): computes :math:`S(\omega)` at arbitrary
+       frequencies via :meth:`__call__`, incorporating kernel variance and
+       lengthscale.
+
+    Args:
+        distribution: NumPyro distribution to sample from (used by RFF).
+        evaluate_fn: Callable ``(omega, variance, lengthscale) -> S(omega)``
+            that evaluates the spectral density at given frequencies.
     """
-    dist = npd.StudentT(df=nu, loc=0.0, scale=1.0)
-    return dist
+
+    def __init__(
+        self,
+        distribution: npd.Distribution,
+        evaluate_fn: tp.Callable[
+            [Float[Array, " M"], ScalarFloat, ScalarFloat], Float[Array, " M"]
+        ],
+    ):
+        self._distribution = distribution
+        self._evaluate_fn = evaluate_fn
+
+    def sample(self, key: Array, sample_shape: tuple[int, ...]) -> Float[Array, "..."]:
+        """Draw frequency samples from the spectral distribution (used by RFF).
+
+        Args:
+            key: JAX PRNG key.
+            sample_shape: Shape of the sample array.
+
+        Returns:
+            Sampled frequencies.
+        """
+        return self._distribution.sample(key=key, sample_shape=sample_shape)
+
+    def __call__(
+        self,
+        omega: Float[Array, " M"],
+        variance: ScalarFloat,
+        lengthscale: ScalarFloat,
+    ) -> Float[Array, " M"]:
+        r"""Evaluate :math:`S(\omega)` at the given frequencies (used by HSGP).
+
+        Args:
+            omega: Frequencies at which to evaluate the spectral density.
+            variance: Kernel variance :math:`\sigma^2`.
+            lengthscale: Kernel lengthscale :math:`\ell`.
+
+        Returns:
+            Spectral density values :math:`S(\omega)`.
+        """
+        return self._evaluate_fn(omega, variance, lengthscale)
 
 
 def squared_distance(x: Float[Array, " D"], y: Float[Array, " D"]) -> ScalarFloat:
-    r"""Compute the squared distance between a pair of inputs.
+    r"""Squared Euclidean distance :math:`\lVert x - y \rVert^2`.
 
     Args:
-        x (Float[Array, " D"]): First input.
-        y (Float[Array, " D"]): Second input.
+        x: First input vector.
+        y: Second input vector.
 
-    Returns
-    -------
-        ScalarFloat: The squared distance between the inputs.
+    Returns:
+        The squared distance between the inputs.
     """
     return jnp.sum((x - y) ** 2)
 
 
 def euclidean_distance(x: Float[Array, " D"], y: Float[Array, " D"]) -> ScalarFloat:
-    r"""Compute the euclidean distance between a pair of inputs.
+    r"""Euclidean distance :math:`\lVert x - y \rVert`, clamped for stability.
 
     Args:
-        x (Float[Array, " D"]): First input.
-        y (Float[Array, " D"]): Second input.
+        x: First input vector.
+        y: Second input vector.
 
-    Returns
-    -------
-        ScalarFloat: The euclidean distance between the inputs.
+    Returns:
+        The Euclidean distance between the inputs.
     """
     return jnp.sqrt(jnp.maximum(squared_distance(x, y), 1e-36))
diff --git a/gpjax/linalg/__init__.py b/gpjax/linalg/__init__.py
index 489ac2cbd..366caebac 100644
--- a/gpjax/linalg/__init__.py
+++ b/gpjax/linalg/__init__.py
@@ -13,6 +13,7 @@
     Identity,
     Kronecker,
     LinearOperator,
+    LowRank,
     Triangular,
 )
 from gpjax.linalg.utils import (
@@ -28,6 +29,7 @@
     "Identity",
     "Kronecker",
     "LinearOperator",
+    "LowRank",
     "Triangular",
     "diag",
     "logdet",
diff --git a/gpjax/linalg/operations.py b/gpjax/linalg/operations.py
index d287d49a9..1ccb69fd2 100644
--- a/gpjax/linalg/operations.py
+++ b/gpjax/linalg/operations.py
@@ -12,6 +12,7 @@
     Identity,
     Kronecker,
     LinearOperator,
+    LowRank,
     Triangular,
 )
 from gpjax.typing import ScalarFloat
@@ -219,6 +220,10 @@ def _handle_blockdiag(A):
     def _handle_dense(A):
         return jnp.diag(A.array)
 
+    def _handle_lowrank(A):
+        # diag(W W^T)_i = sum_j W_{ij}^2
+        return jnp.sum(A.factor**2, axis=1)
+
     def _handle_default(A):
         return jnp.diag(A.to_dense())
 
@@ -229,6 +234,7 @@ def _handle_default(A):
         Kronecker: _handle_kronecker,
         BlockDiag: _handle_blockdiag,
         Dense: _handle_dense,
+        LowRank: _handle_lowrank,
     }
 
     handler = dispatch_table.get(type(A), _handle_default)
diff --git a/gpjax/linalg/operators.py b/gpjax/linalg/operators.py
index db49a9e43..98f14dabe 100644
--- a/gpjax/linalg/operators.py
+++ b/gpjax/linalg/operators.py
@@ -407,6 +407,52 @@ def _kronecker_tree_unflatten(aux_data, children):
 
 jtu.register_pytree_node(Kronecker, _kronecker_tree_flatten, _kronecker_tree_unflatten)
 
+
+class LowRank(LinearOperator):
+    r"""Low-rank matrix K = W W^T where W has shape (N, m) with m << N.
+
+    Basis-function kernel approximations (HSGP, RFF) produce a kernel
+    matrix that factors as K = W W^T.  Storing only the (N, m) factor W
+    rather than the full (N, N) matrix enables O(N m^2) inference via the
+    Woodbury identity instead of O(N^3) Cholesky-based inference.
+    """
+
+    def __init__(self, factor: Float[Array, "N m"]):
+        super().__init__()
+        self.factor = factor
+
+    @property
+    def shape(self) -> tuple[int, int]:
+        num_data = self.factor.shape[0]
+        return (num_data, num_data)
+
+    @property
+    def rank(self) -> int:
+        return self.factor.shape[1]
+
+    @property
+    def dtype(self) -> jnp.dtype:
+        return self.factor.dtype
+
+    def to_dense(self) -> Float[Array, "N N"]:
+        return self.factor @ self.factor.T
+
+    @property
+    def T(self) -> "LowRank":
+        # W W^T is symmetric, so the transpose is itself.
+        return self
+
+
+def _lowrank_tree_flatten(lowrank):
+    return (lowrank.factor,), None
+
+
+def _lowrank_tree_unflatten(aux_data, children):
+    return LowRank(children[0])
+
+
+jtu.register_pytree_node(LowRank, _lowrank_tree_flatten, _lowrank_tree_unflatten)
+
 __all__ = [
     "BlockDiag",
     "Dense",
@@ -414,5 +460,6 @@ def _kronecker_tree_unflatten(aux_data, children):
     "Identity",
     "Kronecker",
     "LinearOperator",
+    "LowRank",
     "Triangular",
 ]
diff --git a/gpjax/linalg/woodbury.py b/gpjax/linalg/woodbury.py
new file mode 100644
index 000000000..c96d74d36
--- /dev/null
+++ b/gpjax/linalg/woodbury.py
@@ -0,0 +1,121 @@
+r"""Woodbury identity helpers for efficient low-rank + diagonal solves.
+
+When a kernel is approximated by a low-rank factorisation K \approx W W^T
+(e.g. via HSGP or RFF), the marginal covariance becomes
+
+    \Sigma = W W^T + D,   where  D = diag(noise),  W is (N, m),  m << N.
+
+Naively inverting \Sigma costs O(N^3).  The *Woodbury matrix identity* reduces
+this to O(N m^2 + m^3) by working with the small (m x m) *capacitance matrix*
+
+    A = I_m + W^T D^{-1} W.
+
+The three operations exposed here --- solve, log-determinant, and quadratic
+form --- are the building blocks for the GP marginal likelihood and posterior
+prediction when the kernel has low-rank structure.
+"""
+
+import jax.numpy as jnp
+import jax.scipy as jsp
+from jaxtyping import Float
+
+from gpjax.typing import Array, ScalarFloat
+
+
+def _capacitance_cholesky(
+    W: Float[Array, "N m"],
+    noise_inv: Float[Array, " N"],
+) -> Float[Array, "m m"]:
+    r"""Cholesky factor of the capacitance matrix A = I_m + W^T D^{-1} W.
+
+    This is the shared computation underlying all Woodbury operations.
+
+    Args:
+        W: Factor matrix, shape (N, m).
+        noise_inv: Element-wise reciprocal of the diagonal noise, shape (N,).
+
+    Returns:
+        Lower-triangular Cholesky factor L_A such that A = L_A L_A^T.
+    """
+    m = W.shape[1]
+    Dinv_W = noise_inv[:, None] * W
+    A = jnp.eye(m) + W.T @ Dinv_W
+    return jnp.linalg.cholesky(A)
+
+
+def woodbury_solve(
+    W: Float[Array, "N m"],
+    noise: Float[Array, " N"],
+    b: Float[Array, "N ..."],
+) -> Float[Array, "N ..."]:
+    r"""Solve (W W^T + D) x = b via the Woodbury identity.
+
+    .. math::
+
+        x = D^{-1} b  -  D^{-1} W \, A^{-1} \, W^T D^{-1} b
+
+    where D = diag(noise) and A = I_m + W^T D^{-1} W.
+
+    Args:
+        W: Factor matrix, shape (N, m).
+        noise: Diagonal noise vector, shape (N,).
+        b: Right-hand side, shape (N,) or (N, K).
+
+    Returns:
+        Solution x with same shape as b.
+    """
+    noise_inv = 1.0 / noise
+    Dinv_W = noise_inv[:, None] * W
+    Dinv_b = noise_inv * b if b.ndim == 1 else noise_inv[:, None] * b
+
+    L_A = _capacitance_cholesky(W, noise_inv)
+
+    WtDinv_b = Dinv_W.T @ b
+    forward = jsp.linalg.solve_triangular(L_A, WtDinv_b, lower=True)
+    Ainv_WtDinv_b = jsp.linalg.solve_triangular(L_A.T, forward, lower=False)
+
+    return Dinv_b - Dinv_W @ Ainv_WtDinv_b
+
+
+def woodbury_logdet(
+    W: Float[Array, "N m"],
+    noise: Float[Array, " N"],
+) -> ScalarFloat:
+    r"""Log-determinant of W W^T + D via the matrix determinant lemma.
+
+    .. math::
+
+        \log|\Sigma| = \log|A| + \sum_i \log(\text{noise}_i)
+
+    where A = I_m + W^T D^{-1} W.
+
+    Args:
+        W: Factor matrix, shape (N, m).
+        noise: Diagonal noise vector, shape (N,).
+
+    Returns:
+        Log-determinant as a scalar.
+    """
+    noise_inv = 1.0 / noise
+    L_A = _capacitance_cholesky(W, noise_inv)
+    logdet_A = 2.0 * jnp.sum(jnp.log(jnp.diag(L_A)))
+    return logdet_A + jnp.sum(jnp.log(noise))
+
+
+def woodbury_quad(
+    W: Float[Array, "N m"],
+    noise: Float[Array, " N"],
+    diff: Float[Array, " N"],
+) -> ScalarFloat:
+    r"""Quadratic form diff^T \Sigma^{-1} diff where \Sigma = W W^T + D.
+
+    Args:
+        W: Factor matrix, shape (N, m).
+        noise: Diagonal noise vector, shape (N,).
+        diff: Vector, shape (N,).
+
+    Returns:
+        Quadratic form as a scalar.
+    """
+    solved = woodbury_solve(W, noise, diff)
+    return diff @ solved
diff --git a/gpjax/objectives.py b/gpjax/objectives.py
index 236383b93..7c4d36004 100644
--- a/gpjax/objectives.py
+++ b/gpjax/objectives.py
@@ -23,7 +23,9 @@
     psd,
     solve,
 )
+from gpjax.linalg.operators import LowRank
 from gpjax.linalg.utils import add_jitter
+from gpjax.linalg.woodbury import woodbury_logdet, woodbury_quad
 from gpjax.typing import (
     Array,
     ScalarFloat,
@@ -120,12 +122,25 @@ def conjugate_mll(posterior: ConjugatePosterior, data: Dataset) -> ScalarFloat:
     noise = posterior.likelihood.noise_vector(data.n)
 
     Kxx = kernel.gram(x)
-    Kxx_dense = add_jitter(Kxx.to_dense(), posterior.prior.jitter)
-    Sigma_dense = Kxx_dense + jnp.diag(noise)
-    Sigma = psd(Dense(Sigma_dense))
+    diff = jnp.atleast_1d((y_flat - mx_flat).squeeze())
+
+    if isinstance(Kxx, LowRank):
+        # Low-rank path: Sigma = W W^T + D.
+        # Use the Woodbury identity for O(N m^2) instead of O(N^3) Cholesky.
+        W = Kxx.factor
+        noise_with_jitter = noise + posterior.prior.jitter
+        num_datapoints = x.shape[0] * posterior.likelihood.num_outputs
+
+        log_det = woodbury_logdet(W, noise_with_jitter)
+        quadratic = woodbury_quad(W, noise_with_jitter, diff)
 
-    mll = GaussianDistribution(jnp.atleast_1d(mx_flat.squeeze()), Sigma)
-    return mll.log_prob(jnp.atleast_1d(y_flat.squeeze())).squeeze()
+        return -0.5 * (num_datapoints * jnp.log(2.0 * jnp.pi) + log_det + quadratic)
+
+    # Dense path: Sigma = Kxx + D.  Standard Cholesky-based log-probability.
+    Kxx_dense = add_jitter(Kxx.to_dense(), posterior.prior.jitter)
+    Sigma = psd(Dense(Kxx_dense + jnp.diag(noise)))
+    marginal = GaussianDistribution(jnp.atleast_1d(mx_flat.squeeze()), Sigma)
+    return marginal.log_prob(diff).squeeze()
 
 
 def conjugate_loocv(posterior: ConjugatePosterior, data: Dataset) -> ScalarFloat:
diff --git a/mkdocs.yml b/mkdocs.yml
index 527f18c0c..3c7d4bf3a 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -31,6 +31,7 @@ nav:
     - Multi-Output GPs: _examples/multioutput.md
     - Accelerated Multi-Output GPs: _examples/oilmm.md
     - Orthogonal Additive GPs: _examples/oak.md
+    - Hilbert Space GPs: _examples/hsgp.md
   - 🧪 Experimental:
     - Numpyro Integration: _examples/numpyro_integration.md
     - Spatial Linear Gaussian Process: _examples/spatial_linear_gp.md
diff --git a/pyproject.toml b/pyproject.toml
index acd01cae4..dfb9f7655 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -4,7 +4,7 @@ build-backend = "hatchling.build"
 
 [project]
 name = "gpjax"
-dynamic = ["version"]
+version = "0.13.6"
 description = 'Gaussian processes in JAX.'
 readme = "README.md"
 requires-python = ">=3.11"
@@ -57,6 +57,7 @@ docs = [
   "markdown-katex>=202406.1035",
   "scikit-learn>=1.5.1",
   "ucimlrepo>=0.0.7",
+  "rdata>=1.0.0",
 ]
 
 [tool.uv]
diff --git a/tests/test_heteroscedastic.py b/tests/test_heteroscedastic.py
index 94c99cf5e..f46cb73cd 100644
--- a/tests/test_heteroscedastic.py
+++ b/tests/test_heteroscedastic.py
@@ -167,11 +167,13 @@ def test_softplus_transform_numerical_accuracy(mean: float, variance: float, see
     # E[1/sigma^2]
     mc_inv_variance = jnp.mean(1.0 / transformed_samples)
 
-    # Allow for some MC error and quadrature approximation error
+    # Allow for some MC error and quadrature approximation error.
+    # atol covers near-zero values where rtol alone is unreliable.
     rtol = 0.15
-    assert jnp.allclose(moments.variance, mc_variance, rtol=rtol)
-    assert jnp.allclose(moments.log_variance, mc_log_variance, rtol=rtol)
-    assert jnp.allclose(moments.inv_variance, mc_inv_variance, rtol=rtol)
+    atol = 0.01
+    assert jnp.allclose(moments.variance, mc_variance, rtol=rtol, atol=atol)
+    assert jnp.allclose(moments.log_variance, mc_log_variance, rtol=rtol, atol=atol)
+    assert jnp.allclose(moments.inv_variance, mc_inv_variance, rtol=rtol, atol=atol)
 
 
 def test_heteroscedastic_variational_predict(prior, noise_prior, dataset):
diff --git a/tests/test_kernels/test_approximations.py b/tests/test_kernels/test_approximations.py
index c78d66120..1771b631d 100644
--- a/tests/test_kernels/test_approximations.py
+++ b/tests/test_kernels/test_approximations.py
@@ -13,7 +13,7 @@
     RationalQuadratic,
     StationaryKernel,
 )
-from gpjax.linalg.operators import Dense
+from gpjax.linalg.operators import LowRank
 import jax
 from jax import config
 import jax.numpy as jnp
@@ -52,7 +52,7 @@ def test_gram(
     linop = approximate.gram(x)
 
     # Check the return type
-    assert isinstance(linop, Dense)
+    assert isinstance(linop, LowRank)
 
     Kxx = linop.to_dense() + jnp.eye(n_data) * _jitter
 
diff --git a/tests/test_kernels/test_computations.py b/tests/test_kernels/test_computations.py
index d1883ab2b..45cfdcde2 100644
--- a/tests/test_kernels/test_computations.py
+++ b/tests/test_kernels/test_computations.py
@@ -15,6 +15,7 @@
     PSD,
     Dense,
     Diagonal,
+    LowRank,
 )
 import jax.numpy as jnp
 import jax.random as jr
@@ -67,7 +68,7 @@ def test_gram_shape(self, rff_kernel):
         """Gram matrix should be (N, N)."""
         x = jr.normal(jr.key(1), (10, 2))
         gram = rff_kernel.gram(x)
-        assert isinstance(gram, Dense)
+        assert isinstance(gram, LowRank)
         assert PSD in gram.annotations
         assert gram.shape == (10, 10)
 
diff --git a/tests/test_kernels/test_hsgp.py b/tests/test_kernels/test_hsgp.py
new file mode 100644
index 000000000..82bbf28e8
--- /dev/null
+++ b/tests/test_kernels/test_hsgp.py
@@ -0,0 +1,309 @@
+"""Tests for the Hilbert Space Gaussian Process (HSGP) kernel approximation."""
+
+from gpjax.kernels.approximations.hsgp import HSGP
+from gpjax.kernels.stationary import RBF, Matern12, Matern32, Matern52
+from gpjax.linalg.operators import LowRank
+import jax
+from jax import config
+import jax.numpy as jnp
+import jax.random as jr
+import numpy.testing as npt
+import pytest
+
+config.update("jax_enable_x64", True)
+
+STATIONARY_KERNELS = [RBF, Matern12, Matern32, Matern52]
+
+
+def _make_hsgp(
+    kernel_class=RBF,
+    num_basis_fns: int = 20,
+    domain_half_width: float = 5.0,
+    center: float = 0.0,
+) -> HSGP:
+    """Create an HSGP with sensible defaults for testing."""
+    base_kernel = kernel_class(n_dims=1)
+    return HSGP(
+        base_kernel=base_kernel,
+        num_basis_fns=num_basis_fns,
+        domain_half_width=domain_half_width,
+        center=center,
+    )
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Eigenvalues
+# ──────────────────────────────────────────────────────────────────────
+
+
+def test_eigenvalue_count():
+    hsgp = _make_hsgp(num_basis_fns=10, domain_half_width=3.0)
+    eigenvalues = hsgp.eigenvalues()
+    assert eigenvalues.shape == (10,)
+
+
+def test_eigenvalue_formula():
+    """sqrt(lambda_j) = j * pi / (2 * L)."""
+    half_width = 5.0
+    num_basis = 4
+    hsgp = _make_hsgp(num_basis_fns=num_basis, domain_half_width=half_width)
+    eigenvalues = hsgp.eigenvalues()
+    indices = jnp.arange(1, num_basis + 1)
+    expected = indices * jnp.pi / (2.0 * half_width)
+    npt.assert_allclose(eigenvalues, expected)
+
+
+def test_eigenvalues_are_strictly_increasing():
+    hsgp = _make_hsgp(num_basis_fns=20, domain_half_width=3.0)
+    eigenvalues = hsgp.eigenvalues()
+    assert jnp.all(jnp.diff(eigenvalues) > 0)
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Eigenfunctions
+# ──────────────────────────────────────────────────────────────────────
+
+
+def test_eigenfunction_shape():
+    hsgp = _make_hsgp(num_basis_fns=10, domain_half_width=3.0)
+    inputs = jnp.linspace(-2, 2, 50)[:, None]
+    basis_matrix = hsgp.eigenfunctions(inputs)
+    assert basis_matrix.shape == (50, 10)
+
+
+def test_eigenfunctions_are_approximately_orthonormal():
+    """Eigenfunctions should be approximately orthonormal over [-L, L]."""
+    half_width = 3.0
+    num_basis = 5
+    hsgp = _make_hsgp(num_basis_fns=num_basis, domain_half_width=half_width, center=0.0)
+    num_quadrature_points = 10_000
+    inputs = jnp.linspace(-half_width, half_width, num_quadrature_points)[:, None]
+    basis_matrix = hsgp.eigenfunctions(inputs)
+    spacing = 2.0 * half_width / num_quadrature_points
+    gram_matrix = basis_matrix.T @ basis_matrix * spacing
+    npt.assert_allclose(gram_matrix, jnp.eye(num_basis), atol=1e-2)
+
+
+def test_eigenfunctions_vanish_at_boundaries():
+    """Eigenfunctions should be zero at x = -L and x = L."""
+    half_width = 3.0
+    hsgp = _make_hsgp(num_basis_fns=5, domain_half_width=half_width, center=0.0)
+    boundary_points = jnp.array([[-half_width], [half_width]])
+    basis_at_boundary = hsgp.eigenfunctions(boundary_points)
+    npt.assert_allclose(basis_at_boundary, 0.0, atol=1e-12)
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Centering
+# ──────────────────────────────────────────────────────────────────────
+
+
+def test_explicit_center_is_stored():
+    hsgp = HSGP(
+        base_kernel=RBF(n_dims=1),
+        num_basis_fns=5,
+        domain_half_width=3.0,
+        center=1.0,
+    )
+    assert hsgp._center == 1.0
+
+
+def test_auto_center_uses_midpoint_of_input_range():
+    hsgp = _make_hsgp(num_basis_fns=5, domain_half_width=3.0, center=None)
+    inputs = jnp.linspace(2.0, 8.0, 50)[:, None]
+    _ = hsgp.eigenfunctions(inputs)
+    assert hsgp._center == pytest.approx(5.0)
+
+
+def test_auto_center_persists_across_calls():
+    """Once set, auto-center should not change on subsequent calls."""
+    hsgp = _make_hsgp(num_basis_fns=5, domain_half_width=3.0, center=None)
+    first_inputs = jnp.linspace(0, 10, 50)[:, None]
+    second_inputs = jnp.linspace(-5, 5, 50)[:, None]
+    _ = hsgp.eigenfunctions(first_inputs)
+    center_after_first_call = hsgp._center
+    _ = hsgp.eigenfunctions(second_inputs)
+    assert hsgp._center == center_after_first_call
+
+
+# ──────────────────────────────────────────────────────────────────────
+# compute_basis
+# ──────────────────────────────────────────────────────────────────────
+
+
+def test_compute_basis_returns_correct_shapes():
+    hsgp = _make_hsgp(num_basis_fns=10, domain_half_width=3.0)
+    inputs = jnp.linspace(-2, 2, 50)[:, None]
+    basis_matrix, sqrt_spectral_weights = hsgp.compute_basis(inputs)
+    assert basis_matrix.shape == (50, 10)
+    assert sqrt_spectral_weights.shape == (10,)
+
+
+def test_compute_basis_sqrt_psd_is_positive():
+    hsgp = _make_hsgp(num_basis_fns=10, domain_half_width=3.0)
+    inputs = jnp.linspace(-2, 2, 50)[:, None]
+    _, sqrt_spectral_weights = hsgp.compute_basis(inputs)
+    assert jnp.all(sqrt_spectral_weights > 0)
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Gram, cross-covariance, and diagonal
+# ──────────────────────────────────────────────────────────────────────
+
+
+@pytest.mark.parametrize("KernelClass", STATIONARY_KERNELS)
+def test_gram_returns_psd_low_rank_matrix(KernelClass):
+    hsgp = _make_hsgp(kernel_class=KernelClass)
+    inputs = jnp.linspace(-3, 3, 30)[:, None]
+    gram_operator = hsgp.gram(inputs)
+
+    assert isinstance(gram_operator, LowRank)
+    gram_dense = gram_operator.to_dense()
+    assert gram_dense.shape == (30, 30)
+
+    eigenvalues_of_gram, _ = jnp.linalg.eigh(gram_dense + 1e-6 * jnp.eye(30))
+    assert jnp.all(eigenvalues_of_gram > 0)
+
+
+@pytest.mark.parametrize("KernelClass", STATIONARY_KERNELS)
+def test_cross_covariance_shape(KernelClass):
+    hsgp = _make_hsgp(kernel_class=KernelClass)
+    inputs_a = jnp.linspace(-3, 3, 30)[:, None]
+    inputs_b = jnp.linspace(-2, 2, 20)[:, None]
+    cross_covariance = hsgp.cross_covariance(inputs_a, inputs_b)
+    assert cross_covariance.shape == (30, 20)
+
+
+def test_gram_is_symmetric():
+    hsgp = _make_hsgp(kernel_class=RBF)
+    inputs = jnp.linspace(-3, 3, 30)[:, None]
+    gram_dense = hsgp.gram(inputs).to_dense()
+    npt.assert_allclose(gram_dense, gram_dense.T, atol=1e-12)
+
+
+@pytest.mark.parametrize("KernelClass", STATIONARY_KERNELS)
+def test_diagonal_matches_gram_diagonal(KernelClass):
+    hsgp = _make_hsgp(kernel_class=KernelClass)
+    inputs = jnp.linspace(-3, 3, 30)[:, None]
+    diagonal_dense = hsgp.diagonal(inputs).to_dense()
+    gram_dense = hsgp.gram(inputs).to_dense()
+    npt.assert_allclose(jnp.diag(diagonal_dense), jnp.diag(gram_dense), atol=1e-10)
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Convergence to exact kernel
+# ──────────────────────────────────────────────────────────────────────
+
+
+@pytest.mark.parametrize("KernelClass", [RBF, Matern32, Matern52])
+def test_gram_error_decreases_with_more_basis_functions(KernelClass):
+    """With more basis functions, the HSGP Gram matrix should converge to exact."""
+    base_kernel = KernelClass(n_dims=1)
+    inputs = jnp.linspace(-1, 1, 30)[:, None]
+    exact_gram = base_kernel.gram(inputs).to_dense()
+
+    hsgp_coarse = _make_hsgp(kernel_class=KernelClass, num_basis_fns=10)
+    hsgp_fine = _make_hsgp(kernel_class=KernelClass, num_basis_fns=80)
+
+    error_coarse = jnp.linalg.norm(exact_gram - hsgp_coarse.gram(inputs).to_dense())
+    error_fine = jnp.linalg.norm(exact_gram - hsgp_fine.gram(inputs).to_dense())
+
+    assert error_fine < error_coarse
+
+
+def test_rbf_gram_closely_matches_exact():
+    """RBF with many basis functions should be very close to exact."""
+    base_kernel = RBF(n_dims=1)
+    inputs = jnp.linspace(-1, 1, 20)[:, None]
+    exact_gram = base_kernel.gram(inputs).to_dense()
+
+    hsgp = _make_hsgp(kernel_class=RBF, num_basis_fns=100)
+    approximate_gram = hsgp.gram(inputs).to_dense()
+
+    max_absolute_error = jnp.max(jnp.abs(exact_gram - approximate_gram))
+    assert max_absolute_error < 0.01
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Input validation
+# ──────────────────────────────────────────────────────────────────────
+
+
+def test_nonstationary_kernel_is_rejected():
+    from gpjax.kernels.nonstationary import Linear
+
+    with pytest.raises(TypeError):
+        HSGP(base_kernel=Linear(1), num_basis_fns=10, domain_half_width=3.0)
+
+
+def test_pointwise_call_raises():
+    hsgp = _make_hsgp(num_basis_fns=10, domain_half_width=3.0)
+    with pytest.raises(RuntimeError):
+        hsgp(jnp.array([1.0]), jnp.array([2.0]))
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Integration with Prior/Posterior pipeline
+# ──────────────────────────────────────────────────────────────────────
+
+
+@pytest.fixture
+def training_data():
+    """Synthetic sinusoidal regression dataset."""
+    key = jr.key(42)
+    num_points = 50
+    inputs = jnp.linspace(-3, 3, num_points)[:, None]
+    targets = jnp.sin(inputs) + 0.1 * jr.normal(key, (num_points, 1))
+
+    from gpjax.dataset import Dataset
+
+    return Dataset(X=inputs, y=targets), num_points
+
+
+@pytest.fixture
+def hsgp_posterior(training_data):
+    """Conjugate posterior with an HSGP-RBF kernel."""
+    from gpjax.gps import Prior
+    from gpjax.likelihoods import Gaussian
+    from gpjax.mean_functions import Zero
+
+    _, num_points = training_data
+    hsgp = _make_hsgp(kernel_class=RBF, num_basis_fns=20)
+    prior = Prior(kernel=hsgp, mean_function=Zero())
+    likelihood = Gaussian(num_datapoints=num_points)
+    return prior * likelihood
+
+
+def test_conjugate_mll_returns_finite_scalar(training_data, hsgp_posterior):
+    from gpjax.objectives import conjugate_mll
+
+    dataset, _ = training_data
+    mll = conjugate_mll(hsgp_posterior, dataset)
+    assert jnp.isfinite(mll)
+
+
+def test_posterior_predict_returns_finite_moments(training_data, hsgp_posterior):
+    dataset, _ = training_data
+    test_inputs = jnp.linspace(-2.5, 2.5, 30)[:, None]
+    prediction = hsgp_posterior.predict(test_inputs, dataset)
+
+    assert jnp.all(jnp.isfinite(prediction.mean))
+    assert jnp.all(jnp.isfinite(prediction.covariance()))
+
+
+def test_conjugate_mll_is_differentiable(training_data, hsgp_posterior):
+    """conjugate_mll with HSGP must be differentiable w.r.t. kernel parameters."""
+    from flax import nnx
+    from gpjax.objectives import conjugate_mll
+
+    dataset, _ = training_data
+    graphdef, state = nnx.split(hsgp_posterior)
+
+    def negative_mll(state):
+        model = nnx.merge(graphdef, state)
+        return -conjugate_mll(model, dataset)
+
+    gradients = jax.grad(negative_mll)(state)
+    flat_gradients = jax.tree.leaves(gradients)
+    for grad_leaf in flat_gradients:
+        assert jnp.all(jnp.isfinite(grad_leaf)), f"Non-finite gradient: {grad_leaf}"
diff --git a/tests/test_kernels/test_spectral_density.py b/tests/test_kernels/test_spectral_density.py
new file mode 100644
index 000000000..fd29581f2
--- /dev/null
+++ b/tests/test_kernels/test_spectral_density.py
@@ -0,0 +1,131 @@
+"""Tests for the SpectralDensity class and kernel spectral density evaluation."""
+
+from gpjax.kernels.stationary import RBF, Matern12, Matern32, Matern52
+from gpjax.kernels.stationary.utils import SpectralDensity
+from jax import config
+import jax.numpy as jnp
+import jax.random as jr
+import numpy.testing as npt
+import pytest
+
+config.update("jax_enable_x64", True)
+
+UNIT_VARIANCE = jnp.array(1.0)
+UNIT_LENGTHSCALE = jnp.array(1.0)
+TEST_FREQUENCIES = jnp.array([0.0, 1.0, 5.0])
+
+
+# ──────────────────────────────────────────────────────────────────────
+# SpectralDensity interface
+# ──────────────────────────────────────────────────────────────────────
+
+
+def test_spectral_density_has_sample_method():
+    """SpectralDensity must expose sample() for RFF compatibility."""
+    spectral_density = RBF(n_dims=1).spectral_density
+    assert isinstance(spectral_density, SpectralDensity)
+    samples = spectral_density.sample(key=jr.key(0), sample_shape=(10, 1))
+    assert samples.shape == (10, 1)
+
+
+def test_spectral_density_is_callable():
+    """SpectralDensity must be callable with (omega, variance, lengthscale)."""
+    spectral_density = RBF(n_dims=1).spectral_density
+    frequencies = jnp.array([0.5, 1.0, 2.0])
+    values = spectral_density(frequencies, UNIT_VARIANCE, UNIT_LENGTHSCALE)
+    assert values.shape == (3,)
+    assert jnp.all(values > 0)
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Closed-form spectral density formulae
+# ──────────────────────────────────────────────────────────────────────
+
+
+def test_rbf_spectral_density_matches_closed_form():
+    """S(w) = variance * sqrt(2*pi) * lengthscale * exp(-0.5 * lengthscale^2 * w^2)."""
+    variance = jnp.array(2.0)
+    lengthscale = jnp.array(0.5)
+    spectral_density = RBF(n_dims=1, variance=2.0, lengthscale=0.5).spectral_density
+
+    result = spectral_density(TEST_FREQUENCIES, variance, lengthscale)
+    expected = (
+        variance
+        * jnp.sqrt(2 * jnp.pi)
+        * lengthscale
+        * jnp.exp(-0.5 * lengthscale**2 * TEST_FREQUENCIES**2)
+    )
+    npt.assert_allclose(result, expected, atol=1e-12)
+
+
+def test_rbf_spectral_density_peaks_at_zero_and_decays():
+    """RBF spectral density peaks at omega=0 and decays monotonically."""
+    spectral_density = RBF(n_dims=1).spectral_density
+    frequencies = jnp.linspace(0, 10, 100)
+    values = spectral_density(frequencies, UNIT_VARIANCE, UNIT_LENGTHSCALE)
+    assert values[0] == jnp.max(values)
+    assert jnp.all(jnp.diff(values) <= 0)
+
+
+def test_matern12_spectral_density_matches_closed_form():
+    """S(w) = variance * 2/ell * 1/(1/ell^2 + w^2)."""
+    variance = jnp.array(1.5)
+    lengthscale = jnp.array(0.8)
+    spectral_density = Matern12(
+        n_dims=1, variance=1.5, lengthscale=0.8
+    ).spectral_density
+
+    result = spectral_density(TEST_FREQUENCIES, variance, lengthscale)
+    expected = (
+        variance * (2.0 / lengthscale) / (1.0 / lengthscale**2 + TEST_FREQUENCIES**2)
+    )
+    npt.assert_allclose(result, expected, atol=1e-12)
+
+
+def test_matern32_spectral_density_matches_closed_form():
+    """S(w) = variance * 4*(sqrt(3)/ell)^3 / (3/ell^2 + w^2)^2."""
+    variance = jnp.array(2.0)
+    lengthscale = jnp.array(1.5)
+    spectral_density = Matern32(
+        n_dims=1, variance=2.0, lengthscale=1.5
+    ).spectral_density
+
+    result = spectral_density(TEST_FREQUENCIES, variance, lengthscale)
+    alpha = jnp.sqrt(3.0) / lengthscale
+    expected = (
+        variance * 4.0 * alpha**3 / (3.0 / lengthscale**2 + TEST_FREQUENCIES**2) ** 2
+    )
+    npt.assert_allclose(result, expected, atol=1e-12)
+
+
+def test_matern52_spectral_density_matches_closed_form():
+    """S(w) = variance * (16/3)*(sqrt(5)/ell)^5 / (5/ell^2 + w^2)^3."""
+    variance = jnp.array(3.0)
+    lengthscale = jnp.array(2.0)
+    spectral_density = Matern52(
+        n_dims=1, variance=3.0, lengthscale=2.0
+    ).spectral_density
+
+    result = spectral_density(TEST_FREQUENCIES, variance, lengthscale)
+    alpha = jnp.sqrt(5.0) / lengthscale
+    expected = (
+        variance
+        * (16.0 / 3.0)
+        * alpha**5
+        / (5.0 / lengthscale**2 + TEST_FREQUENCIES**2) ** 3
+    )
+    npt.assert_allclose(result, expected, atol=1e-12)
+
+
+# ──────────────────────────────────────────────────────────────────────
+# Positivity across all stationary kernels
+# ──────────────────────────────────────────────────────────────────────
+
+
+@pytest.mark.parametrize("KernelClass", [RBF, Matern12, Matern32, Matern52])
+def test_spectral_density_is_positive_everywhere(KernelClass):
+    """All spectral densities must be positive for all omega."""
+    spectral_density = KernelClass(n_dims=1).spectral_density
+    frequencies = jnp.linspace(0, 20, 200)
+    values = spectral_density(frequencies, UNIT_VARIANCE, UNIT_LENGTHSCALE)
+    assert jnp.all(values > 0)
diff --git a/tests/test_linalg/__init__.py b/tests/test_linalg/__init__.py
new file mode 100644
index 000000000..e69de29bb
diff --git a/tests/test_linalg/test_lowrank.py b/tests/test_linalg/test_lowrank.py
new file mode 100644
index 000000000..0a3657166
--- /dev/null
+++ b/tests/test_linalg/test_lowrank.py
@@ -0,0 +1,81 @@
+"""Tests for the LowRank linear operator."""
+
+import jax
+from jax import config
+import jax.numpy as jnp
+import jax.random as jr
+import numpy.testing as npt
+
+config.update("jax_enable_x64", True)
+
+from gpjax.linalg.operations import diag
+from gpjax.linalg.operators import LowRank
+
+
+class TestLowRankProperties:
+    def test_shape(self):
+        W = jnp.ones((10, 3))
+        op = LowRank(W)
+        assert op.shape == (10, 10)
+
+    def test_rank(self):
+        W = jnp.ones((10, 3))
+        op = LowRank(W)
+        assert op.rank == 3
+
+    def test_dtype(self):
+        W = jnp.ones((10, 3), dtype=jnp.float64)
+        op = LowRank(W)
+        assert op.dtype == jnp.float64
+
+    def test_to_dense(self):
+        key = jr.key(0)
+        W = jr.normal(key, (10, 3))
+        op = LowRank(W)
+        npt.assert_allclose(op.to_dense(), W @ W.T, atol=1e-12)
+
+    def test_transpose_is_self(self):
+        W = jnp.ones((10, 3))
+        op = LowRank(W)
+        assert op.T is op
+
+    def test_diag_dispatch(self):
+        key = jr.key(0)
+        W = jr.normal(key, (10, 3))
+        op = LowRank(W)
+        expected = jnp.sum(W**2, axis=1)
+        npt.assert_allclose(diag(op), expected, atol=1e-12)
+
+
+class TestLowRankPyTree:
+    def test_flatten_unflatten_roundtrip(self):
+        key = jr.key(0)
+        W = jr.normal(key, (10, 3))
+        op = LowRank(W)
+        leaves, treedef = jax.tree.flatten(op)
+        restored = treedef.unflatten(leaves)
+        npt.assert_allclose(restored.factor, op.factor)
+
+    def test_jit_compatible(self):
+        key = jr.key(0)
+        W = jr.normal(key, (10, 3))
+        op = LowRank(W)
+
+        @jax.jit
+        def fn(op):
+            return op.to_dense()
+
+        result = fn(op)
+        npt.assert_allclose(result, W @ W.T, atol=1e-12)
+
+    def test_grad_through_to_dense(self):
+        key = jr.key(0)
+        W = jr.normal(key, (10, 3))
+
+        @jax.grad
+        def fn(W):
+            op = LowRank(W)
+            return jnp.sum(op.to_dense())
+
+        grads = fn(W)
+        assert jnp.all(jnp.isfinite(grads))
diff --git a/tests/test_linalg.py b/tests/test_linalg/test_operators.py
similarity index 100%
rename from tests/test_linalg.py
rename to tests/test_linalg/test_operators.py
diff --git a/tests/test_linalg/test_woodbury.py b/tests/test_linalg/test_woodbury.py
new file mode 100644
index 000000000..0c32ff9f1
--- /dev/null
+++ b/tests/test_linalg/test_woodbury.py
@@ -0,0 +1,379 @@
+"""Tests for Woodbury identity helper functions."""
+
+import jax
+from jax import config
+import jax.numpy as jnp
+import jax.random as jr
+import numpy.testing as npt
+import pytest
+
+config.update("jax_enable_x64", True)
+
+from gpjax.linalg.woodbury import woodbury_logdet, woodbury_quad, woodbury_solve
+
+
+def _dense_reference(W, noise):
+    """Build the dense matrix W W^T + diag(noise) for reference."""
+    return W @ W.T + jnp.diag(noise)
+
+
+class TestWoodburySolve:
+    @pytest.mark.parametrize("N,m", [(20, 3), (50, 5), (100, 10)])
+    def test_matches_dense_vector(self, N, m):
+        key = jr.key(0)
+        k1, k2, _k3 = jr.split(key, 3)
+        W = jr.normal(k1, (N, m))
+        noise = jnp.ones(N) * 0.1
+        b = jr.normal(k2, (N,))
+
+        result = woodbury_solve(W, noise, b)
+        expected = jnp.linalg.solve(_dense_reference(W, noise), b)
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    @pytest.mark.parametrize("N,m", [(20, 3), (50, 5)])
+    def test_matches_dense_matrix(self, N, m):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        W = jr.normal(k1, (N, m))
+        noise = jnp.ones(N) * 0.1
+        B = jr.normal(k2, (N, 7))
+
+        result = woodbury_solve(W, noise, B)
+        expected = jnp.linalg.solve(_dense_reference(W, noise), B)
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    def test_heterogeneous_noise(self):
+        key = jr.key(0)
+        k1, k2, k3 = jr.split(key, 3)
+        N, m = 30, 4
+        W = jr.normal(k1, (N, m))
+        noise = jnp.abs(jr.normal(k2, (N,))) + 0.01
+        b = jr.normal(k3, (N,))
+
+        result = woodbury_solve(W, noise, b)
+        expected = jnp.linalg.solve(_dense_reference(W, noise), b)
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    def test_jit(self):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        W = jr.normal(k1, (20, 3))
+        noise = jnp.ones(20) * 0.1
+        b = jr.normal(k2, (20,))
+
+        result = jax.jit(woodbury_solve)(W, noise, b)
+        expected = jnp.linalg.solve(_dense_reference(W, noise), b)
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    def test_grad(self):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        W = jr.normal(k1, (20, 3))
+        noise = jnp.ones(20) * 0.1
+        b = jr.normal(k2, (20,))
+
+        @jax.grad
+        def fn(W):
+            return jnp.sum(woodbury_solve(W, noise, b))
+
+        grads = fn(W)
+        assert jnp.all(jnp.isfinite(grads))
+
+
+class TestWoodburyLogdet:
+    @pytest.mark.parametrize("N,m", [(20, 3), (50, 5), (100, 10)])
+    def test_matches_dense(self, N, m):
+        key = jr.key(0)
+        W = jr.normal(key, (N, m))
+        noise = jnp.ones(N) * 0.1
+
+        result = woodbury_logdet(W, noise)
+        expected = jnp.linalg.slogdet(_dense_reference(W, noise))[1]
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    def test_heterogeneous_noise(self):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        N, m = 30, 4
+        W = jr.normal(k1, (N, m))
+        noise = jnp.abs(jr.normal(k2, (N,))) + 0.01
+
+        result = woodbury_logdet(W, noise)
+        expected = jnp.linalg.slogdet(_dense_reference(W, noise))[1]
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    def test_jit(self):
+        key = jr.key(0)
+        W = jr.normal(key, (20, 3))
+        noise = jnp.ones(20) * 0.1
+
+        result = jax.jit(woodbury_logdet)(W, noise)
+        expected = jnp.linalg.slogdet(_dense_reference(W, noise))[1]
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    def test_grad(self):
+        key = jr.key(0)
+        W = jr.normal(key, (20, 3))
+        noise = jnp.ones(20) * 0.1
+
+        @jax.grad
+        def fn(W):
+            return woodbury_logdet(W, noise)
+
+        grads = fn(W)
+        assert jnp.all(jnp.isfinite(grads))
+
+
+class TestWoodburyQuad:
+    @pytest.mark.parametrize("N,m", [(20, 3), (50, 5)])
+    def test_matches_dense(self, N, m):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        W = jr.normal(k1, (N, m))
+        noise = jnp.ones(N) * 0.1
+        diff = jr.normal(k2, (N,))
+
+        result = woodbury_quad(W, noise, diff)
+        Sigma = _dense_reference(W, noise)
+        expected = diff @ jnp.linalg.solve(Sigma, diff)
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    def test_jit(self):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        W = jr.normal(k1, (20, 3))
+        noise = jnp.ones(20) * 0.1
+        diff = jr.normal(k2, (20,))
+
+        result = jax.jit(woodbury_quad)(W, noise, diff)
+        Sigma = _dense_reference(W, noise)
+        expected = diff @ jnp.linalg.solve(Sigma, diff)
+        npt.assert_allclose(result, expected, atol=1e-6)
+
+    def test_grad(self):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        W = jr.normal(k1, (20, 3))
+        noise = jnp.ones(20) * 0.1
+        diff = jr.normal(k2, (20,))
+
+        @jax.grad
+        def fn(W):
+            return woodbury_quad(W, noise, diff)
+
+        grads = fn(W)
+        assert jnp.all(jnp.isfinite(grads))
+
+
+class TestNumericalStability:
+    def test_large_N_small_m(self):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        N, m = 1000, 10
+        W = jr.normal(k1, (N, m)) * 0.5
+        noise = jnp.ones(N) * 0.1
+        b = jr.normal(k2, (N,))
+
+        result = woodbury_solve(W, noise, b)
+        assert jnp.all(jnp.isfinite(result))
+
+        ld = woodbury_logdet(W, noise)
+        assert jnp.isfinite(ld)
+
+    def test_small_noise(self):
+        key = jr.key(0)
+        k1, k2 = jr.split(key)
+        N, m = 50, 5
+        W = jr.normal(k1, (N, m))
+        noise = jnp.ones(N) * 1e-8
+        b = jr.normal(k2, (N,))
+
+        result = woodbury_solve(W, noise, b)
+        assert jnp.all(jnp.isfinite(result))
+
+
+class TestConjugateMllIntegration:
+    """Verify Woodbury path in conjugate_mll matches dense path."""
+
+    @pytest.mark.parametrize("KernelClass", ["RBF", "Matern52"])
+    def test_mll_matches_dense(self, KernelClass):
+        from gpjax.dataset import Dataset
+        from gpjax.gps import Prior
+        from gpjax.kernels.approximations.hsgp import HSGP
+        from gpjax.kernels.stationary import RBF, Matern52
+        from gpjax.likelihoods import Gaussian
+        from gpjax.mean_functions import Zero
+        from gpjax.objectives import conjugate_mll
+
+        kernel_cls = {"RBF": RBF, "Matern52": Matern52}[KernelClass]
+
+        key = jr.key(42)
+        n = 50
+        x = jnp.linspace(-3, 3, n)[:, None]
+        y = jnp.sin(x) + 0.1 * jr.normal(key, (n, 1))
+        D = Dataset(X=x, y=y)
+
+        base_kernel = kernel_cls(n_dims=1)
+        hsgp = HSGP(
+            base_kernel=base_kernel,
+            num_basis_fns=40,
+            domain_half_width=5.0,
+            center=0.0,
+        )
+        prior = Prior(kernel=hsgp, mean_function=Zero())
+        likelihood = Gaussian(num_datapoints=n)
+        posterior = prior * likelihood
+
+        # Woodbury path (automatic via LowRank gram)
+        mll_woodbury = conjugate_mll(posterior, D)
+
+        # Dense reference: manually compute via dense gram
+        dense_gram = hsgp.gram(x).to_dense()
+        from gpjax.distributions import GaussianDistribution
+        from gpjax.linalg import Dense, psd
+        from gpjax.linalg.utils import add_jitter
+
+        noise = likelihood.noise_vector(n)
+        mx = prior.mean_function(x)
+        y_flat, mx_flat = likelihood.prepare_targets(y, mx)
+        Kxx_dense = add_jitter(dense_gram, prior.jitter)
+        Sigma_dense = Kxx_dense + jnp.diag(noise)
+        Sigma = psd(Dense(Sigma_dense))
+        mll_dense = (
+            GaussianDistribution(jnp.atleast_1d(mx_flat.squeeze()), Sigma)
+            .log_prob(jnp.atleast_1d(y_flat.squeeze()))
+            .squeeze()
+        )
+
+        npt.assert_allclose(mll_woodbury, mll_dense, atol=1e-5)
+
+    def test_mll_differentiable_woodbury(self):
+        from flax import nnx
+        from gpjax.dataset import Dataset
+        from gpjax.gps import Prior
+        from gpjax.kernels.approximations.hsgp import HSGP
+        from gpjax.kernels.stationary import RBF
+        from gpjax.likelihoods import Gaussian
+        from gpjax.mean_functions import Zero
+        from gpjax.objectives import conjugate_mll
+
+        key = jr.key(42)
+        n = 50
+        x = jnp.linspace(-3, 3, n)[:, None]
+        y = jnp.sin(x) + 0.1 * jr.normal(key, (n, 1))
+        D = Dataset(X=x, y=y)
+
+        base_kernel = RBF(n_dims=1)
+        hsgp = HSGP(
+            base_kernel=base_kernel,
+            num_basis_fns=20,
+            domain_half_width=5.0,
+            center=0.0,
+        )
+        prior = Prior(kernel=hsgp, mean_function=Zero())
+        likelihood = Gaussian(num_datapoints=n)
+        posterior = prior * likelihood
+
+        graphdef, state = nnx.split(posterior)
+
+        def loss(state):
+            model = nnx.merge(graphdef, state)
+            return -conjugate_mll(model, D)
+
+        grad_fn = jax.grad(loss)
+        grads = grad_fn(state)
+
+        flat_grads = jax.tree.leaves(grads)
+        for g in flat_grads:
+            assert jnp.all(jnp.isfinite(g))
+
+
+class TestPredictIntegration:
+    """Verify Woodbury predict path matches dense predict path."""
+
+    def test_predict_mean_matches_dense(self):
+        from gpjax.dataset import Dataset
+        from gpjax.gps import Prior
+        from gpjax.kernels.approximations.hsgp import HSGP
+        from gpjax.kernels.stationary import RBF
+        from gpjax.likelihoods import Gaussian
+        from gpjax.mean_functions import Zero
+
+        key = jr.key(42)
+        n = 50
+        x = jnp.linspace(-3, 3, n)[:, None]
+        y = jnp.sin(x) + 0.1 * jr.normal(key, (n, 1))
+        D = Dataset(X=x, y=y)
+
+        base_kernel = RBF(n_dims=1)
+        hsgp = HSGP(
+            base_kernel=base_kernel,
+            num_basis_fns=30,
+            domain_half_width=5.0,
+            center=0.0,
+        )
+        prior = Prior(kernel=hsgp, mean_function=Zero())
+        likelihood = Gaussian(num_datapoints=n)
+        posterior = prior * likelihood
+
+        x_test = jnp.linspace(-2.5, 2.5, 20)[:, None]
+        pred = posterior.predict(x_test, D)
+
+        assert jnp.all(jnp.isfinite(pred.mean))
+        assert jnp.all(jnp.isfinite(pred.covariance()))
+
+    def test_predict_diagonal(self):
+        from gpjax.dataset import Dataset
+        from gpjax.gps import Prior
+        from gpjax.kernels.approximations.hsgp import HSGP
+        from gpjax.kernels.stationary import RBF
+        from gpjax.likelihoods import Gaussian
+        from gpjax.mean_functions import Zero
+
+        key = jr.key(42)
+        n = 50
+        x = jnp.linspace(-3, 3, n)[:, None]
+        y = jnp.sin(x) + 0.1 * jr.normal(key, (n, 1))
+        D = Dataset(X=x, y=y)
+
+        base_kernel = RBF(n_dims=1)
+        hsgp = HSGP(
+            base_kernel=base_kernel,
+            num_basis_fns=30,
+            domain_half_width=5.0,
+            center=0.0,
+        )
+        prior = Prior(kernel=hsgp, mean_function=Zero())
+        likelihood = Gaussian(num_datapoints=n)
+        posterior = prior * likelihood
+
+        x_test = jnp.linspace(-2.5, 2.5, 20)[:, None]
+        pred = posterior.predict(x_test, D, return_covariance_type="diagonal")
+
+        assert jnp.all(jnp.isfinite(pred.mean))
+        assert jnp.all(jnp.isfinite(pred.variance))
+
+    def test_predict_rff(self):
+        from gpjax.dataset import Dataset
+        from gpjax.gps import Prior
+        from gpjax.kernels.approximations import RFF
+        from gpjax.kernels.stationary import RBF
+        from gpjax.likelihoods import Gaussian
+        from gpjax.mean_functions import Zero
+
+        key = jr.key(42)
+        n = 50
+        x = jnp.linspace(-3, 3, n)[:, None]
+        y = jnp.sin(x) + 0.1 * jr.normal(key, (n, 1))
+        D = Dataset(X=x, y=y)
+
+        rff = RFF(base_kernel=RBF(n_dims=1), num_basis_fns=50, key=jr.key(0))
+        prior = Prior(kernel=rff, mean_function=Zero())
+        likelihood = Gaussian(num_datapoints=n)
+        posterior = prior * likelihood
+
+        x_test = jnp.linspace(-2.5, 2.5, 20)[:, None]
+        pred = posterior.predict(x_test, D)
+
+        assert jnp.all(jnp.isfinite(pred.mean))
+        assert jnp.all(jnp.isfinite(pred.covariance()))
diff --git a/uv.lock b/uv.lock
index daa54f696..4042addeb 100644
--- a/uv.lock
+++ b/uv.lock
@@ -17,7 +17,7 @@ resolution-markers = [
 ]
 
 [options]
-exclude-newer = "2026-02-06T09:56:46.847909Z"
+exclude-newer = "2026-02-13T18:59:31.785479Z"
 exclude-newer-span = "P7D"
 
 [[package]]
@@ -951,6 +951,7 @@ wheels = [
 
 [[package]]
 name = "gpjax"
+version = "0.13.6"
 source = { editable = "." }
 dependencies = [
     { name = "beartype" },
@@ -985,6 +986,7 @@ docs = [
     { name = "networkx" },
     { name = "pandas" },
     { name = "pymdown-extensions" },
+    { name = "rdata" },
     { name = "scikit-learn" },
     { name = "seaborn" },
     { name = "ucimlrepo" },
@@ -1041,6 +1043,7 @@ requires-dist = [
     { name = "optax", specifier = ">0.2.1" },
     { name = "pandas", marker = "extra == 'docs'", specifier = ">=1.5.3" },
     { name = "pymdown-extensions", marker = "extra == 'docs'", specifier = ">=10.7.1" },
+    { name = "rdata", marker = "extra == 'docs'", specifier = ">=1.0.0" },
     { name = "scikit-learn", marker = "extra == 'docs'", specifier = ">=1.5.1" },
     { name = "seaborn", marker = "extra == 'docs'", specifier = ">=0.12.2" },
     { name = "tensorstore", marker = "sys_platform == 'darwin'", specifier = "!=0.1.76" },
@@ -3111,6 +3114,21 @@ wheels = [
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 ]
 
+[[package]]
+name = "rdata"
+version = "1.0.0"
+source = { registry = "https://pypi.org/simple" }
+dependencies = [
+    { name = "numpy" },
+    { name = "pandas" },
+    { name = "typing-extensions" },
+    { name = "xarray" },
+]
+sdist = { url = "https://files.pythonhosted.org/packages/5b/e5/c5626257359fde4662f9c2f658bcb37873ffad1490c8991640a361b8cf8e/rdata-1.0.0.tar.gz", hash = "sha256:d924d7657c9eeaee4b86a0ae87999f5beb0070ed47bd491c85bc79ee9644596d", size = 56618, upload-time = "2025-08-15T17:17:10.832Z" }
+wheels = [
+    { url = "https://files.pythonhosted.org/packages/29/b6/aec7624eb1db90e58b7fa34ab6491ee06385020bbb9598d4b8a28051da81/rdata-1.0.0-py3-none-any.whl", hash = "sha256:b671e31676e158dc215297595d5085aee1674b91be3c26e38638ca056167d402", size = 72291, upload-time = "2025-08-15T17:17:09.404Z" },
+]
+
 [[package]]
 name = "readme-renderer"
 version = "44.0"
@@ -4121,6 +4139,20 @@ wheels = [
     { url = "https://files.pythonhosted.org/packages/3f/0e/fa3b193432cfc60c93b42f3be03365f5f909d2b3ea410295cf36df739e31/widgetsnbextension-4.0.15-py3-none-any.whl", hash = "sha256:8156704e4346a571d9ce73b84bee86a29906c9abfd7223b7228a28899ccf3366", size = 2196503, upload-time = "2025-11-01T21:15:53.565Z" },
 ]
 
+[[package]]
+name = "xarray"
+version = "2026.1.0"
+source = { registry = "https://pypi.org/simple" }
+dependencies = [
+    { name = "numpy" },
+    { name = "packaging" },
+    { name = "pandas" },
+]
+sdist = { url = "https://files.pythonhosted.org/packages/f5/85/113ff1e2cde9e8a5b13c2f0ef4e9f5cd6ca3a036b6452f4dd523419289b5/xarray-2026.1.0.tar.gz", hash = "sha256:0c9814761f9d9a9545df37292d3fda89f83201f3e02ae0f09f03313d9cfdd5e2", size = 3107024, upload-time = "2026-01-28T17:49:03.822Z" }
+wheels = [
+    { url = "https://files.pythonhosted.org/packages/7d/8e/952a351c10df395d9bab850f611f4368834ae9104d6449049f5a49e00925/xarray-2026.1.0-py3-none-any.whl", hash = "sha256:5fcc03d3ed8dfb662aa254efe6cd65efc70014182bbc2126e4b90d291d970d41", size = 1403009, upload-time = "2026-01-28T17:49:01.538Z" },
+]
+
 [[package]]
 name = "xdoctest"
 version = "1.3.0"