Metal FFT for powers of 2 up to 2048 (ml-explore#915)

* add Metal FFT for powers of 2

* skip GPU test on linux

* fix contiguity bug

* address comments

* Update mlx/backend/metal/fft.cpp

* Update mlx/backend/metal/fft.cpp

* fix bug in synch

---------

Co-authored-by: Alex Barron <abarron22@apple.com>
Co-authored-by: Awni Hannun <awni.hannun@gmail.com>
Co-authored-by: Awni Hannun <awni@apple.com>

Author: 4 people

benchmarks/python/fft_bench.py

@@ -0,0 +1,57 @@
+# Copyright © 2024 Apple Inc.
+
+import matplotlib
+import mlx.core as mx
+import numpy as np
+from time_utils import measure_runtime
+
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt
+
+
+def bandwidth_gb(runtime_ms, system_size):
+    bytes_per_fft = np.dtype(np.complex64).itemsize * 2
+    bytes_per_gb = 1e9
+    ms_per_s = 1e3
+    return system_size * bytes_per_fft / runtime_ms * ms_per_s / bytes_per_gb
+
+
+def run_bench(system_size):
+    def fft(x):
+        out = mx.fft.fft(x)
+        mx.eval(out)
+        return out
+
+    bandwidths = []
+    for k in range(4, 12):
+        n = 2**k
+        x = mx.random.uniform(shape=(system_size // n, n)).astype(mx.float32)
+        x = x.astype(mx.complex64)
+        mx.eval(x)
+        runtime_ms = measure_runtime(fft, x=x)
+        bandwidths.append(bandwidth_gb(runtime_ms, system_size))
+
+    return bandwidths
+
+
+def time_fft():
+
+    with mx.stream(mx.cpu):
+        cpu_bandwidths = run_bench(system_size=int(2**22))
+
+    with mx.stream(mx.gpu):
+        gpu_bandwidths = run_bench(system_size=int(2**29))
+
+    # plot bandwidths
+    x = [2**k for k in range(4, 12)]
+    plt.scatter(x, gpu_bandwidths, color="green", label="GPU")
+    plt.scatter(x, cpu_bandwidths, color="red", label="CPU")
+    plt.title("MLX FFT Benchmark")
+    plt.xlabel("N")
+    plt.ylabel("Bandwidth (GB/s)")
+    plt.legend()
+    plt.savefig("fft_plot.png")
+
+
+if __name__ == "__main__":
+    time_fft()

mlx/backend/metal/fft.cpp

@@ -1,12 +1,106 @@
 // Copyright © 2023 Apple Inc.
-
+#include "mlx/backend/metal/copy.h"
+#include "mlx/backend/metal/utils.h"
+#include "mlx/mlx.h"
 #include "mlx/primitives.h"
 
 namespace mlx::core {
 
 void FFT::eval_gpu(const std::vector<array>& inputs, array& out) {
+  auto& s = out.primitive().stream();
+  auto& d = metal::device(s.device);
+
   auto& in = inputs[0];
-  throw std::runtime_error("[FFT] NYI for Metal backend.");
+
+  if (axes_.size() == 0 || axes_.size() > 1 || inverse_ ||
+      in.dtype() != complex64 || out.dtype() != complex64) {
+    // Could also fallback to CPU implementation here.
+    throw std::runtime_error(
+        "GPU FFT is only implemented for 1D, forward, complex FFTs.");
+  }
+
+  size_t n = in.shape(axes_[0]);
+
+  if (!is_power_of_2(n) || n > 2048 || n < 4) {
+    throw std::runtime_error(
+        "GPU FFT is only implemented for the powers of 2 from 4 -> 2048");
+  }
+
+  // Make sure that the array is contiguous and has stride 1 in the FFT dim
+  std::vector<array> copies;
+  auto check_input = [this, &copies, &s](const array& x) {
+    // TODO: Pass the strides to the kernel so
+    // we can avoid the copy when x is not contiguous.
+    bool no_copy = x.strides()[axes_[0]] == 1 && x.flags().row_contiguous ||
+        x.flags().col_contiguous;
+    if (no_copy) {
+      return x;
+    } else {
+      array x_copy(x.shape(), x.dtype(), nullptr, {});
+      std::vector<size_t> strides;
+      size_t cur_stride = x.shape(axes_[0]);
+      for (int axis = 0; axis < x.ndim(); axis++) {
+        if (axis == axes_[0]) {
+          strides.push_back(1);
+        } else {
+          strides.push_back(cur_stride);
+          cur_stride *= x.shape(axis);
+        }
+      }
+
+      auto flags = x.flags();
+      size_t f_stride = 1;
+      size_t b_stride = 1;
+      flags.col_contiguous = true;
+      flags.row_contiguous = true;
+      for (int i = 0, ri = x.ndim() - 1; i < x.ndim(); ++i, --ri) {
+        flags.col_contiguous &= (strides[i] == f_stride || x.shape(i) == 1);
+        f_stride *= x.shape(i);
+        flags.row_contiguous &= (strides[ri] == b_stride || x.shape(ri) == 1);
+        b_stride *= x.shape(ri);
+      }
+      // This is probably over-conservative
+      flags.contiguous = false;
+
+      x_copy.set_data(
+          allocator::malloc_or_wait(x.nbytes()), x.data_size(), strides, flags);
+      copy_gpu_inplace(x, x_copy, CopyType::GeneralGeneral, s);
+      copies.push_back(x_copy);
+      return x_copy;
+    }
+  };
+  const array& in_contiguous = check_input(inputs[0]);
+
+  // TODO: allow donation here
+  out.set_data(
+      allocator::malloc_or_wait(out.nbytes()),
+      in_contiguous.data_size(),
+      in_contiguous.strides(),
+      in_contiguous.flags());
+
+  // We use n / 4 threads by default since radix-4
+  // is the largest single threaded radix butterfly
+  // we currently implement.
+  size_t m = n / 4;
+  size_t batch = in.size() / in.shape(axes_[0]);
+
+  auto& compute_encoder = d.get_command_encoder(s.index);
+  {
+    std::ostringstream kname;
+    kname << "fft_" << n;
+    auto kernel = d.get_kernel(kname.str());
+
+    bool donated = in.data_shared_ptr() == nullptr;
+    compute_encoder->setComputePipelineState(kernel);
+    compute_encoder.set_input_array(in_contiguous, 0);
+    compute_encoder.set_output_array(out, 1);
+
+    auto group_dims = MTL::Size(1, m, 1);
+    auto grid_dims = MTL::Size(batch, m, 1);
+    compute_encoder->dispatchThreads(grid_dims, group_dims);
+  }
+  d.get_command_buffer(s.index)->addCompletedHandler(
+      [copies](MTL::CommandBuffer*) mutable { copies.clear(); });
 }
 
 } // namespace mlx::core

mlx/backend/metal/kernels/CMakeLists.txt

@@ -21,6 +21,7 @@ set(
   "binary_two"
   "conv"
   "copy"
+  "fft"
   "gemv"
   "quantized"
   "random"

mlx/backend/metal/kernels/fft.metal

@@ -0,0 +1,195 @@
+// Copyright © 2024 Apple Inc.
+
+// Metal FFT using Stockham's algorithm
+//
+// References:
+// - VkFFT (https://github.com/DTolm/VkFFT)
+// - Eric Bainville's excellent page (http://www.bealto.com/gpu-fft.html)
+
+#include <metal_math>
+#include <metal_common>
+
+
+#include "mlx/backend/metal/kernels/defines.h"
+#include "mlx/backend/metal/kernels/utils.h"
+
+using namespace metal;
+
+float2 complex_mul(float2 a, float2 b) {
+  float2 c;
+  c.x = a.x * b.x - a.y * b.y;
+  c.y = a.x * b.y + a.y * b.x;
+  return c;
+}
+
+float2 get_twiddle(int k, int p) {
+  float theta = -1.0f * k * M_PI_F / (2*p);
+
+  float2 twiddle;
+  twiddle.x = metal::fast::cos(theta);
+  twiddle.y = metal::fast::sin(theta);
+  return twiddle;
+}
+
+// single threaded radix2 implemetation
+void radix2(int i, int p, int m, threadgroup float2* read_buf, threadgroup float2* write_buf) {
+  float2 x_0 = read_buf[i];
+  float2 x_1 = read_buf[i + m];
+
+  // The index within this sub-DFT
+  int k = i & (p - 1);
+
+  float2 twiddle = get_twiddle(k, p);
+
+  float2 z = complex_mul(x_1, twiddle);
+
+  float2 y_0 = x_0 + z;
+  float2 y_1 = x_0 - z;
+
+  int j = (i << 1) - k;
+
+  write_buf[j] = y_0;
+  write_buf[j + p] = y_1;
+}
+
+// single threaded radix4 implemetation
+void radix4(int i, int p, int m, threadgroup float2* read_buf, threadgroup float2* write_buf) {
+  float2 x_0 = read_buf[i];
+  float2 x_1 = read_buf[i + m];
+  float2 x_2 = read_buf[i + 2*m];
+  float2 x_3 = read_buf[i + 3*m];
+
+  // The index within this sub-DFT
+  int k = i & (p - 1);
+
+  float2 twiddle = get_twiddle(k, p);
+  // e^a * e^b = e^(a + b)
+  float2 twiddle_2 = complex_mul(twiddle, twiddle);
+  float2 twiddle_3 = complex_mul(twiddle, twiddle_2);
+
+  x_1 = complex_mul(x_1, twiddle);
+  x_2 = complex_mul(x_2, twiddle_2);
+  x_3 = complex_mul(x_3, twiddle_3);
+
+  float2 minus_i;
+  minus_i.x = 0;
+  minus_i.y = -1;
+
+  // Hard coded twiddle factors for DFT4
+  float2 z_0 = x_0 + x_2;
+  float2 z_1 = x_0 - x_2;
+  float2 z_2 = x_1 + x_3;
+  float2 z_3 = complex_mul(x_1 - x_3, minus_i);
+
+  float2 y_0 = z_0 + z_2;
+  float2 y_1 = z_1 + z_3;
+  float2 y_2 = z_0 - z_2;
+  float2 y_3 = z_1 - z_3;
+
+  int j = ((i - k) << 2) + k;
+
+  write_buf[j] = y_0;
+  write_buf[j + p] = y_1;
+  write_buf[j + 2*p] = y_2;
+  write_buf[j + 3*p] = y_3;
+}
+
+
+// Each FFT is computed entirely in shared GPU memory.
+//
+// N is decomposed into radix-2 and radix-4 DFTs:
+// e.g. 128 = 2 * 4 * 4 * 4
+//
+// At each step we use n / 4 threads, each performing
+// a single-threaded radix-4 or radix-2 DFT.
+//
+// We provide the number of radix-2 and radix-4
+// steps at compile time for a ~20% performance boost.
+template <size_t n, size_t radix_2_steps, size_t radix_4_steps>
+[[kernel]] void fft(
+    const device float2 *in [[buffer(0)]],
+    device float2 * out [[buffer(1)]],
+    uint3 thread_position_in_grid [[thread_position_in_grid]],
+    uint3 threads_per_grid [[threads_per_grid]]) {
+
+  // Index of the DFT in batch
+  int batch_idx = thread_position_in_grid.x * n;
+  // The index in the DFT we're working on
+  int i = thread_position_in_grid.y;
+  // The number of the threads we're using for each DFT
+  int m = threads_per_grid.y;
+
+  // Allocate 2 shared memory buffers for Stockham.
+  // We alternate reading from one and writing to the other at each radix step.
+  threadgroup float2 shared_in[n];
+  threadgroup float2 shared_out[n];
+
+  // Pointers to facilitate Stockham buffer swapping
+  threadgroup float2* read_buf = shared_in;
+  threadgroup float2* write_buf = shared_out;
+  threadgroup float2* tmp;
+
+  // Copy input into shared memory
+  shared_in[i] = in[batch_idx + i];
+  shared_in[i + m] = in[batch_idx + i + m];
+  shared_in[i + 2*m] = in[batch_idx + i + 2*m];
+  shared_in[i + 3*m] = in[batch_idx + i + 3*m];
+
+  threadgroup_barrier(mem_flags::mem_threadgroup);
+
+  int p = 1;
+
+  for (size_t r = 0; r < radix_2_steps; r++) {
+    radix2(i, p, m*2, read_buf, write_buf);
+    radix2(i + m, p, m*2, read_buf, write_buf);
+    p *= 2;
+
+    threadgroup_barrier(mem_flags::mem_threadgroup);
+
+    // Stockham switch of buffers
+    tmp = write_buf;
+    write_buf = read_buf;
+    read_buf = tmp;
+  }
+
+  for (size_t r = 0; r < radix_4_steps; r++) {
+    radix4(i, p, m, read_buf, write_buf);
+    p *= 4;
+
+    threadgroup_barrier(mem_flags::mem_threadgroup);
+
+    // Stockham switch of buffers
+    tmp = write_buf;
+    write_buf = read_buf;
+    read_buf = tmp;
+  }
+
+  // Copy shared memory to output
+  out[batch_idx + i] = read_buf[i];
+  out[batch_idx + i + m] = read_buf[i + m];
+  out[batch_idx + i + 2*m] = read_buf[i + 2*m];
+  out[batch_idx + i + 3*m] = read_buf[i + 3*m];
+}
+
+#define instantiate_fft(name, n, radix_2_steps, radix_4_steps) \
+  template [[host_name("fft_" #name)]] \
+  [[kernel]] void fft<n, radix_2_steps, radix_4_steps>( \
+      const device float2* in [[buffer(0)]], \
+      device float2* out [[buffer(1)]], \
+    uint3 thread_position_in_grid [[thread_position_in_grid]], \
+    uint3 threads_per_grid [[threads_per_grid]]);
+
+
+// Explicitly define kernels for each power of 2.
+instantiate_fft(4, /* n= */ 4, /* radix_2_steps= */ 0, /* radix_4_steps= */ 1)
+instantiate_fft(8, 8, 1, 1)
+instantiate_fft(16, 16, 0, 2)
+instantiate_fft(32, 32, 1, 2)
+instantiate_fft(64, 64, 0, 3)
+instantiate_fft(128, 128, 1, 3)
+instantiate_fft(256, 256, 0, 4)
+instantiate_fft(512, 512, 1, 4)
+instantiate_fft(1024, 1024, 0, 5)
+// 2048 is the max that will fit into 32KB of threadgroup memory.
+// TODO: implement 4 step FFT for larger n.
+instantiate_fft(2048, 2048, 1, 5)

mlx/backend/metal/utils.h

@@ -130,6 +130,10 @@ inline void debug_set_primitive_buffer_label(
 #endif
 }
 
+bool is_power_of_2(int n) {
+  return ((n & (n - 1)) == 0) && n != 0;
+}
+
 } // namespace
 
 } // namespace mlx::core