geomlearn

Companion Code Library for Learning as Geometry Discovery

Author: Agus Sudjianto | Date: March 2026

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About the Book

Learning is geometry discovery under task constraints.

A model does not merely map inputs to outputs. It discovers or imposes a geometry — a system of distances, directions, comparisons, and admissible transformations — in which the relevant structure becomes visible and actionable. The fitted function is the visible outcome; the geometry that makes it stable, useful, and intelligible is the deeper achievement.

Learning as Geometry Discovery is a geometry-first interpretation of machine learning and data science. It offers a unifying framework that connects classical statistical models, modern deep learning, memory systems, and reasoning architectures through a shared geometric vocabulary.

Four Organizing Principles

1. Representation is a hypothesis. Every choice of features, coordinates, or embedding is a hypothesis about what structure matters.
2. Metric geometry is not enough. Distance, similarity, and inner products capture enormous structure — but direction, order, flow, and constrained composition require richer tools.
3. Local simplicity is the source of interpretability. Models become interpretable not by being globally simple, but by being locally simple in a discoverable way.
4. Reasoning requires constrained movement through state. Memory, knowledge retrieval, and logical inference are forms of directed, constrained movement through a structured state space.

Structure

The book has nineteen chapters in eight parts:

Part	Title	Chapters
I	Why Geometry Is the Real Object of Learning	1 -- 2
II	Expanding the Geometric Vocabulary	3 -- 6
III	Geometric Algebra in Learning	7 -- 10
IV	Attention, Interaction, and Learned Geometry	11 -- 12
V	Models as Geometric Mechanisms	13 -- 14
VI	Time, Causality, and Directional Systems	15 -- 16
VII	Reasoning, Memory, and Knowledge	17 -- 18
VIII	Synthesis and Research Agenda	19

Parts I--II build the geometric ladder: coordinates, inner products, similarity, kernels, partition geometry, manifolds, and curvature — covering most of classical machine learning. Part III introduces directional structure: geometric algebra, rotation, and the symmetric/antisymmetric operator decomposition. Part IV develops learned geometry: attention and interaction. Part V reinterprets familiar models as geometric mechanisms. Part VI extends geometry to time, causality, transport, and flow. Part VII treats knowledge, memory, and reasoning as geometrically structured state evolution. Part VIII synthesizes the framework and proposes a research agenda.

Reading Paths

• Practical ML and statistics: Chapters 1--6, 9--14, 19
• Representation and deep learning: Chapters 1--4, 7--12, 16, 18--19
• Memory, reasoning, and agentic systems: Chapters 7--12, 15--19

Who This Book Is For

• ML and data science practitioners who want deeper structural understanding of why models work
• Statisticians seeking a broader geometric language beyond classical covariance-and-projection
• Advanced undergraduates and graduate students looking for a unifying conceptual framework
• Researchers in interpretability, reasoning, memory, and trustworthy AI

The main text is accessible with linear algebra and basic probability. Graduate extensions deepen the mathematics. Computational labs make the ideas concrete.

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About This Package

geomlearn is the companion PyTorch library implementing the geometric algorithms, decompositions, and diagnostics from each chapter of the book.

Installation

pip install geomlearn

Or install from source:

git clone https://github.com/asudjianto-xml/geomlearn.git
cd geomlearn
pip install .

Requirements

• Python >= 3.10
• PyTorch >= 2.0

API

Each module provides a consistent interface:

• analyze() — extract geometric quantities from data or models
• diagnose() — run health checks, return structured diagnostics
• Classes use fit() / transform() pattern where applicable
• All tensors are PyTorch tensors; GPU-aware via device parameter

Modules

Module	Topic
ch01_geometry_discovery	Learning as geometry discovery
ch02_everyday_geometry	Geometry in everyday data science
ch03_basic_geometry	Basic language of geometry
ch04_kernels	Kernels, similarity, and comparison
ch05_partitions	Partition geometry
ch06_manifolds	Manifolds and nonlinear geometry
ch07_direction_flow	Direction, asymmetry, and flow
ch08_geometric_algebra	Orientation, antisymmetry, geometric algebra
ch09_rotations	Rotation, direction, small structured change
ch10_operator_decomposition	Symmetric and antisymmetric structure
ch11_attention	Attention as learned geometry
ch12_interaction	Interaction geometry
ch13_classical_models	Classical models revisited as geometry
ch14_tree_stretch	Tree-routed stretch models
ch15_time_series	Time series, causality, directional geometry
ch16_transport	Transport, propagation, and flow
ch17_knowledge_memory	Knowledge and memory as geometric state
ch18_reasoning	Reasoning as constrained movement
ch19_synthesis	Synthesis and open problems

Quick Start

import torch
import geomlearn

# Example: analyze geometric structure of a dataset
X = torch.randn(100, 5)
result = geomlearn.ch03_basic_geometry.analyze(X)

Companion Notebooks

The companion_book/ directory contains Jupyter notebooks (one per chapter) with worked examples and computational labs that demonstrate the library in action. These notebooks correspond to the computational labs described in the book.

License

MIT