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Geometry, Symmetry, And Trajectories

This page covers topology families that often show up in modern ML systems but are not the same as persistent homology.

Status: mixed. Homotopy, stratified spaces, and finite group actions now have small prototype diagnostics. Knot/link, low-dimensional, and categorical topology remain docs-only until they have datasets and baselines that make the signal actionable.

Stratified And Singular Spaces

Real ML data is often not a smooth manifold. ReLU networks create regions, decision boundaries have corners, embeddings contain seams, and failure surfaces can be lower-dimensional.

A stratified space decomposes a space into pieces:

\[ X = \bigcup_i S_i \]

Each stratum \(S_i\) is easier to model than the whole space. The frontier condition says if one stratum touches the closure of another, their dimensions must fit a controlled hierarchy.

ML use cases:

  • inspect ReLU regions;
  • describe decision-boundary strata;
  • detect non-manifold embedding artifacts;
  • separate boundary failures from bulk failures.

Prototype: topoml.activation_strata records activation sign-pattern strata and boundary fractions. It is a ReLU-region diagnostic, not a complete stratified-space implementation.

Homotopy

Homotopy studies when one path, map, or shape can be continuously deformed into another.

Two paths \(\gamma_0,\gamma_1 : [0,1] \to X\) are path-homotopic when there is a continuous map:

\[ H : [0,1] \times [0,1] \to X \]

with \(H(s,0)=\gamma_0(s)\), \(H(s,1)=\gamma_1(s)\), and endpoints fixed.

For ML, homotopy can distinguish optimization paths or planning trajectories that homology may compress into the same count.

flowchart LR
  A["Same start/end"] --> B["Path around obstacle"]
  A --> C["Path through corridor"]
  B --> D["Different deformation class"]
  C --> D
  D --> E["Different safety or routing decision"]

Prototype: topoml.path_homotopy_signature computes the winding number of a finite 2D loop around a basepoint:

\[ w = \mathrm{round}\left(\Delta \theta / 2\pi\right) \]

This catches simple loop classes around an obstacle. It does not compute a full fundamental group.

Cohomology Beyond Betti Counts

Betti numbers count how many independent features exist. Cohomology can encode interactions, consistency constraints, and obstructions.

A cochain complex has maps:

\[ C^0 \xrightarrow{d} C^1 \xrightarrow{d} C^2 \]

with:

\[ d \circ d = 0 \]

A cocycle satisfies \(d\alpha = 0\). A nontrivial cohomology class can indicate global structure that local patches cannot remove.

ML use cases:

  • feature interaction diagnostics;
  • distributed consistency checks;
  • circular coordinates;
  • obstruction-style tests for incompatible local explanations.

Groups, Quotients, And Equivariance

A group action describes a symmetry:

\[ G \times X \to X \]

The orbit of a point is:

\[ Gx = \{g \cdot x : g \in G\} \]

The quotient \(X/G\) identifies points that differ only by the group action. For ML, this is the language behind rotation invariance, permutation equivariance, gauge-equivariant models, and symmetry-aware tests.

Prototype APIs:

  • topoml.finite_orbit_signature summarizes a finite sampled orbit, stabilizer count, and quotient representative.
  • topoml.equivariance_residual tests finite sampled invariance or equivariance residuals across model outputs.

These APIs test sampled actions only. They do not prove a model is equivariant for a continuous group.

Bundles And Sections

A fiber bundle has a projection:

\[ \pi : E \to B \]

The base \(B\) represents positions or contexts. The fiber over \(b\) contains local values above that context:

\[ \pi^{-1}(b) \]

A section chooses one value from each fiber. In ML language, bundles are useful for layerwise parameter transport, local coordinate systems, gauge fields, distributed model alignment, and symmetry defects.

flowchart TB
  B["Base space: inputs, positions, contexts"] --> F1["Fiber: local representation"]
  B --> F2["Fiber: local gauge or frame"]
  F1 --> S["Section: chosen local state"]
  F2 --> S
  S --> Q["Check transport, holonomy, and mismatch"]

Knots and braids study embedded curves and crossings. A braid word is built from generators such as:

\[ \sigma_i \]

where \(\sigma_i\) swaps neighboring strands with a crossing.

ML and systems use cases are early-stage:

  • multi-agent trajectory entanglement;
  • robotics path planning;
  • thread interleavings;
  • attention-head coupling over depth;
  • recurrent trajectory crossings in latent space.

Status: Docs-only until there is a dataset, invariant, and baseline that makes the signal actionable.

Low-Dimensional And Geometric Topology

For surfaces, the Euler characteristic is:

\[ \chi = V - E + F \]

For an orientable closed surface of genus \(g\):

\[ \chi = 2 - 2g \]

This matters for meshes, scenes, robotics maps, simulation surfaces, and any ML pipeline that predicts geometry. If a model changes genus, breaks orientability, or creates invalid handles, topology can catch it.

Status: Docs-only first. A prototype should start with mesh diagnostics and compare against ordinary geometry-only validation.

Categorical And Pointless Topology

Pointless topology describes spaces through open sets instead of points. A locale or frame treats opens as the primary object, with operations similar to intersection and union.

For distributed systems and observability, this is useful because a system may know regions, constraints, logs, or covers without knowing exact points.

Status: Docs-only. This is foundation material for future sheaf and observability APIs, not an active implementation claim.