Comparison Spaces

The idea of a comparison space is to have a space $Y$ and a map $\Gamma \rightarrow Y$ such that the maps

$H_1( O_r(\gamma) \rightarrow Y_r)$

yield a meaningful classifier for a cycle in the thickening of $\gamma$. Currently, only static spaces $Y$ are supported.

Comparison Space Interface

The comparison space interface is defined by:

abstract type AbstractComparisonSpace end

The following methods are required:

• betti_1(cs): Returns the first Betti number of the comparison space.
• map_cycle(cs, points, simplices, coeffs): Returns a vector of length betti_1(cs) representing the image of the cycle mapped to the comparison space.

Cubical Comparison Space

Cubical comparison spaces are comparison spaces which are cubical complexes. We implemented cubical comparison spaces in space alone and in the unit tangent bundle. Furthermore, we provide an interface which can be extended, e.g. to support different domains or periodic boundaries.

Interface

The interface for cubical comparison spaces is defined by:

abstract type AbstractCubicalComparisonSpace <: AbstractComparisonSpace end

It assumes that a cubical acyclic carrier (i.e. a subtype of AbstractCubicalAcyclicCarrier) is used to map the cycle.

The following methods are required:

• edge_boxes(cs, p1, p2): Compute the sequence of boxes covering the edge between points p1 and p2.
• carrier(cs): Returns the cubical acyclic carrier.
• betti_1(comparison_space): Returns the first Betti number of the comparison space.

The implementation of map_cycle for AbstractCubicalComparisonSpaces does the following:

1. For each edge:
   1. Compute the boxes which it intersects (using edge_boxes).
   2. Use the carrier from carrier(cs) to get a 1-chain in the comparison space.
2. Sum and return the 1-chains.

CubicalComparisonSpace and SBCubicalComparisonSpace

We provide two implementations of cubical comparison spaces:

• CubicalComparisonSpace: The cubes are assumed to cover the data in space only.
• SBCubicalComparisonSpace: The cubes are assumed to cover the data in the unit tangent bundle.

Acyclic Carrier Interface

The logic of representing cubical complexes is separated from the logic of the inclusion map. Currently, only a Vietoris–Rips type implementation is supported.

Sampleable Trajectory

Under Construction.

Subsegment Experiments

Use RandomSubsegmentExperiment to evaluate cycling signatures on randomly sampled trajectory subsegments. For backend comparisons, sample starts once and pass them into each run:

exp = RandomSubsegmentExperiment(traj_space, [20, 40, 80], 25, 42)
starts = sample_segment_starts(exp)

dm = run_experiment(exp; alg=Val(:DistanceMatrix), segment_starts=starts, progress=false)
rp = run_experiment(exp; alg=Val(:Ripserer), segment_starts=starts, progress=false)
comparison = compare_experiment_results(dm, rp)

When Ripserer is loaded, the package exposes three Ripserer-backed algorithms: Val(:Ripserer) uses a thresholded filtration with Ripserer's involuted representatives, Val(:RipsererNoThreshold) uses the same representative path without passing a filtration threshold to Ripserer, and Val(:RipsererManualReconstruct) uses a thresholded filtration plus Ripserer.reconstruct_cycle.

run_timed_experiment records one time_ns() measurement per cycling_signature call and returns a TimedRandomSubsegmentResult. Use run_paired_timed_experiments to sample starts once and run multiple backends on those identical subsegments. The package returns plain Julia results; downstream scripts can convert summarize_timings and summarize_agreement output to tables.