PairwiseDistance

class gtda.diagrams.PairwiseDistance(metric='landscape', metric_params=None, order=2.0, n_jobs=None)

Distances between pairs of persistence diagrams.

Given two collections of persistence diagrams consisting of birth-death-dimension triples [b, d, q], a collection of distance matrices or a single distance matrix between pairs of diagrams is calculated according to the following steps:

1. All diagrams are partitioned into subdiagrams corresponding to distinct homology dimensions.

2. Pairwise distances between subdiagrams of equal homology dimension are calculated according to the parameters metric and metric_params. This gives a collection of distance matrices, \(\mathbf{D} = (D_{q_1}, \ldots, D_{q_n})\).

3. The final result is either \(\mathbf{D}\) itself as a three-dimensional array, or a single distance matrix constructed by taking norms of the vectors of distances between diagram pairs.

Important notes:

• Input collections of persistence diagrams for this transformer must satisfy certain requirements, see e.g. fit.

• The shape of outputs of transform depends on the value of the order parameter.

Parameters

• metric ('bottleneck' | 'wasserstein' | 'betti' | 'landscape' | 'silhouette' | 'heat' | 'persistence_image', optional, default: 'landscape') –

  Distance or dissimilarity function between subdiagrams:

  • 'bottleneck' and 'wasserstein' refer to the identically named perfect-matching–based notions of distance.

  • 'betti' refers to the \(L^p\) distance between Betti curves.

  • 'landscape' refers to the \(L^p\) distance between persistence landscapes.

  • 'silhouette' refers to the \(L^p\) distance between silhouettes.

  • 'heat' refers to the \(L^p\) distance between Gaussian-smoothed diagrams.

  • 'persistence_image' refers to the \(L^p\) distance between Gaussian-smoothed diagrams represented on birth-persistence axes.

• metric_params (dict or None, optional, default: None) –

  Additional keyword arguments for the metric function (passing None is equivalent to passing the defaults described below):

  • If metric == 'bottleneck' the only argument is delta (float, default: 0.01). When equal to 0., an exact algorithm is used; otherwise, a faster approximate algorithm is used and symmetry is not guaranteed.

  • If metric == 'wasserstein' the available arguments are p (float, default: 2.) and delta (float, default: 0.01). Unlike the case of 'bottleneck', delta cannot be set to 0. and an exact algorithm is not available.

  • If metric == 'betti' the available arguments are p (float, default: 2.) and n_bins (int, default: 100).

  • If metric == 'landscape' the available arguments are p (float, default: 2.), n_bins (int, default: 100) and n_layers (int, default: 1).

  • If metric == 'silhouette' the available arguments are p (float, default: 2.), power (float, default: 1.) and n_bins (int, default: 100).

  • If metric == 'heat' the available arguments are p (float, default: 2.), sigma (float, default: 0.1) and n_bins (int, default: 100).

  • If metric == 'persistence_image' the available arguments are p (float, default: 2.), sigma (float, default: 0.1), n_bins (int, default: 100) and weight_function (callable or None, default: None).

• order (float or None, optional, default: 2.) – If None, transform returns for each pair of diagrams a vector of distances corresponding to the dimensions in homology_dimensions_. Otherwise, the \(p\)-norm of these vectors with \(p\) equal to order is taken.

• n_jobs (int or None, optional, default: None) – The number of jobs to use for the computation. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors.

effective_metric_params_

Dictionary containing all information present in metric_params as well as relevant quantities computed in fit.

Type

dict

homology_dimensions_

Homology dimensions seen in fit, sorted in ascending order.

Type

tuple

See also

Amplitude, Scaler, Filtering, BettiCurve, PersistenceLandscape, PersistenceImage, HeatKernel, Silhouette, gtda.homology.VietorisRipsPersistence

Notes

To compute distances without first splitting the computation between different homology dimensions, data should be first transformed by an instance of ForgetDimension.

Hera is used as a C++ backend for computing bottleneck and Wasserstein distances between persistence diagrams. Python bindings were modified for performance from the Dyonisus 2 package.

__init__(metric='landscape', metric_params=None, order=2.0, n_jobs=None)

Initialize self. See help(type(self)) for accurate signature.

fit(X, y=None)

Store all observed homology dimensions in homology_dimensions_ and compute effective_metric_params. Then, return the estimator.

This method is here to implement the usual scikit-learn API and hence work in pipelines.

Parameters

• X (ndarray of shape (n_samples_fit, n_features, 3)) – Input data. Array of persistence diagrams, each a collection of triples [b, d, q] representing persistent topological features through their birth (b), death (d) and homology dimension (q). It is important that, for each possible homology dimension, the number of triples for which q equals that homology dimension is constants across the entries of X.

• y (None) – There is no need for a target in a transformer, yet the pipeline API requires this parameter.

Returns

self

Return type

object

fit_transform(X, y=None, **fit_params)

Fit to data, then transform it.

Fits transformer to X and y with optional parameters fit_params and returns a transformed version of X.

Parameters

• X (ndarray of shape (n_samples_fit, n_features, 3)) – Input data. Array of persistence diagrams, each a collection of triples [b, d, q] representing persistent topological features through their birth (b), death (d) and homology dimension (q). It is important that, for each possible homology dimension, the number of triples for which q equals that homology dimension is constants across the entries of X.

• y (None) – There is no need for a target in a transformer, yet the pipeline API requires this parameter.

Returns

Xt – Distance matrix or collection of distance matrices between diagrams in X and diagrams seen in fit. In the second case, index i along axis 2 corresponds to the i-th homology dimension in homology_dimensions_.

Return type

ndarray of shape (n_samples, n_samples_fit, n_homology_dimensions) if order is None, else (n_samples, n_samples_fit)

get_params(deep=True)

Get parameters for this estimator.

Parameters

deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns

params – Parameter names mapped to their values.

Return type

mapping of string to any

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters

**params (dict) – Estimator parameters.

Returns

self – Estimator instance.

Return type

object

transform(X, y=None)

Computes a distance or vector of distances between the diagrams in X and the diagrams seen in fit.

Parameters

• X (ndarray of shape (n_samples, n_features, 3)) – Input data. Array of persistence diagrams, each a collection of triples [b, d, q] representing persistent topological features through their birth (b), death (d) and homology dimension (q). It is important that, for each possible homology dimension, the number of triples for which q equals that homology dimension is constants across the entries of X.

• y (None) – There is no need for a target in a transformer, yet the pipeline API requires this parameter.

Returns

Xt – Distance matrix or collection of distance matrices between diagrams in X and diagrams seen in fit. In the second case, index i along axis 2 corresponds to the i-th homology dimension in homology_dimensions_.

Return type

ndarray of shape (n_samples, n_samples_fit, n_homology_dimensions) if order is None, else (n_samples, n_samples_fit)