HeatKernel

class gtda.diagrams.HeatKernel(sigma=0.1, n_bins=100, n_jobs=None)

Convolution of persistence diagrams with a Gaussian kernel.

Based on ideas in 1. Given a persistence diagram consisting of birth-death-dimension triples [b, d, q], subdiagrams corresponding to distinct homology dimensions are considered separately and regarded as sums of Dirac deltas. Then, the convolution with a Gaussian kernel is computed over a rectangular grid of locations evenly sampled from appropriate ranges of the filtration parameter. The same is done with the reflected images of the subdiagrams about the diagonal, and the difference between the results of the two convolutions is computed. The result can be thought of as a (multi-channel) raster image.

Important note:

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

Parameters

• sigma (float, optional default 0.1) – Standard deviation for Gaussian kernel.

• n_bins (int, optional, default: 100) – The number of filtration parameter values, per available homology dimension, to sample during fit.

• 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.

homology_dimensions_

Homology dimensions seen in fit.

Type

tuple

samplings_

For each number in homology_dimensions_, a discrete sampling of filtration parameters, calculated during fit according to the minimum birth and maximum death values observed across all samples.

Type

dict

See also

BettiCurve, PersistenceLandscape, PersistenceEntropy, Amplitude, PairwiseDistance, Silhouette, PersistenceImage, gtda.homology.VietorisRipsPersistence

Notes

The samplings in samplings_ are in general different between different homology dimensions. This means that the (i, j)-th pixel of an image in homology dimension q typically arises from a different pair of parameter values to the (i, j)-th pixel of an image in dimension q’.

References

1

J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt, “A Stable Multi-Scale Kernel for Topological Machine Learning”; 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4741–4748, 2015; DOI: 10.1109/CVPR.2015.7299106.

__init__(sigma=0.1, n_bins=100, 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, for each dimension separately, store evenly sample filtration parameter values in samplings_. 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, 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, 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 – Multi-channel raster images: one image per sample and one channel per homology dimension seen in fit. Index i along axis 1 corresponds to the i-th homology dimension in homology_dimensions_.

Return type

ndarray of shape (n_samples, n_homology_dimensions, n_bins, n_bins)

fit_transform_plot(X, y=None, sample=0, **plot_params)

Fit to data, then apply transform_plot.

Parameters

• X (ndarray of shape (n_samples, ..)) – Input data.

• y (ndarray of shape (n_samples,) or None) – Target values for supervised problems.

• sample (int) – Sample to be plotted.

• **plot_params – Optional plotting parameters.

Returns

Xt – Transformed one-sample slice from the input.

Return type

ndarray of shape (1, ..)

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

plot(Xt, sample=0, homology_dimension_idx=0, colorscale='blues', plotly_params=None)

Plot a single channel –- corresponding to a given homology dimension – in a sample from a collection of heat kernel images.

Parameters

• Xt (ndarray of shape (n_samples, n_homology_dimensions, n_bins, n_bins)) – Collection of multi-channel raster images, such as returned by transform.

• sample (int, optional, default: 0) – Index of the sample in Xt to be selected.

• homology_dimension_idx (int, optional, default: 0) – Index of the channel in the selected sample to be plotted. If Xt is the result of a call to transform and this index is i, the plot corresponds to the homology dimension given by the i-th entry in homology_dimensions_.

• colorscale (str, optional, default: "blues") – Color scale to be used in the heat map. Can be anything allowed by plotly.graph_objects.Heatmap.

• plotly_params (dict or None, optional, default: None) – Custom parameters to configure the plotly figure. Allowed keys are "trace" and "layout", and the corresponding values should be dictionaries containing keyword arguments as would be fed to the update_traces and update_layout methods of plotly.graph_objects.Figure.

Returns

fig – Plotly figure.

Return type

plotly.graph_objects.Figure object

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)

Compute multi-channel raster images from diagrams in X by convolution with a Gaussian kernel and reflection about the diagonal.

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 – Multi-channel raster images: one image per sample and one channel per homology dimension seen in fit. Index i along axis 1 corresponds to the i-th homology dimension in homology_dimensions_.

Return type

ndarray of shape (n_samples, n_homology_dimensions, n_bins, n_bins)

transform_plot(X, sample=0, **plot_params)

Take a one-sample slice from the input collection and transform it. Before returning the transformed object, plot the transformed sample.

Parameters

• X (ndarray of shape (n_samples, ..)) – Input data.

• sample (int) – Sample to be plotted.

• **plot_params – Optional plotting parameters.

Returns

Xt – Transformed one-sample slice from the input.

Return type

ndarray of shape (1, ..)