scott.topology.distances

class scott.topology.distances.TopologicalDistance(diagram1, diagram2, norm=2)[source]

Bases: ABC

Abstract base class for computing topological distances.

Takes in two persistence diagrams (or lists of persistence diagrams) and handles necessary transformations to compute a specific distance function.

diagram 1

The List[PersistenceDiagram] to be compared with diagram2.

Type:

List[PersistenceDiagram]

diagram 2

The List[PersistenceDiagram] to be compared with diagram1.

Type:

List[PersistenceDiagram]

norm

Defines what norm will be used for calculations. Default is 2.

Type:

int, default=2.

__init__(diagram1, diagram2, norm=2) → None[source]

Initializes an instance of the TopologicalDistance class, and converts diagram1 and diagram2 to lists if they are not already.

Parameters:

  • diagram1 (PersistenceDiagram or List[PersistenceDiagram]) – The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram2.

  • 2 (diagram) – The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram1.

  • norm (int, default=2.) – Defines what norm will be used for calculations. Default is 2.

Return type:

None

compute_norm(x)[source]

Compute norm of a vector x according to the specified order.

abstract supports_distribution() → bool[source]

Indicates if the distance type supports distributions of persistence diagrams.

abstract fit(**kwargs)[source]

Converts persistence diagrams to topological descriptors.

abstract transform(descriptor1, descriptor2) → float[source]

Defines how to compute distances between topological descriptors.

class scott.topology.distances.ImageDistance(diagram1, diagram2, norm=2, bandwidth=1.0, weight=<function ImageDistance.<lambda>>, resolution=[20, 20])[source]

Bases: TopologicalDistance

Takes in persistence diagrams, converts them to persistence images, and computes the distance between them. This TopologicalDistance subclass supports comparison of distributions (i.e. input of lists of persistence diagrams).

diagram 1

The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram2.

Type:

PersistenceDiagram or List[PersistenceDiagram]

diagram 2

The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram1.

Type:

PersistenceDiagram or List[PersistenceDiagram]

norm

Defines what norm will be used for calculations. Default is 2.

Type:

int, default=2.

bandwidth

Controls the Gaussian kernel for the probability distribution calculated for each point on the birth/persistence diagram. See gudhi documentation for more information.

Type:

double, default = 1.0

weight

Defines the weight function used to compute the weighted sum of the probability distributions for each point on the bith/persistence diagram, i.e. the persistence surface. Default is a constant function. Other common choices are a linear function or a bump function, which put greater emphasis on points with longer persistence. See gudhi documentation for more information.

Type:

function, default = lambda x: 1

resolution

The dimensions of the persistence image in pixels.

Type:

List[int, int], default = [20,20]

image_transformer

The object that powers the transformation of persistence diagrams into persistence images.

Type:

gudhi package object

__init__(diagram1, diagram2, norm=2, bandwidth=1.0, weight=<function ImageDistance.<lambda>>, resolution=[20, 20]) → None[source]

Initializes an instance of the TopologicalDistance class, and converts diagram1 and diagram2 to lists if they are not already.

Parameters:

  • diagram1 (PersistenceDiagram or List[PersistenceDiagram]) – The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram2.

  • 2 (diagram) – The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram1.

  • norm (int, default=2.) – Defines what norm will be used for calculations. Default is 2.

Return type:

None

supports_distribution() → bool[source]

Indicates support for distributions of persistence images.

fit()[source]

Computes and returns persistence images from the persistence diagrams, diagram1 and diagram2 (or the average persistence images, if diagram1 and/or diagram2 are distributions.

Returns:

  • avg1 (PersistenceImage) – A wrapper object for the (average) persistence landscape for diagram1. Attribute called ‘pixels’ stores a Dict[int, np.array] that a maps homology dimension key to a np.array that stores image data.

  • avg2 (PersistenceImage) – A wrapper object for the (average) persistence landscape for diagram1. Attribute called ‘pixels’ stores a Dict[int, np.array] that a maps homology dimension key to a np.array that stores image data.

transform(image1, image2) → float[source]

Computes the norm-based distance between two persistence images. Cannot be executed before fit() method has been run to generate persistence images.

Parameters:

  • image1 (PersistenceImage) – A wrapper object for the first persistence image to be compared. Attribute called ‘pixels’ stores a Dict[int, np.array] that a maps homology dimension key to a np.array with shape (resolution[0] * resolution[1], 1), which contains the values of the persistence surface at each pixel.

  • image2 (PersistenceImage) – A wrapper object for the second persistence image to be compared. Attribute called ‘pixels’ stores a Dict[int, np.array] that a maps homology dimension key to a np.array with shape (resolution[0] * resolution[1], 1), which contains the values of the persistence surface at each pixel.

Returns:

The norm-based distance between the two given persistence landscapes.

Return type:

float

fit_transform() → float[source]

Generates persistence images for diagram1, diagram2 and computes the distance between them. :returns: The distance between diagram1 and diagram2, computed via image vectorization. :rtype: float

class scott.topology.distances.LandscapeDistance(diagram1, diagram2, norm=2, resolution=1000, num_functions=5)[source]

Bases: TopologicalDistance

Takes in persistence diagrams, converts them to persistence landscapes, and computes the distance between them. This TopologicalDistance subclass supports comparison of distributions (i.e. input of lists of persistence diagrams).

diagram 1

The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram2.

Type:

PersistenceDiagram or List[PersistenceDiagram]

diagram 2

The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram1.

Type:

PersistenceDiagram or List[PersistenceDiagram]

norm

Defines what norm will be used for calculations. Default is 2.

Type:

int, default=2.

resolution

The resolution/number of samples in each landscape function.

Type:

int, default=1000

num_functions

The number of landscape functions to be computed.

Type:

int, default=5.

landscape_transformer

The object that powers the transformation of persistence diagrams to persistence landscapes.

Type:

gudhi package object

__init__(diagram1, diagram2, norm=2, resolution=1000, num_functions=5) → None[source]

Creates a LandscapeDistance object for two persistence diagrams (or lists of persistence diagrams).

Parameters:

  • 1 (diagram) – The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram2.

  • 2 (diagram) – The PersistenceDiagram or List[PersistenceDiagram] to be compared with diagram1.

  • norm (int, default=2.) – Defines what norm will be used for calculations. Default is 2.

  • resolution (int, default=1000) – The resolution/number of samples in each landscape function.

  • num_functions (int, default=5.) – The number of landscape functions to be computed.

supports_distribution() → bool[source]

Indicates support for distributions of persistence diagrams.

fit()[source]

Computes and returns persistence landscapes from the persistence diagrams, diagram1 and diagram2 (or the average persistence landscapes, if diagram1 and/or diagram2 are distributions.

Returns:

  • avg1 (PersistenceLandscape) – A wrapper object for the (average) persistence landscape for diagram1. Attribute called ‘functions’ stores a Dict[int, np.array] that a maps homology dimension key to a np.array with shape (num_functions * resolution, 1) which concatenates the samples from all landscape functions.

  • avg2 (PersistenceLandscape) – A wrapper object for the (average) persistence landscape for diagram1. Attribute called ‘functions’ stores a Dict[int, np.array] that a maps homology dimension key to a np.array with shape (num_functions * resolution, 1) which concatenates the samples from all landscape functions.

transform(landscape1: PersistenceLandscape, landscape2: PersistenceLandscape) → float[source]

Computes the norm-based distance between two persistence landscapes. Cannot be executed before fit() method has been run to generate persistence landscapes.

Parameters:

  • landscape1 (PersistenceLandscape) – A wrapper object for the first persistence landscape to be compared. Attribute called ‘functions’ stores a Dict[int, np.array] that a maps homology dimension key to a np.array with shape (num_functions * resolution, 1) which concatenates the samples from all landscape functions.

  • landscape1 – A wrapper object for the second persistence landscape to be compared. Attribute called ‘functions’ stores a Dict[int, np.array] that a maps homology dimension key to a np.array with shape (num_functions * resolution, 1) which concatenates the samples from all landscape functions.

Returns:

The norm-based distance between the two given persistence landscapes.

Return type:

float

fit_transform() → float[source]

Generates persistence landscapes for diagram1, diagram2 and computes the distance between them. :returns: The distance between diagram1 and diagram2, computed via landscape vectorization. :rtype: float