This type synonym exists to make other synonyms more concise. Each simplex in a filtration is represented as a triple: its filtration index, the indices of its vertices in the original data, and the indices of its faces in the next lowest dimension. Edges do not have reference to their faces, as it would be redundant with their vertices. All simplices are sorted according to filtration index upon construction of the filtration. In each dimension, all simplices are sorted in increasing order of filtration index, and every simplices face indices are sorted in decreasing order; both of these facts are critical to the computation of persistent homology.

A type representing a filtration whose vertices all have filtration index 0. Slightly faster and slightly less memory usage. The first component is simply the number of vertices. The second component is a vector with an entry for each dimension of simplices, starting at dimension 1 for edges.

Representation of a filtration which, unlike SimpleFiltration, can cope with vertices that have a non-zero filtration index. Vertices of the filtration are represented like all other simplices except that they don't their own have vertices or faces.

Note that, since this library currently only deals with static pointcloud data, all of the filtration construction functions produce vertices whose filtration index is 0. Thus, if you want to use this type you will have to construct the instances yourself.

A type extending the number line to positive and negative infinity. Used for representing infinite barcodes, bottleneck distance, and persistence landscapes.

(x, Finite y) is a topological feature that appears at the index or scale x and disappears at the index or scale y. (x, Infinity) begins at x and doesn't disappear.

A Persistence landscape is a certain type of piecewise linear function based on a barcode diagram. It can be represented as a list of critical points paired with critical values. Useful for taking averages and differences between barcode diagrams.

Shows all the information in a simplex.

Shows all the information in a filtration.

Gets the simplicial complex specified by the filtration index. This is O(n) with respect to the number of simplices.

Return the dimension of the highest dimensional simplex in the filtration (constant time).

Convert a simple filtration into an ordinary filtration.

This function creates a filtration out of a simplicial complex by removing simplices that contain edges that are too long for each scale in the list. This is really a helper function to be called by makeRipsFiltrationFast, but I decided to expose it in case you have a simplicial complex and weighted graph lying around. The scales MUST be in decreasing order.

This function constructs a filtration of the Vietoris-Rips complexes associated with the scales. Note that this a fast function, meaning it uses O(n^2) memory to quickly access distances where n is the number of data points.

Same as above except it uses parallelism when computing the Vietoris-Rips complex of the largest scale.

The same as filterbyWeightsFast except it uses far less memory at the cost of speed. Note that the scales must be in decreasing order.

Constructs the filtration of Vietoris-Rips complexes corresponding to each of the scales.

Same as above except it uses parallelism when computing the Vietoris-Rips complex of the largest scale.

The nth entry in the list will describe the n-dimensional topology of the filtration. That is, the first list will represent clusters, the second list will represent tunnels or punctures, the third will represent hollow volumes, and the nth index list will represent n-dimensional holes in the data. Features are encoded by the filtration indices where they appear and disappear.

Same as above except this function acts on filtrations whose vertices all have filtration index zero (for a very slight speedup).

The nth entry in the list will describe the n-dimensional topology of the filtration. However, features are encoded by the scales where they appear and disappear. For consistency, scales must be in decreasing order.

Same as above except acts only on filtrations whose vertices all have filtration index 0. Note that scales must be in decreasing order.

The standard (Euclidean) metric between index barcodes. The distance between infinite and finite barcodes is infinite, and the distance between two infinite barcodes is the absolute value of the difference of their fst component.

Given a metric, return the Hausdorff distance (referred to as bottleneck distance in TDA) between the two sets. Returns nothing if either list of barcodes is empty.

Get's all the bottleneck distances; a good way to determine the similarity of the topology of two filtrations.

Compute the persistence landscape of the barcodes for a single dimension.

Evaluate the nth function in the landscape for the given point.

Evaluate all the real-valued functions in the landscape.

Compute a linear combination of the landscapes. If the coefficient list is too short, the rest of the coefficients are assumed to be zero. If it is too long, the extra coefficients are discarded.

Average the persistence landscapes.

Subtract the second landscape from the first.

If p>=1 then it will compute the L^p norm on the given interval. Uses trapezoidal approximation. You should ensure that the stepsize partitions the interval evenly.

Given the same information as above, computes the L^p distance between the two landscapes. One way to compare the topologies of two filtrations.