computeComplexPolynomial: Compute Complex Polynomial Coefficients from a Persistence...

In TDAvec: Vector Summaries of Persistence Diagrams

Compute Complex Polynomial Coefficients from a Persistence Diagram

Description

For a given persistence diagram D=\{(b_i,d_i)\}_{i=1}^N (corresponding to a specified homological dimension), computeComplexPolynomial() computes the coefficients of a complex polynomial

C_X(z) = \prod_{i=1}^N [z - X(b_i,d_i)],

where X:\mathbb{R}^2\rightarrow\mathbb{C} is any one of the following three functions:

• R(x,y)=x+iy.

• S(x, y) = \begin{cases} \frac{y - x}{\alpha \sqrt{2}}(x+iy) & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{otherwise.} \end{cases}

• T(x, y) = \frac{y - x}{2} [(\cos\alpha - \sin\alpha)+i(\cos\alpha + \sin\alpha)],

where \alpha=\sqrt{x^2+y^2}. Points in D with infinite death values are ignored.

Usage

computeComplexPolynomial(D, homDim, m = 1, polyType = "R")

Arguments

D	a persistence diagram: a matrix with three columns containing the homological dimension, birth and death values respectively.
homDim	the homological dimension (0 for H_0, 1 for H_1, etc.). Rows in D are filtered based on this value.
m	the number of coefficients to return (default is 1). Must be less than or equal to the number of points in the diagram.
polyType	a string specifying the polynomial type to use. Options are "R" (default), "S", or "T".

Details

The function extracts rows from D where the first column equals homDim, and computes values based on the filtered data and polyType. If D does not contain any points corresponding to homDim, a matrix of zeros is returned.

Value

A numeric matrix of dimension m \times 2. Each row corresponds to a coefficient of the polynomial C_X(z):

• Column 1: Real part of the coefficient.

• Column 2: Imaginary part of the coefficient.

Author(s)

Umar Islambekov

References

1. Ferri, M. and Landi, C., (1999). Representing size functions by complex polynomials. Proc. Math. Met. in Pattern Recognition, 9, pp.16-19.

2. Di Fabio, B. and Ferri, M., (2015). Comparing persistence diagrams through complex vectors. In Image Analysis and Processing—ICIAP 2015: 18th International Conference, Genoa, Italy, September 7-11, 2015, Proceedings, Part I 18 (pp. 294-305). Springer International Publishing.

3. Ali, D., Asaad, A., Jimenez, M.J., Nanda, V., Paluzo-Hidalgo, E. and Soriano-Trigueros, M., (2023). A survey of vectorization methods in topological data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence.

Examples

N <- 100 # The number of points to sample

set.seed(123) # Set a random seed for reproducibility

# Sample N points uniformly from the unit circle and add Gaussian noise
theta <- runif(N, min = 0, max = 2 * pi)
X <- cbind(cos(theta), sin(theta)) + rnorm(2 * N, mean = 0, sd = 0.2)

# Compute the persistence diagram using the Rips filtration built on top of X
# The 'threshold' parameter specifies the maximum distance for building simplices
D <- TDAstats::calculate_homology(X, threshold = 2)

# Compute first 5 coefficients of the complex polynomial of type "R" for dimension H_0
computeComplexPolynomial(D, homDim = 0, m = 5, polyType = "R")

# Compute the first 5 coefficients of the complex polynomial of type "S"
#  for homological dimension H_0
computeComplexPolynomial(D, homDim = 0, m = 5, polyType = "S")

TDAvec documentation built on April 4, 2025, 1:37 a.m.