vprusso
Feature: Check if set of vectors constitute a tight frame
Provide a function that takes in as input a matrix (as a numpy array) and returns True if the matrix constitutes an equilangular tight frame and False otherwise.
A function like the following could be written:
import numpy as np
def is_etf(mat: np.ndarray) -> bool:
"""Determine if matrix forms an equilangular tight frame (ETF).
Definition taken from the condition of:
http://users.cms.caltech.edu/~jtropp/conf/Tro05-Complex-Equiangular-SPIE-preprint.pdf
"""
# Each column has unit norm.
nrows, ncols = mat.shape[0], mat.shape[1]
for col in range(ncols):
if not np.isclose(np.linalg.norm(mat[:][col]), 1):
return False
# Columns are equilangular.
vals = []
for i in range(ncols):
for j in range(ncols):
if i != j:
vals.append(np.abs(inner_product(mat[:][i], mat[:][j])))
if len(set(vals)) > 1:
return False
# Matrix forms a tight frame.
return np.allclose(mat @ mat.conj().T, (ncols / nrows) * np.identity(nrows * ncols))
One would also need to provide proper documentation, examples, and unit tests for this function.
Additional Option
Provide a function that takes in as input a matrix (as a numpy array) and returns True if the set of vectors constitute as a tight frame and False otherwise.
A function like the following could be written:
import numpy as np
def is_tight_frame(vectors: list[np.ndarray]) -> bool:
"""Check if list of vectors constitutes a tight frame."""
n, d = len(vectors), vectors[0].shape[0]
col_sum = 0
for i in range(n):
col_sum += np.linalg.norm(vectors[i])
return np.isclose(sum(overlaps(vectors)), 1/np.sqrt(d) * col_sum**2)
One would also need to provide proper documentation, examples, and unit tests for this function.
Note that this requires the overlaps function to be defined from:
https://github.com/vprusso/toqito/issues/556
PR #999 solves this issue. Please take a look at your convenience @vprusso @purva-thakre