Dictionary Utilities

This module provides utility functions for dictionary learning, such as computing coherence and normalization checks.

dictionary_utils.py - Utility functions for matrix operations used in dictionary learning.

This module provides various utility functions for working with matrices, such as: 1. Computing the independent columns of a matrix. 2. Checking if the columns of a matrix are normalized. 3. Computing the coherence of a matrix. 4. Checking various properties of a matrix, such as full rank, normalization, and coherence.

These utilities are useful for tasks in dictionary learning, signal processing, and sparse coding.

Example usage:

>>> from dictionary_utils import compute_independent_columns, check_normalization, compute_coherence, check_matrix_properties

>>> A = np.random.randn(5, 5)
>>> independent_columns = compute_independent_columns(A)
>>> is_normalized = check_normalization(A)
>>> coherence = compute_coherence(A)
>>> check_matrix_properties(A)

CompSensePack.dictionaries.dictionary_utils.check_matrix_properties(A)

Checks various properties of a matrix.

This function checks if the matrix A is full rank, if its columns and rows are normalized, and computes the coherence of the matrix. It also prints the results for each property.

Parameters

Anumpy.ndarray

The matrix to check.

Returns

None

Example

>>> A = np.array([[1, 2], [3, 4]])
>>> check_matrix_properties(A)
Is full rank: True
Are columns normalized: False
Are rows normalized: False
Coherence: 0.9999999999999999

CompSensePack.dictionaries.dictionary_utils.check_normalization(A)

Checks if the columns of a matrix are normalized (i.e., each column has a unit norm).

This function calculates the norm of each column in the matrix A and checks if all column norms are close to 1.0, which indicates normalization.

Parameters

Anumpy.ndarray

The matrix to check for normalization.

Returns

is_normalizedbool

True if all columns of A are normalized, False otherwise.

Example

>>> A = np.array([[1, 0], [0, 1]])
>>> check_normalization(A)
True

CompSensePack.dictionaries.dictionary_utils.compute_coherence(matrix)

Computes the coherence of the given matrix.

Coherence is a measure of the maximum correlation between any two columns of a matrix. It is useful in various applications, such as signal processing and compressed sensing, to assess the degree of similarity between different columns of the matrix.

Parameters

matrixnumpy.ndarray

An (N x M) matrix where coherence is to be calculated.

Returns

coherencefloat

The coherence of the matrix, defined as the maximum absolute value of the off-diagonal elements in the Gram matrix of the column-normalized input matrix.

Example

>>> matrix = np.array([[1, 0], [0, 1]])
>>> compute_coherence(matrix)
0.0

CompSensePack.dictionaries.dictionary_utils.compute_independent_columns(A, tol=1e-10)

Computes the independent columns of a matrix using the QR decomposition.

This function identifies the independent columns of a given matrix A by performing a QR decomposition. It selects columns corresponding to non-zero diagonal elements of the R matrix, which are considered linearly independent.

Parameters

Anumpy.ndarray

The matrix for which to compute the independent columns.

tolfloat, optional (default=1e-10)

The tolerance value for considering diagonal elements of R as non-zero.

Returns

ind_colsnumpy.ndarray

A matrix containing the independent columns of A.

Notes

• The QR decomposition is used to determine the rank of the matrix A.

• Columns corresponding to non-zero diagonal elements of the R matrix are considered independent.

Example

>>> A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> compute_independent_columns(A)
array([[1, 2],
       [4, 5],
       [7, 8]])