sparse.coherence.babel¶
- sparse.coherence.babel(mat)[source]¶
For an arbitrary input matrix \(\boldsymbol{A}\) of size N x M and normalized columns \(\{ a_i \mid i=1,2,...,M \}\), the Babel-Function is defined by
(1)¶\[\mu_1(k) = \max_{\mid \Lambda \mid = k} \left[ \max_{j \notin \Lambda} \sum_{i \in \Lambda}{\mid a_i^\top a_j \mid} \right]\]If \(\mu_1(k-1) < 1\), this implies that any set of \(k\) columns from \(\boldsymbol{A}\) are linearly dependent. In this case, the Spark necessarily satisfies
(2)¶\[\text{Spark}(\boldsymbol{A}) > k = 1 + \arg \min_k \left({\mu_1(k) > 1}\right)\]- Parameters
- mat(N, M) np.ndarray
The input matrix \(\boldsymbol{A}\).
- Returns
- CoherenceSpark
- A namedtuple with two attributes:
.coherence - a list of M-1 elements of \(\mu_1(k), \ k=1,2,...,M-1\);
.spark - Spark lower bound (2) of mat.
Notes