Combinatorial Separable Convolutional Dictionaries

Recent works have considered the use of a linear combination of separable filters to approximate a non-separable filter bank (FB) to obtain computational advantages in CNNs and convolutional sparse representations/coding (CSR/CSC). However, it has been recently shown that there are advantages to directly solving the convolutional dictionary learning (CDL) problem considering a separable FB. A separable filter bank of M 2-d filters is typically constructed from a paired set of M horizontal filters and M vertical filters. In contrast, here we propose an outer product construction involving all possible combinations of vertical and horizontal filters, so that M vertical and M horizontal filters generate M
²
2-d filters. Our computational experiments show that this alternative form results in a reduction in computation time of 10% and 80% for the CDL and CSC problems respectively, while matching the reconstruction performance of the typical separable FB approach for the same cardinality.