Learning Explicitly Conditioned Sparsifying Transforms
Andrei Pătraşcu, Cristian Rusu, Paul Irofti
TL;DR
The paper tackles the instability and suboptimal representation of learned sparsifying transforms by introducing explicit conditioning control, enforcing $\kappa(W)\le \rho$ and $\|W\|_F=\tau$. It develops an Alternating Minimization framework over the SVD factors $W=U\Sigma V^T$ and sparse codes $X$, with a novel, efficient spectrum-projection step that reduces to a 1D convex problem, plus Procrustes-based and thresholding updates. Empirical results on synthetic and real image datasets demonstrate improved representation error and competitive denoising performance compared to prior Bresler-type methods, showcasing the practical benefits of explicit conditioning. The work offers a numerically stable, parameter-light approach with potential for scaling to larger datasets and online settings in sparse-transforms-based processing.
Abstract
Sparsifying transforms became in the last decades widely known tools for finding structured sparse representations of signals in certain transform domains. Despite the popularity of classical transforms such as DCT and Wavelet, learning optimal transforms that guarantee good representations of data into the sparse domain has been recently analyzed in a series of papers. Typically, the conditioning number and representation ability are complementary key features of learning square transforms that may not be explicitly controlled in a given optimization model. Unlike the existing approaches from the literature, in our paper, we consider a new sparsifying transform model that enforces explicit control over the data representation quality and the condition number of the learned transforms. We confirm through numerical experiments that our model presents better numerical behavior than the state-of-the-art.