A Singular Value Relaxation Technique for Learning Sparsifying Transforms

Subhadip Mukherjee, Chandra Sekhar Seelamantula

Abstract: We address the problem of learning data-adaptive square sparsifying transforms subject to a condition number constraint and adopt an alternating minimization (alt. min.) strategy to solve it. We propose a quadratic program based approach in every iteration of alt. min. to update the singular values of the transform so that the condition number constraint is satisfied. The set of updated singular values, as it turns out after applying the Karush-Kuhn-Tucker conditions for optimality, can be expressed as an affine transformation applied to the current set of singular values. We refer to the resulting technique as singular value relaxation (SVR). The SVR-based transform learning algorithm is employed in signal sparsification and denoising applications. Performance evaluations of SVR show that it is about three times faster than K-SVD for denoising images of size 512×512 and results in a PSNR gain of about 0.5 to 1 dB over K-SVD for synthesized signals, and about 0.2 to 0.3 dB for natural images. The PSNR gains of SVR are shown to be comparable with a recently proposed transform learning algorithm that employs a closed-form transform-update rule.