Transformed Schatten-1 iterative thresholding algorithms for low rank matrix completion

Abstract

We study a non-convex low-rank promoting penalty function, the transformed Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter a ∈ (0,+∞). We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed point representation for its global minimizer. The TS1 thresholding functions are in closed analytical form for all parameter values. The TS1 threshold values differ in subcritical (supercritical) parameter regime where the TS1 threshold functions are continuous (discontinuous). We propose TS1 iterative thresholding algorithms and compare them with some state-ofthe-art algorithms on matrix completion test problems. For problems with known rank, a fully adaptive TS1 iterative thresholding algorithm consistently performs the best under different conditions, where ground truth matrices are generated by multivariate Gaussian, (0,1) uniform and Chi-square distributions. For problems with unknown rank, TS1 algorithms with an additional rank estimation procedure approach the level of IRucL-q which is an iterative reweighted algorithm, non-convex in nature and best in performance.

DAC: Data-Free Automatic Acceleration of Convolutional Networks

“…Another research direction is to combine low rank constraints with weight decomposition. These constraints could be convex regularizations like nuclear norm and Frobenius norm, or non-convex quasi-norms like Schatten p and TS1 [30,29,31].…”

Minimization of transformed $l_1$ penalty: Closed form representation and iterative thresholding algorithms

“…In order to reduce computation time, we shall explore thresholding property for TL1 penalty. In another paper [26], we expand TL1 thresholding and representation theories to low rank matrix completion problems via Schatten-1 quasi-norm.…”

Enhanced Low-Rank Matrix Approximation

“…The use of these non-convex penalty functions makes the overall LRMA problem non-convex. As such, iterative algorithms aiming to reach a stationary point (i.e., not necessarily global optimum) of the non-convex objective function have been developed [21], [55]. Also, a non-iterative locally optimal solution for the LRMA problem using the proximal p-norm is reported in [7].…”

“…We use the normalized root square error (RSE) defined as RSE = X − M F / M F , as a performance measure. We compare the proposed ELMA method to the weighted nuclear norm minimization (WNNM) [21], standard nuclear norm minimization (NNM) [6], p-shrinkage (PS) [7] and the TS1 [55] LRMA methods. For the ELMA, NNM and PS methods, we set λ = βσ, where β is manually set to optimize the RSE for each method.…”