OptShrink: An Algorithm for Improved Low-Rank Signal Matrix Denoising by Optimal, Data-Driven Singular Value Shrinkage

Abstract

Abstract. The truncated singular value decomposition (SVD) of the measurement matrix is the optimal solution to the representation problem of how to best approximate a noisy measurement matrix using a low-rank matrix. Here, we consider the (unobservable) denoising problem of how to best approximate a low-rank signal matrix buried in noise by optimal (re)weighting of the singular vectors of the measurement matrix. We exploit recent results from random matrix theory to exactly characterize the large matrix limit of the optimal weighting coefficients and show that they can be computed directly from data for a large class of noise models that includes the i.i.d. Gaussian noise case.Our analysis brings into sharp focus the shrinkage-and-thresholding form of the optimal weights, the non-convex nature of the associated shrinkage function (on the singular values) and explains why matrix regularization via singular value thresholding with convex penalty functions (such as the nuclear norm) will always be suboptimal. We validate our theoretical predictions with numerical simulations, develop an implementable algorithm (OptShrink) that realizes the predicted performance gains and show how our methods can be used to improve estimation in the setting where the measured matrix has missing entries.

Integrating cell biology, image analysis, and computational mechanical modeling to analyze the contributions of cellulose and xyloglucan to stomatal function

“…22 Noise due to point spread functions can be reduced via vessel enhancing diffusion and low rank representation. 23,24 Post-processing on vesselness maps can reveal MT positions (Fig. 1b).…”

New rank detection methods for reduced-rank MIMO systems

“…When the channel matrix has reduced rank, the number of its entries is larger than its real dimension, and thus designs based on full-rank channels become inefficient. This motivates the research on reduced-rank technologies for MIMO systems [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

“…For reduced-rank channel estimation, these schemes will cause performance degradation. A more natural and efficient approach is to use SVD-based channel estimation methods [15][16][17][18][19][20][21][22][23]. It was shown in [16,17] that the maximum-likelihood (ML) estimation of the reduced-rank MIMO channel with Gaussian noise is the truncated SVD method.…”

“…In truncated SVD, if the channel rank is known to be r, the MIMO channel matrix is estimated from the SVD of the received signal-plus-noise matrix, by keeping the largest r singular values and their corresponding singular vectors. While the truncated SVD method traces back to the 1930s [28], it was rediscovered in [16,17] for MIMO channel estimation and further improved to reduce the mean squared error (MSE) of the estimation [18][19][20][21][22]. One improved scheme is shrinkage-and-threshold SVD, where the truncated singular values are further shrunk to remove the noise effect.…”

“…Various rank detection methods are available in the literature [17][18][19][20][21][22][23]. In [17], a minimum description length (MDL)-based rank detection was used for MIMO channels.…”