David L. Donoho

Abstract: An unknown $m$ by $n$ matrix $X_0$ is to be estimated from noisy measurements $Y=X_0+Z$, where the noise matrix $Z$ has i.i.d. Gaussian entries. A popular matrix denoising scheme solves the nuclear norm penalization problem $\operatorname {min}_X\|Y-X\|_F^2/2+\lambda\|X\|_*$, where $\|X\|_*$ denotes the nuclear norm (sum of singular values). This is the analog, for matrices, of $\ell_1$ penalization in the vector case. It has been empirically observed that if $X_0$ has low rank, it may be recovered quite accurately from the noisy measurement $Y$. In a proportional growth framework where the rank $r_n$, number of rows $m_n$ and number of columns $n$ all tend to $\infty$ proportionally to each other ($r_n/m_n\rightarrow \rho$, $m_n/n\rightarrow \beta$), we evaluate the asymptotic minimax MSE $\mathcal {M}(\rho,\beta)=\lim_{m_n,n\rightarrow \infty}\inf_{\lambda}\sup_{\operatorname {rank}(X)\leq r_n}\operatorname {MSE}(X_0,\hat{X}_{\lambda})$. Our formulas involve incomplete moments of the quarter- and semi-circle laws ($\beta=1$, square case) and the Marcenko-Pastur law ($\beta<1$, nonsquare case). For finite $m$ and $n$, we show that MSE increases as the nonzero singular values of $X_0$ grow larger. As a result, the finite-$n$ worst-case MSE, a quantity which can be evaluated numerically, is achieved when the signal $X_0$ is "infinitely strong." The nuclear norm penalization problem is solved by applying soft thresholding to the singular values of $Y$. We also derive the minimax threshold, namely the value $\lambda^*(\rho)$, which is the optimal place to threshold the singular values. All these results are obtained for general (nonsquare, nonsymmetric) real matrices. Comparable results are obtained for square symmetric nonnegative-definite matrices.

TL;DR: It is observed that convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian random matrices, and this finding is established for each of the four associated phase transitions.

TL;DR: In this paper, the authors present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the 1 norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets.

TL;DR: In this article, the authors presented at the International Conference on Wavelets and Applications, Toulouse, France, June, 1992, based on presentation at the NSF DMS 92-09130.

TL;DR: The Rare/Weak (RW) model is a theoretical framework simultaneously controlling the size and prevalence of useful/significant items among the useless/null bulk, and shows that HC has important advantages over better known procedures such as False Discovery Rate (FDR) control and Family-wise Error control (FwER), in particular, certain optimality properties.