William Leeb

William Leeb, Vladimir Rokhlin

This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method reformulates the equation as a collection of second-kind integral equations defined on local subdomains. Each such equation can be stably discretized and solved. The boundary values of these local solutions are matched by solving a banded linear system. The method of deferred corrections is then used to increase the accuracy of the scheme. Deferred corrections requires applying the integral operator to a function on the entire domain, for which we provide an algorithm with linear cost. We illustrate the performance of our method on several numerical examples.

William Leeb

Accurate detection of signal components is a frequently-encountered challenge in statistical applications with low signal-to-noise ratio. This problem is particularly challenging in settings with heteroscedastic noise. In certain signal-plus-noise models of data, such as the classical spiked covariance model and its variants, there are closed formulas for the spectral signal detection threshold (the largest sample eigenvalue attributable solely to noise) for isotropic noise in the limit of infinitely large data matrices. However, more general noise models currently lack provably fast and accurate methods for numerically evaluating the threshold. In this work, we introduce a rapid algorithm for evaluating the spectral signal detection threshold in the limit of infinitely large data matrices. We consider noise matrices with a separable variance profile (whose variance matrix is rank one), as these arise often in applications. The solution is based on nested applications of Newton's method. We also devise a new algorithm for evaluating the Stieltjes transform of the spectral distribution at real values exceeding the threshold. The Stieltjes transform on this domain is known to be a key quantity in parameter estimation for spectral denoising methods. The correctness of both algorithms is proven from a detailed analysis of the master equations characterizing the Stieltjes transform, and their performance is demonstrated in numerical experiments.

William Leeb

We analyze the Laplacian pyramids algorithm of Rabin and Coifman for extending and denoising a function sampled on a discrete set of points. We provide mild conditions under which the algorithm converges, and prove stability bounds on the extended function. We also consider the iterative application of truncated Laplacian pyramids kernels for denoising signals by non-local means.

William Leeb

We consider the problem of estimating a low-rank matrix from a noisy observed matrix. Previous work has shown that the optimal method depends crucially on the choice of loss function. In this paper, we use a family of weighted loss functions, which arise naturally for problems such as submatrix denoising, denoising with heteroscedastic noise, and denoising with missing data. However, weighted loss functions are challenging to analyze because they are not orthogonally-invariant. We derive optimal spectral denoisers for these weighted loss functions. By combining different weights, we then use these optimal denoisers to construct a new denoiser that exploits heterogeneity in the signal matrix to boost estimation with unweighted loss.