Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices

Super-resolution is a fundamental task in imaging, where the goal is to extract fine-grained structure from coarse-grained measurements. Here we are interested in a popular mathematical abstraction of this problem that has been widely studied in the statistics, signal processing and machine learning communities. We exactly resolve the threshold at which noisy super-resolution is possible. In particular, we establish a sharp phase transition for the relationship between the cutoff frequency ($m$) and the separation ($\Delta$). If $m > 1/\Delta + 1$, our estimator converges to the true values at an inverse polynomial rate in terms of the magnitude of the noise. And when $m < (1-\epsilon) /\Delta$ no estimator can distinguish between a particular pair of $\Delta$-separated signals even if the magnitude of the noise is exponentially small. Our results involve making novel connections between {\em extremal functions} and the spectral properties of Vandermonde matrices. We establish a sharp phase transition for their condition number which in turn allows us to give the first noise tolerance bounds for the matrix pencil method. Moreover we show that our methods can be interpreted as giving preconditioners for Vandermonde matrices, and we use this observation to design faster algorithms for super-resolution. We believe that these ideas may have other applications in designing faster algorithms for other basic tasks in signal processing.Comment: 19 page

A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs

A self-learning algebraic multigrid method for dominant and minimal singular triplets and eigenpairs is described. The method consists of two multilevel phases. In the first, multiplicative phase (setup phase), tentative singular triplets are calculated along with a multigrid hierarchy of interpolation operators that approximately fit the tentative singular vectors in a collective and self-learning manner, using multiplicative update formulas. In the second, additive phase (solve phase), the tentative singular triplets are improved up to the desired accuracy by using an additive correction scheme with fixed interpolation operators, combined with a Ritz update. A suitable generalization of the singular value decomposition is formulated that applies to the coarse levels of the multilevel cycles. The proposed algorithm combines and extends two existing multigrid approaches for symmetric positive definite eigenvalue problems to the case of dominant and minimal singular triplets. Numerical tests on model problems from different areas show that the algorithm converges to high accuracy in a modest number of iterations, and is flexible enough to deal with a variety of problems due to its self-learning properties.Comment: 29 page

Stability of the replica symmetric solution in diluted perceptron learning

We study the role played by the dilution in the average behavior of a perceptron model with continuous coupling with the replica method. We analyze the stability of the replica symmetric solution as a function of the dilution field for the generalization and memorization problems. Thanks to a Gardner like stability analysis we show that at any fixed ratio $\alpha$ between the number of patterns M and the dimension N of the perceptron ($\alpha=M/N$), there exists a critical dilution field $h_c$ above which the replica symmetric ansatz becomes unstable.Comment: Stability of the solution in arXiv:0907.3241, 13 pages, (some typos corrected