arXiv subjects

Simone Brugiapaglia

Publications and source records attributed to Simone Brugiapaglia.

42 records · Page 3Linked to original sources

arXiv2019

Correcting for unknown errors in sparse high-dimensional function approximation

We consider sparsity-based techniques for the approximation of high-dimensional functions from random pointwise evaluations. To date, almost all the works published in this field contain some a priori assumptions about the error corrupting the samples that are hard to verify in practice. In this paper, we instead focus on the scenario where the error is unknown. We study the performance of four sparsity-promoting optimization problems: weighted quadratically-constrained basis pursuit, weighted LASSO, weighted square-root LASSO, and weighted LAD-LASSO. From the theoretical perspective, we prove uniform recovery guarantees for these decoders, deriving recipes for the optimal choice of the respective tuning parameters. On the numerical side, we compare them in the pure function approximation case and in applications to uncertainty quantification of ODEs and PDEs with random inputs. Our main conclusion is that the lesser-known square-root LASSO is better suited for high-dimensional approximation than the other procedures in the case of bounded noise, since it avoids (both theoretically and numerically) the need for parameter tuning.

math.NA↗

arXiv2019

Sparse approximation of multivariate functions from small datasets via weighted orthogonal matching pursuit

We show the potential of greedy recovery strategies for the sparse approximation of multivariate functions from a small dataset of pointwise evaluations by considering an extension of the orthogonal matching pursuit to the setting of weighted sparsity. The proposed recovery strategy is based on a formal derivation of the greedy index selection rule. Numerical experiments show that the proposed weighted orthogonal matching pursuit algorithm is able to reach accuracy levels similar to those of weighted $\ell^1$ minimization programs while considerably improving the computational efficiency for small values of the sparsity level.

math.NA↗

arXiv2018

A compressive spectral collocation method for the diffusion equation under the restricted isometry property

We propose a compressive spectral collocation method for the numerical approximation of Partial Differential Equations (PDEs). The approach is based on a spectral Sturm-Liouville approximation of the solution and on the collocation of the PDE in strong form at randomized points, by taking advantage of the compressive sensing principle. The proposed approach makes use of a number of collocation points substantially less than the number of basis functions when the solution to recover is sparse or compressible. Focusing on the case of the diffusion equation, we prove that, under suitable assumptions on the diffusion coefficient, the matrix associated with the compressive spectral collocation approach satisfies the restricted isometry property of compressive sensing with high probability. Moreover, we demonstrate the ability of the proposed method to reduce the computational cost associated with the corresponding full spectral collocation approach while preserving good accuracy through numerical illustrations.

math.NA↗

arXiv2017

Robustness to unknown error in sparse regularization

Quadratically-constrained basis pursuit has become a popular device in sparse regularization; in particular, in the context of compressed sensing. However, the majority of theoretical error estimates for this regularizer assume an a priori bound on the noise level, which is usually lacking in practice. In this paper, we develop stability and robustness estimates which remove this assumption. First, we introduce an abstract framework and show that robust instance optimality of any decoder in the noise-aware setting implies stability and robustness in the noise-blind setting. This is based on certain sup-inf constants referred to as quotients, strictly related to the quotient property of compressed sensing. We then apply this theory to prove the robustness of quadratically-constrained basis pursuit under unknown error in the cases of random Gaussian matrices and of random matrices with heavy-tailed rows, such as random sampling matrices from bounded orthonormal systems. We illustrate our results in several cases of practical importance, including subsampled Fourier measurements and recovery of sparse polynomial expansions.

cs.IT↗

arXiv2017

Compressed sensing approaches for polynomial approximation of high-dimensional functions

In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the computation of polynomial approximations of high-dimensional functions on $d$-dimensional hypercubes. We show that smooth, multivariate functions possess expansions in orthogonal polynomial bases that are not only approximately sparse, but possess a particular type of structured sparsity defined by so-called lower sets. This structure can be exploited via the use of weighted $\ell^1$ minimization techniques, and, as we demonstrate, doing so leads to sample complexity estimates that are at most logarithmically dependent on the dimension $d$. Hence the curse of dimensionality - the bane of high-dimensional approximation - is mitigated to a significant extent. We also discuss several practical issues, including unknown noise (due to truncation or numerical error), and highlight a number of open problems and challenges.

math.NA↗

arXiv2017

Recovery guarantees for compressed sensing with unknown errors

From a numerical analysis perspective, assessing the robustness of l1-minimization is a fundamental issue in compressed sensing and sparse regularization. Yet, the recovery guarantees available in the literature usually depend on a priori estimates of the noise, which can be very hard to obtain in practice, especially when the noise term also includes unknown discrepancies between the finite model and data. In this work, we study the performance of l1-minimization when these estimates are not available, providing robust recovery guarantees for quadratically constrained basis pursuit and random sampling in bounded orthonormal systems. Several applications of this work are approximation of high-dimensional functions, infinite-dimensional sparse regularization for inverse problems, and fast algorithms for non-Cartesian Magnetic Resonance Imaging.

math.NA↗