Correcting for unknown errors in sparse high-dimensional function approximation

TL;DR

This paper investigates sparsity-based methods for high-dimensional function approximation with unknown errors, providing theoretical guarantees and numerical comparisons, highlighting the advantages of square-root LASSO in bounded noise scenarios.

Contribution

It offers uniform recovery guarantees for four sparsity-promoting decoders and demonstrates the practical benefits of square-root LASSO without parameter tuning.

Findings

Square-root LASSO performs better with bounded noise.

Theoretical recovery guarantees are established for all four methods.

Numerical experiments compare methods in function approximation and uncertainty quantification.

Abstract

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.