Recovery guarantees for polynomial approximation from dependent data with outliers
This work addresses the problem of robust function approximation from noisy and dependent data for fields like statistics and engineering, offering theoretical guarantees but is incremental in extending existing methods to dependent data settings.
The paper tackles learning nonlinear functions from corrupted and dependent data by recasting it as a sparse robust linear regression problem, providing reconstruction guarantees for an ℓ₁-optimization framework with dependent data, and verifying results through numerical simulations.
Learning non-linear systems from noisy, limited, and/or dependent data is an important task across various scientific fields including statistics, engineering, computer science, mathematics, and many more. In general, this learning task is ill-posed; however, additional information about the data's structure or on the behavior of the unknown function can make the task well-posed. In this work, we study the problem of learning nonlinear functions from corrupted and dependent data. The learning problem is recast as a sparse robust linear regression problem where we incorporate both the unknown coefficients and the corruptions in a basis pursuit framework. The main contribution of our paper is to provide a reconstruction guarantee for the associated $\ell_1$-optimization problem where the sampling matrix is formed from dependent data. Specifically, we prove that the sampling matrix satisfies the null space property and the stable null space property, provided that the data is compact and satisfies a suitable concentration inequality. We show that our recovery results are applicable to various types of dependent data such as exponentially strongly $α$-mixing data, geometrically $\mathcal{C}$-mixing data, and uniformly ergodic Markov chain. Our theoretical results are verified via several numerical simulations.