Robust polynomial regression up to the information theoretic limit

We consider the problem of robust polynomial regression, where one receives samples $(x_i, y_i)$ that are usually within $\sigma$ of a polynomial $y = p(x)$, but have a $\rho$ chance of being arbitrary adversarial outliers. Previously, it was known how to efficiently estimate $p$ only when $\rho < \frac{1}{\log d}$. We give an algorithm that works for the entire feasible range of $\rho < 1/2$, while simultaneously improving other parameters of the problem. We complement our algorithm, which gives a factor 2 approximation, with impossibility results that show, for example, that a $1.09$ approximation is impossible even with infinitely many samples.Comment: 19 Pages. To appear in FOCS 201

Model order selection from noisy polynomial data without using any polynomial coefficients

(c) 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.Given a set of noisy data values from a polynomial, determining the degree and coefficients of the polynomial is a problem of polynomial regressions. Polynomial regressions are very common in engineering, science, and other disciplines, and it is at the heart of data science. Linear regressions and the least squares method have been around for two hundred years. Existing techniques select a model, which includes both the degree and coefficients of a polynomial, from a set of candidate models which have already been fitted to the data. The philosophy behind the proposed method is fundamentally different to what have been practised in the last two hundred years. In the first stage only the degree of a polynomial to represent the noisy data is selected without any knowledge or reference to its coefficient values. Having selected the degree, polynomial coefficients are estimated in the second stage. The development of the first stage has been inspired by the very recent results that all polynomials of degree q give rise to the same set of known time-series coefficients of autoregressive models and a constant term μ. Computer experiments have been carried out with simulated noisy data from polynomials using four well known model selection criteria as well as the proposed method (PTS1). The results obtained from the proposed method for degree selection and predictions are significantly better than those from the existing methods. Also, it is experimentally observed that the root-mean square (RMS) prediction errors and the variation of the RMS prediction errors from the proposed method scale linearly with the standard deviations of the noise for each degree of a polynomial