High Dimensional Linear Regression Using Lattice Basis Reduction

Authors:
Ilias Zadik MIT
David Gamarnik Massachusetts Institute of Technology

Introduction:

The authors consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector eta^* from n noisy linear observations Y=X eta^+W in R^n, for known X in R^{n imes p} and unknown W in R^n.The authors propose a new polynomial-time algorithm for this task which is based on the seminal Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction algorithm.

Abstract:

We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector \beta^* from n noisy linear observations Y=X \beta^+W in R^n, for known X in R^{n \times p} and unknown W in R^n. Unlike most of the literature on this model we make no sparsity assumption on \beta^. Instead we adopt a regularization based on assuming that the underlying vectors \beta^* have rational entries with the same denominator Q. We call this Q-rationality assumption. We propose a new polynomial-time algorithm for this task which is based on the seminal Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction algorithm. We establish that under the Q-rationality assumption, our algorithm recovers exactly the vector \beta^* for a large class of distributions for the iid entries of X and non-zero noise W. We prove that it is successful under small noise, even when the learner has access to only one observation (n=1). Furthermore, we prove that in the case of the Gaussian white noise for W, n=o(p/\log p) and Q sufficiently large, our algorithm tolerates a nearly optimal information-theoretic level of the noise.

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