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High Dimensional Linear Regression using Lattice Basis Reduction
Ilias Zadik · David Gamarnik

Wed Dec 05 07:45 AM -- 09:45 AM (PST) @ Room 210 #94

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.

Author Information

Ilias Zadik (MIT)

I am a CDS Moore-Sloan (postdoctoral) fellow at the Center for Data Science of NYU and a member of it's Math and Data (MaD) group. I received my PhD on September 2019 from MIT , where I was advised by David Gamarnik. My research lies broadly in the interface of high dimensional statistics, the theory of machine learning and applied probability.

David Gamarnik (Massachusetts Institute of Technology)

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