Neural Information Processing Systems http://nips.cc/

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 focus on the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector $\beta^*\in\mathbb{R}^p$ from its linear measurements, using a small number $n$ of samples. Unlike most of the literature, we make no sparsity assumption on $\beta^*$, but instead adopt a different regularization: In the noiseless setting, we assume $\beta^*$ consists of entries, which are either rational numbers with a common denominator $Q\in\mathbb{Z}^+$ (referred to as $Q$-rationality); or irrational numbers supported on a rationally independent set of bounded cardinality, known to learner; collectively called as the mixed-support assumption. Using a novel combination of the PSLQ integer relation detection, and LLL lattice basis reduction algorithms, we propose a polynomial-time algorithm which provably recovers a $\beta^*\in\mathbb{R}^p$ enjoying the mixed-support assumption, from its linear measurements $Y=X\beta^*\in\mathbb{R}^n$ for a large class of distributions for the random entries of $X$, even with one measurement $(n=1)$. In the noisy setting, we propose a polynomial-time, lattice-based algorithm, which recovers a $\beta^*\in\mathbb{R}^p$ enjoying $Q$-rationality, from its noisy measurements $Y=X\beta^*+W\in\mathbb{R}^n$, even with a single sample $(n=1)$. We further establish for large $Q$, and normal noise, this algorithm tolerates information-theoretically optimal level of noise. We then apply these ideas to develop a polynomial-time, single-sample algorithm for the phase retrieval problem. Our methods address the single-sample $(n=1)$ regime, where the sparsity-based methods such as LASSO and Basis Pursuit are known to fail. Furthermore, our results also reveal an algorithmic connection between the high-dimensional linear regression problem, and the integer relation detection, randomized subset-sum, and shortest vector problems.

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.

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.

We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector $\beta^*$ from $n$ noisy linear observations $Y=X\beta^*+W \in \mathbb{R}^n$, for known $X \in \mathbb{R}^{n \times p}$ and unknown $W \in \mathbb{R}^n$. 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 $Q \in \mathbb{Z}_{>0}$. 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 $W$, $n=o\left(p/\log p\right)$ and $Q$ sufficiently large, our algorithm tolerates a nearly optimal information-theoretic level of the noise.