We focus on the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector $beta^*inmathbb{R}^p$ 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 $Qinmathbb{Z}^+$ (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 $beta^*inmathbb{R}^p$ enjoying the mixed-support assumption, from its linear measurements $Y=Xbeta^*inmathbb{R}^n$ 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 $beta^*inmathbb{R}^p$ enjoying $Q$-rationality, from its noisy measurements $Y=Xbeta^*+Winmathbb{R}^n$, 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.
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