High-Dimensional Linear Regression and Phase Retrieval via PSLQ Integer Relation Algorithm

We study high-dimensional linear regression problem without sparsity, and address the question of efficient recovery with small number of measurements. We propose an algorithm which efficiently recovers an unknown feature vector β∗ ∈ ℝp from its linear measurements Y = Xβ∗ in polynomially many steps, with high probability (as p → ∞), even with a single measurement, provided elements of β∗ are supported on a rationally independent set of at most polynomial in p size known to learner. We use a combination of PSLQ integer relation and LLL lattice basis reduction algorithms to achieve our goal. We then apply our ideas to develop an efficient, single-sample algorithm for the phase retrieval problem, where ${\beta ^ * } \in {\mathbb{C}^p}$ is to be recovered from magnitude-only observations Y = |〈X, β∗〉|.