# Phase Retrieval from Random One-Bit Measurements

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Phase retrieval in real or complex Hilbert spaces is the task of recovering a vector, up to an overall unimodular multiplicative constant, from norms of projections onto subspaces. This dissertation deals with phase retrieval of normalized vectors after the norms of projections are quantized by pairwise comparison to retain only one bit of information. In more specific, geometric terms, we choose a sequence of pairs of subspaces in a real or complex Hilbert space and only record which subspace from each pair is closer to the input vector. The main goal of this paper is to find a feasible algorithm for approximate recovery based on the qualitative information gained about the vector from these binary questions, and to establish error bounds for the approximate recovery procedure. The recovery algorithm we define uses the qualitative proximity information encoded in the binary measurement of an input vector to assemble an auxiliary matrix, and then chooses a unit vector in the principal eigenspace of this auxiliary matrix as the estimate for the input vector. For this measurement and recovery procedure, we provide a pointwise bound for fixed input vectors and a uniform bound that controls the worst-case scenario among all inputs. Both bounds hold with high probability with respect to a choice of subspaces from the uniform distribution induced by the action of the orthogonal or unitary group. For real or complex vectors of dimension