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Phase retrieval in real or complex Hilbert spaces is the task of recovering avector, up to an overall unimodular multiplicative constant, from magnitudes oflinear measurements. In this paper, we assume that the vector is normalized,but retain only qualitative, binary information about the measured magnitudesby comparing them with a threshold. In more specific, geometric terms, wechoose a sequence of subspaces in a real or complex Hilbert space and onlyrecord whether a given vector is closer to the subspace than to thecomplementary subspace. The subspaces have half the dimension of the Hilbertspace and are independent, uniformly distributed with respect to the action ofthe orthogonal or unitary groups. The main goal of this paper is to find afeasible algorithm for approximate recovery based on the information gainedabout the vector from these binary questions and to establish error bounds forits approximate recovery. We provide a pointwise bound for fixed input vectorsand a uniform bound that controls the worst-case scenario among all inputs.Both bounds hold with high probability with respect to the choice of thesubspaces. For real or complex vectors of dimension $n$, the pointwise boundrequires $m \ge C \delta^{-2} n \log(n)$ and the uniform bound $m \ge C\delta^{-2} n^2 \log(\delta^{-1} n)$ binary questions in order to achieve anaccuracy of $\delta$. The accuracy $\delta$ is measured by the operator norm ofthe difference between the rank-one orthogonal projections corresponding to thenormalized input vector and its approximate recovery.

Phase retrieval by random binary questions: Which complementary subspace is closer?