We consider the problem of encoding a finite set of vectors into a small number of bits while approximately retaining information on the angular distances between the vectors. By deriving improved variance bounds related to binary Gaussian circulant embeddings, we largely fix a gap in the proof of the best known fast binary embedding method. Our bounds also show that well-spreadness assumptions on the data vectors, which were needed in earlier work on variance bounds, are unnecessary. In addition, we propose a new binary embedding with a faster running time on sparse data.2010 Mathematics Subject Classification. 60B20,68Q87.

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This work is concerned with the problem of recovering high-dimensional signals, which belong to a convex set of lowcomplexity, from a small number of quantized measurements. We propose to estimate the signals via a convex program based on rectified linear units (ReLUs) for two different quantization schemes, namely one-bit and uniform multi-bit quantization. Assuming that the linear measurement process can be modelled by a sensing matrix with i.i.d. subgaussian rows, we obtain for both schemes near-optimal uniform reconstruction guarantees by adding well-designed noise to the linear measurements prior to the quantization step. In the one-bit case, we show that the program is robust against adversarial bit corruptions as well as additive noise on the linear measurements. Further, our analysis quantifies precisely how the rate-distortion relationship of the program changes depending on whether we seek reconstruction accuracies above or below the noise floor. The proofs rely on recent results by Dirksen and Mendelson on non-Gaussian hyperplane tessellations.

This work theoretically studies the problem of estimating a structured high-dimensional signal x 0 ∈ R n from noisy 1-bit Gaussian measurements. Our recovery approach is based on a simple convex program which uses the hinge loss function as data fidelity term. While such a risk minimization strategy is very natural to learn binary output models, such as in classification, its capacity to estimate a specific signal vector is largely unexplored. A major difficulty is that the hinge loss is just piecewise linear, so that its "curvature energy" is concentrated in a single point. This is substantially different from other popular loss functions considered in signal estimation, e.g., the square or logistic loss, which are at least locally strongly convex. It is therefore somewhat unexpected that we can still prove very similar types of recovery guarantees for the hinge loss estimator, even in the presence of strong noise. More specifically, our non-asymptotic error bounds show that stable and robust reconstruction of x 0 can be achieved with the optimal oversampling rate O(m −1/2 ) in terms of the number of measurements m. Moreover, we permit a wide class of structural assumptions on the ground truth signal, in the sense that x 0 can belong to an arbitrary bounded convex set K ⊂ R n . The proofs of our main results rely on some recent advances in statistical learning theory due to Mendelson. In particular, we invoke an adapted version of Mendelson's small ball method that allows us to establish a quadratic lower bound on the error of the first order Taylor approximation of the empirical hinge loss function. 1 Hereafter, we make the convention that sign(v) = +1 for v ≥ 0 and sign(v) = −1 for v < 0. arXiv:1804.04846v2 [math.ST]