We show that a tessellation generated by a small number of random affine hyperplanes can be used to approximate Euclidean distances between any two points in an arbitrary bounded set , where the random hyperplanes are generated by subgaussian or heavy-tailed normal vectors and uniformly distributed shifts. The number of hyperplanes needed for constructing such tessellations is determined by natural metric complexity measures of the set and the wanted approximation error. In comparison, previous results in this direction were restricted to Gaussian hyperplane tessellations of subsets of the Euclidean unit sphere. As an application, we obtain new reconstruction results in memoryless one-bit compressed sensing with non-Gaussian measurement matrices: by quantizing at uniformly distributed thresholds, it is possible to accurately reconstruct low-complexity signals from a small number of one-bit quantized measurements, even if the measurement vectors are drawn from a heavy-tailed distribution. These reconstruction results are uniform in nature and robust in the presence of pre-quantization noise on the analog measurements as well as adversarial bit corruptions in the quantization process. Moreover, if the measurement matrix is subgaussian then accurate recovery can be achieved via a convex program.
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Sjoerd Dirksen, Shahar Mendelson, Non-Gaussian hyperplane tessellations and robust one-bit compressed sensing. J. Eur. Math. Soc. 23 (2021), no. 9, pp. 2913–2947DOI 10.4171/JEMS/1066