Angle-preserving Quantized Phase Embeddings

We demonstrate that the phase of randomized complex-valued projections of real-valued signals preserves information about the angle, i.e., the correlation, between signals. This information can be exploited to design quantized angle-preserving embeddings, which represent such correlations using a finite bit-rate. These embeddings generalize known results on binary embeddings and 1-bit compressive sensing and allow us to explore the trade-off between number of measurements and number of bits per measurement, given the bit rate. The freedom provided by this trade-off, which has also been observed in quantized Johnson-Lindenstrauss embeddings, can improve performance at reduced rate in a number of applications.