[STAT/AP] Sjoerd Dirksen: Covariance estimation under one-bit quantization

13 mei 2024 15:45 t/m 16:45 | Zet in mijn agenda

A common task in signal processing is to estimate the correlation matrix or the covariance matrix of a high-dimensional Gaussian distribution from i.i.d. samples that have been quantized to finitely many bits. In my talk I will consider a setup where each entry of each sample is quantized to one bit using an efficient, memoryless quantizer. In this setup, a well-known approach in the engineering literature is to use the arcsin law (also known as Grothendieck’s identity) to estimate the correlation matrix. I will present non-asymptotic, near-optimal error bounds for this type of estimator in terms of the spectral and Frobenius norms. Surprisingly, the bounds reveal that this estimator can outperform the sample covariance matrix (of the samples before quantization) in certain scenarios. I will also show that by using dithering, i.e., adding well-designed noise before quantization, one can estimate the full covariance matrix of any (sub)gaussian distribution from quantized samples at the same (minimax optimal) rate as the sample covariance matrix. This second result is based on a new version of the Burkholder-Rosenthal inequalities for matrix martingales.

Based on joint works with Johannes Maly (LMU Munich), Holger Rauhut (LMU Munich), Guiseppe Caire (TU Berlin), Tianyu Yang (TU Berlin)