We study Sigma-Delta (Σ∆) quantization methods coupled with appropriate reconstruction algorithms for digitizing randomly sampled low-rank matrices. We show that the reconstruction error associated with our methods decays polynomially with the oversampling factor, and we leverage our results to obtain root-exponential accuracy by optimizing over the choice of quantization scheme. Additionally, we show that a random encoding scheme, applied to the quantized measurements, yields a near-optimal exponential bit-rate. As an added benefit, our schemes are robust both to noise and to deviations from the low-rank assumption. In short, we provide a full generalization of analogous results, obtained in the classical setup of bandlimited function acquisition, and more recently, in the finite frame and compressed sensing setups to the case of low-rank matrices sampled with sub-Gaussian linear operators. Finally, we believe our techniques for generalizing results from the compressed sensing setup to the analogous low-rank matrix setup is applicable to other quantization schemes.
Index TermsCompressed sensing, quantization, exponential accuracy, rate-distortion, low-rank, one-bit σ k (Z) * := min rank(V )=k Z − V * denotes the error, measured in the nuclear norm, associated with the best rank k approximation of a matrix. Eric Lybrand and Rayan Saab are with the Mathematics Department at the University of California, San Diego. 2
A. Background and prior workLow-rank matrix recovery has seen a wide range of applications, ranging from quantum state tomography [17] and collaborative filtering [31] to sensor localization [32] and face recognition [6], to name a few.Nuclear norm minimization was proposed by Fazel in [14] as a means of finding a matrix of minimial rank in a given convex set. Fazel motivates this through the observation that nuclear norm minimization is the convex relaxation of the rank minimization problem, which has been shown to be NP-hard. Since then, and with the advent of compressed sensing, there has been much work on recovering low-rank matrices from linear measurements. For example, [33] considers recovering low-rank matrices given random linear measurements, and establishes recovery guarantees given that the sampling scheme satisfies the matrix restricted isometry property, which we define in the subsequent section. Perhaps not surprisingly, this analysis closely follows that of sparse vector recovery under ℓ 1 minimization, as it is known that random ensembles of linear maps satisfy the matrix restricted isometry property with high probability [16], [33]. This led to a flurry of papers on nuclear norm minimization for matrix recovery in various contexts, see [6], [29], [9] for example.While the theoretical results on nuclear norm minimization have been promising, convex optimization practically necessitates the use of digital computers for recovering the underlying matrix. It behooves the theory, therefore, to take into account that the measurements must be converted to bits so that numerical solvers can ha...