Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements

Abstract

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y = Φx of a k-sparse signal x ∈ R N , where Φ satisfies the restricted isometry property, then the approximate recoveryThe simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth order Σ∆ quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduction of the approximation error by a factor of (m/k) (r−1/2)α for any 0 < α < 1, if m r k(log N ) 1/(1−α) . The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x that satisfy a mild size condition on their supports.

Root-Exponential Accuracy for Coarse Quantization of Finite Frame Expansions

“…Consequently, there is an upper bound on admissible values of r for these schemes to work and this does not allow one to optimize the value of r freely as a function of λ. A similar issue arises in [9], where the frames are random. On the other hand, in the bandlimited setting, [4] and [5] proposed Σ∆ schemes that do not suffer from an r-dependent constraint on the input sequence.…”

“…Using a different approach, Blum et al [8] showed that such an error rate can be achieved by using alternative left-inverses, called Sobolev duals, for any frame that arises via uniform sampling from piecewise smooth frame-paths. Recently, Güntürk et al [9] showed that for randomly-generated frames, error bounds of O(λ −(r−1/2)α ), for α ∈ (0, 1), are attainable via the use of Sobolev duals. In particular, the parameter α controls the probability (on the draw of the frame) with which the result holds.…”

“…In particular, the parameter α controls the probability (on the draw of the frame) with which the result holds. This allowed [9] to apply Σ∆ quantization in the context of compressed sensing [10], [11].…”

One‐Bit Compressed Sensing by Linear Programming

“…However, the number of bits per measurement there is not one (or constant); this number depends on the sparsity level s and the dynamic range of the signal x. Similarly, in the work of Gunturk et al [14,15] on sigma-delta quantization, the number of bits per measurement depends on the dynamic range of x. On the other hand, by considering sigmadelta quantization and multiple bits, the Gunturk et al are able to provide excellent guarantees on the speed of decay of the error δ as s/m decreases.…”

A note on finite dual frame pairs

“…However, there are settings where noncanonical (alternative) dual frames perform better than the canonical dual. For example, in Sigma-Delta quantization of finite frame expansions, the best approximation-theoretic properties are obtained using noncanonical dual frames [12,3,13]. Other examples related to the uncertainty principle show that noncanonical dual frames can provide improved time-frequency localization over canonical dual frames; see [11].…”