Root-Exponential Accuracy for Coarse Quantization of Finite Frame Expansions

Abstract

In this note, we show that by quantizing the N -dimensional frame coefficients of signals in R d using r-thorder Sigma-Delta quantization schemes, it is possible to achieve root-exponential accuracy in the oversampling rateIn particular, we construct a family of finite frames tailored specifically for coarse Sigma-Delta quantization that admit themselves as both canonical duals and Sobolev duals. Our construction allows for error guarantees that behave as e −c √ λ , where under a mild restriction on the oversampling rate, the constants are absolute. Moreover, weshow that harmonic frames can be used to achieve the same guarantees, but with the constants now depending on d.

Near-Optimal Encoding for Sigma-Delta Quantization of Finite Frame Expansions

“…Finally, it is important to note that stable r th -order Σ∆ schemes with C ρ,Q (r) = O(r r ) do indeed exist (see, e.g., [14,9]), even when A is a 1-bit alphabet. In particular, we cite the following proposition (see [21]). Proposition 1.…”

“…While the above results progressively improved on the coding efficiency of Σ∆ quantization, it remains true that even the root-exponential performance e −c √ N d of [21,22] is generally sub-optimal from an information theoretic perspective, including in the case where X = B d . To be more precise, any quantization scheme tasked with encoding all possible points in B d to within -accuracy must produce outputs which, in the case of an optimal encoder, each correspond to a unique subset of the unit ball having radius at most .…”

“…Motivated by applications in compressed sensing, a similar result [16] was shown for random frames whose elements are Gaussian random variables. This was followed by the results of [21] and [22] showing that there exist (deterministic and random) frames for which higher order Σ∆ schemes yield approximation errors that behave like e , where c is a constant. Specifically, in [21,22] this root-exponential accuracy is achieved by carefully choosing the order of the scheme as a function of the oversampling rate N/d.…”

“…This was followed by the results of [21] and [22] showing that there exist (deterministic and random) frames for which higher order Σ∆ schemes yield approximation errors that behave like e , where c is a constant. Specifically, in [21,22] this root-exponential accuracy is achieved by carefully choosing the order of the scheme as a function of the oversampling rate N/d. For a more comprehensive review of Σ∆ schemes applied to finite frames, see [26].…”

“…In Section 3 we show that random subsampling of the discretely integrated Σ∆ bit-stream allows a linear decoder to achieve exponentially decaying reconstruction error, uniformly for all x ∈ B d . This result pertains to a particular choice of frames, the Sobolev self-dual frames [21], and is contingent on using 1st order Σ∆ schemes. In Section 4 we instead use a Bernoulli matrix for reducing the dimensionality of the integrated Σ∆-bit stream.…”