Sobolev Duals in Frame Theory and Sigma-Delta Quantization

Abstract

Abstract. A new class of alternative dual frames is introduced in the setting of finite frames for R d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (Σ∆) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order O(N −r ) for a wide class of finite frames of size N . This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.

Root-Exponential Accuracy for Coarse Quantization of Finite Frame Expansions

“…For example, in [7], Bodmann et al proved that with tight frames of special design, r-th-order schemes achieve an error decay rate of O(λ −r ), when the left-inverse of the matrix E used in linear reconstruction is the Moore-Penrose inverse. 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 this note, we combine the techniques of Blum et al [8] and Güntürk [4]/Deift et al [5] to show that it is possible to achieve root-exponential accuracy in the finite frame setting. In particular, we show that for a family of tight frames of special design that admit themselves as Sobolev duals, and for harmonic frames, root-exponential error rates of O(e −C √ λ ) are achievable.…”

Alternative dual frames for digital-to-analog conversion in sigma–delta quantization

“…In these constructions and the associated approximation error estimates, we do not make any assumptions on the distribution of the quantization error. See [7] for further analysis of the ramifications of Bennett's white noise assumption in the case of Σ∆ schemes, and in particular a construction of optimal alternative dual frames that minimize d MSE (Q, R G ) for Σ∆ quantizers. …”

A note on finite dual frame pairs

“…Due to their computational simplicity and robustness features, one direction of focus has been on the construction of tight frames in R d and C d ; see, e.g., [2,7,8]. However, it is also known that general frame pairs can sometimes provide increased flexibility and performance benefits over tight frames; e.g., see [3]. The purpose of this note is to extend certain key results from the setting of tight frames to dual pairs of frames.…”

“…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].…”