Uncertainty Principles and Vector Quantization

Abstract

Abstract-Given a frame in C n which satisfies a form of the uncertainty principle (as introduced by Candes and Tao), it is shown how to quickly convert the frame representation of every vector into a more robust Kashin's representation whose coefficients all have the smallest possible dynamic range O(1/ √ n). The information tends to spread evenly among these coefficients. As a consequence, Kashin's representations have a great power for reduction of errors in their coefficients, including coefficient losses and distortions.

Coefficient quantization for frames in Banach spaces

“…The above presented argument from [21] shows that if there is a quotient Q : N ∞ → X , and a frame (…”

“…2 Remark 5.13. In Section 6 we will recall a result of Lyubarskii and Vershinin [21] which shows that for q > 1 there are ε < 1, δ < 1, C < ∞ so that for any n ∈ N and there is a Hilbert frame (…”

“…Lyubarskii and Vershinin observed in [21] that the column vectors (u i ) N i=1 form a tight frame (with A = B = 1), that the first inclusion in (36) yields that every x ∈ B n can be written as…”

Spread representations

“…These representation where all the coefficients are of the same order of magnitude or more precisely where the range of the coefficients is small, are of high interest in coding and compression. They are known to withstand errors in their coefficients in a strong way [2]. One can show that the representation error due to quantification, transmission errors or losses, gets bounded by the average, rather than the sum, of the errors in the coefficients.…”

“…Representations in which the range of the coefficients is small, have already been considered and are known as Kashin's representation [2]. Minimizing the ∞ -norm of the solution pushes the same idea even further since one explicitly minimizes the range of the coefficients while in the Kashin's this is only done in a loose way.…”