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Register a new userFrame Moments and Welch Bound with Erasures
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Marina Haikin
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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The Welch Bound is a lower bound on the root mean square cross correlation between $n$ unit-norm vectors $f_1,...,f_n$ in the $m$ dimensional space ($mathbb{R} ^m$ or $mathbb{C} ^m$), for $ngeq m$. Letting $F = [f_1|...|f_n]$ denote the $m$-by-$n$ frame matrix, the Welch bound can be viewed as a lower bound on the second moment of $F$, namely on the trace of the squared Gram matrix $(FF)^2$. We consider an erasure setting, in which a reduced frame, composed of a random subset of Bernoulli selected vectors, is of interest. We extend the Welch bound to this setting and present the {em erasure Welch bound} on the expected value of the Gram matrix of the reduced frame. Interestingly, this bound generalizes to the $d$-th order moment of $F$. We provide simple, explicit formulae for the generalized bound for $d=2,3,4$, which is the sum of the $d$-th moment of Wachters classical MANOVA distribution and a vanishing term (as $n$ goes to infinity with $frac{m}{n}$ held constant). The bound holds with equality if (and for $d = 4$ only if) $F$ is an Equiangular Tight Frame (ETF). Our results offer a novel perspective on the superiority of ETFs over other frames in a variety of applications, including spread spectrum communications, compressed sensing and analog coding.
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Analog coding decouples the tasks of protecting against erasures and noise. For erasure correction, it creates an analog redundancy by means of band-limited discrete Fourier transform (DFT) interpolation, or more generally, by an over-complete expansion based on a frame. We examine the analog coding paradigm for the dual setup of a source with erasure side-information (SI) at the encoder. The excess rate of analog coding above the rate-distortion function (RDF) is associated with the energy of the inverse of submatrices of the frame, where each submatrix corresponds to a possible erasure pattern. We give a partial theoretical as well as numerical evidence that a variety of structured frames, in particular DFT frames with difference-set spectrum and more general equiangular tight frames (ETFs), with a common MANOVA limiting spectrum, minimize the excess rate over all possible frames. However, they do not achieve the RDF even in the limit as the dimension goes to infinity.
This paper considers the transmission of an infinite sequence of messages (a streaming source) over a packet erasure channel, where every source message must be recovered perfectly at the destination subject to a fixed decoding delay. While the capacity of a channel that introduces only bursts of erasures is well known, only recently, the capacity of a channel with either one burst of erasures or multiple arbitrary erasures in any fixed-sized sliding window has been established. However, the codes shown to achieve this capacity are either non-explicit constructions (proven to exist) or explicit constructions that require large field size that scales exponentially with the delay. This work describes an explicit rate-optimal construction for admissible channel and delay parameters over a field size that scales only quadratically with the delay.
Analog coding is a low-complexity method to combat erasures, based on linear redundancy in the signal space domain. Previous work examined band-limited discrete Fourier transform (DFT) codes for Gaussian channels with erasures or impulses. We extend this concept to source coding with erasure side-information at the encoder and show that the performance of band-limited DFT can be significantly improved using irregular spectrum, and more generally, using equiangular tight frames (ETF). Frames are overcomplete bases and are widely used in mathematics, computer science, engineering, and statistics since they provide a stable and robust decomposition. Design of frames with favorable properties of random subframes is motivated in variety of applications, including code-devision multiple access (CDMA), compressed sensing and analog coding. We present a novel relation between deterministic frames and random matrix theory. We show empirically that the MANOVA ensemble offers a universal description of the spectra of randomly selected subframes with constant aspect ratios, taken from deterministic near-ETFs. Moreover, we derive an analytic framework and bring a formal validation for some of the empirical results, specifically that the asymptotic form for the moments of high orders of subsets of ETF agree with that of MANOVA. Finally, when exploring over-complete bases, the Welch bound is a lower bound on the root mean square cross correlation between vectors. We extend the Welch bound to an erasure setting, in which a reduced frame, composed of a random subset of Bernoulli selected vectors, is of interest. The lower bound involves moment of the reduced frame, and it is tight for ETFs and asymptotically coincides with the MANOVA moments. This result offers a novel perspective on the superiority of ETFs over other frames.
This paper is focuses on the computation of the positive moments of one-side correlated random Gram matrices. Closed-form expressions for the moments can be obtained easily, but numerical evaluation thereof is prone to numerical stability, especially in high-dimensional settings. This letter provides a numerically stable method that efficiently computes the positive moments in closed-form. The developed expressions are more accurate and can lead to higher accuracy levels when fed to moment based-approaches. As an application, we show how the obtained moments can be used to approximate the marginal distribution of the eigenvalues of random Gram matrices.
Distributed computation is a framework used to break down a complex computational task into smaller tasks and distributing them among computational nodes. Erasure correction codes have recently been introduced and have become a popular workaround to the well known ``straggling nodes problem, in particular, by matching linear coding for linear computation tasks. It was observed that decoding tends to amplify the computation ``noise, i.e., the numerical errors at the computation nodes. We propose taking advantage of the case that more nodes return than minimally required. We show how a clever construction of a polynomial code, inspired by recent results on robust frames, can significantly reduce the amplification of noise, and achieves graceful degradation with the number of straggler nodes.
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