13,625 research outputs found
A vector quantization approach to universal noiseless coding and quantization
A two-stage code is a block code in which each block of data is coded in two stages: the first stage codes the identity of a block code among a collection of codes, and the second stage codes the data using the identified code. The collection of codes may be noiseless codes, fixed-rate quantizers, or variable-rate quantizers. We take a vector quantization approach to two-stage coding, in which the first stage code can be regarded as a vector quantizer that “quantizes” the input data of length n to one of a fixed collection of block codes. We apply the generalized Lloyd algorithm to the first-stage quantizer, using induced measures of rate and distortion, to design locally optimal two-stage codes. On a source of medical images, two-stage variable-rate vector quantizers designed in this way outperform standard (one-stage) fixed-rate vector quantizers by over 9 dB. The tail of the operational distortion-rate function of the first-stage quantizer determines the optimal rate of convergence of the redundancy of a universal sequence of two-stage codes. We show that there exist two-stage universal noiseless codes, fixed-rate quantizers, and variable-rate quantizers whose per-letter rate and distortion redundancies converge to zero as (k/2)n -1 log n, when the universe of sources has finite dimension k. This extends the achievability part of Rissanen's theorem from universal noiseless codes to universal quantizers. Further, we show that the redundancies converge as O(n-1) when the universe of sources is countable, and as O(n-1+ϵ) when the universe of sources is infinite-dimensional, under appropriate conditions
About Adaptive Coding on Countable Alphabets: Max-Stable Envelope Classes
In this paper, we study the problem of lossless universal source coding for
stationary memoryless sources on countably infinite alphabets. This task is
generally not achievable without restricting the class of sources over which
universality is desired. Building on our prior work, we propose natural
families of sources characterized by a common dominating envelope. We
particularly emphasize the notion of adaptivity, which is the ability to
perform as well as an oracle knowing the envelope, without actually knowing it.
This is closely related to the notion of hierarchical universal source coding,
but with the important difference that families of envelope classes are not
discretely indexed and not necessarily nested.
Our contribution is to extend the classes of envelopes over which adaptive
universal source coding is possible, namely by including max-stable
(heavy-tailed) envelopes which are excellent models in many applications, such
as natural language modeling. We derive a minimax lower bound on the redundancy
of any code on such envelope classes, including an oracle that knows the
envelope. We then propose a constructive code that does not use knowledge of
the envelope. The code is computationally efficient and is structured to use an
{E}xpanding {T}hreshold for {A}uto-{C}ensoring, and we therefore dub it the
\textsc{ETAC}-code. We prove that the \textsc{ETAC}-code achieves the lower
bound on the minimax redundancy within a factor logarithmic in the sequence
length, and can be therefore qualified as a near-adaptive code over families of
heavy-tailed envelopes. For finite and light-tailed envelopes the penalty is
even less, and the same code follows closely previous results that explicitly
made the light-tailed assumption. Our technical results are founded on methods
from regular variation theory and concentration of measure
Recovery from Linear Measurements with Complexity-Matching Universal Signal Estimation
We study the compressed sensing (CS) signal estimation problem where an input
signal is measured via a linear matrix multiplication under additive noise.
While this setup usually assumes sparsity or compressibility in the input
signal during recovery, the signal structure that can be leveraged is often not
known a priori. In this paper, we consider universal CS recovery, where the
statistics of a stationary ergodic signal source are estimated simultaneously
with the signal itself. Inspired by Kolmogorov complexity and minimum
description length, we focus on a maximum a posteriori (MAP) estimation
framework that leverages universal priors to match the complexity of the
source. Our framework can also be applied to general linear inverse problems
where more measurements than in CS might be needed. We provide theoretical
results that support the algorithmic feasibility of universal MAP estimation
using a Markov chain Monte Carlo implementation, which is computationally
challenging. We incorporate some techniques to accelerate the algorithm while
providing comparable and in many cases better reconstruction quality than
existing algorithms. Experimental results show the promise of universality in
CS, particularly for low-complexity sources that do not exhibit standard
sparsity or compressibility.Comment: 29 pages, 8 figure
Universal Lossless Compression with Unknown Alphabets - The Average Case
Universal compression of patterns of sequences generated by independently
identically distributed (i.i.d.) sources with unknown, possibly large,
alphabets is investigated. A pattern is a sequence of indices that contains all
consecutive indices in increasing order of first occurrence. If the alphabet of
a source that generated a sequence is unknown, the inevitable cost of coding
the unknown alphabet symbols can be exploited to create the pattern of the
sequence. This pattern can in turn be compressed by itself. It is shown that if
the alphabet size is essentially small, then the average minimax and
maximin redundancies as well as the redundancy of every code for almost every
source, when compressing a pattern, consist of at least 0.5 log(n/k^3) bits per
each unknown probability parameter, and if all alphabet letters are likely to
occur, there exist codes whose redundancy is at most 0.5 log(n/k^2) bits per
each unknown probability parameter, where n is the length of the data
sequences. Otherwise, if the alphabet is large, these redundancies are
essentially at least O(n^{-2/3}) bits per symbol, and there exist codes that
achieve redundancy of essentially O(n^{-1/2}) bits per symbol. Two sub-optimal
low-complexity sequential algorithms for compression of patterns are presented
and their description lengths analyzed, also pointing out that the pattern
average universal description length can decrease below the underlying i.i.d.\
entropy for large enough alphabets.Comment: Revised for IEEE Transactions on Information Theor
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