155,959 research outputs found
Frame Permutation Quantization
Frame permutation quantization (FPQ) is a new vector quantization technique
using finite frames. In FPQ, a vector is encoded using a permutation source
code to quantize its frame expansion. This means that the encoding is a partial
ordering of the frame expansion coefficients. Compared to ordinary permutation
source coding, FPQ produces a greater number of possible quantization rates and
a higher maximum rate. Various representations for the partitions induced by
FPQ are presented, and reconstruction algorithms based on linear programming,
quadratic programming, and recursive orthogonal projection are derived.
Implementations of the linear and quadratic programming algorithms for uniform
and Gaussian sources show performance improvements over entropy-constrained
scalar quantization for certain combinations of vector dimension and coding
rate. Monte Carlo evaluation of the recursive algorithm shows that mean-squared
error (MSE) decays as 1/M^4 for an M-element frame, which is consistent with
previous results on optimal decay of MSE. Reconstruction using the canonical
dual frame is also studied, and several results relate properties of the
analysis frame to whether linear reconstruction techniques provide consistent
reconstructions.Comment: 29 pages, 5 figures; detailed added to proof of Theorem 4.3 and a few
minor correction
Decomposition Methods for Large Scale LP Decoding
When binary linear error-correcting codes are used over symmetric channels, a
relaxed version of the maximum likelihood decoding problem can be stated as a
linear program (LP). This LP decoder can be used to decode error-correcting
codes at bit-error-rates comparable to state-of-the-art belief propagation (BP)
decoders, but with significantly stronger theoretical guarantees. However, LP
decoding when implemented with standard LP solvers does not easily scale to the
block lengths of modern error correcting codes. In this paper we draw on
decomposition methods from optimization theory, specifically the Alternating
Directions Method of Multipliers (ADMM), to develop efficient distributed
algorithms for LP decoding.
The key enabling technical result is a "two-slice" characterization of the
geometry of the parity polytope, which is the convex hull of all codewords of a
single parity check code. This new characterization simplifies the
representation of points in the polytope. Using this simplification, we develop
an efficient algorithm for Euclidean norm projection onto the parity polytope.
This projection is required by ADMM and allows us to use LP decoding, with all
its theoretical guarantees, to decode large-scale error correcting codes
efficiently.
We present numerical results for LDPC codes of lengths more than 1000. The
waterfall region of LP decoding is seen to initiate at a slightly higher
signal-to-noise ratio than for sum-product BP, however an error floor is not
observed for LP decoding, which is not the case for BP. Our implementation of
LP decoding using ADMM executes as fast as our baseline sum-product BP decoder,
is fully parallelizable, and can be seen to implement a type of message-passing
with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the
49th Annual Allerton Conference, September 2011. This version to appear in
IEEE Transactions on Information Theor
A linear approach for sparse coding by a two-layer neural network
Many approaches to transform classification problems from non-linear to
linear by feature transformation have been recently presented in the
literature. These notably include sparse coding methods and deep neural
networks. However, many of these approaches require the repeated application of
a learning process upon the presentation of unseen data input vectors, or else
involve the use of large numbers of parameters and hyper-parameters, which must
be chosen through cross-validation, thus increasing running time dramatically.
In this paper, we propose and experimentally investigate a new approach for the
purpose of overcoming limitations of both kinds. The proposed approach makes
use of a linear auto-associative network (called SCNN) with just one hidden
layer. The combination of this architecture with a specific error function to
be minimized enables one to learn a linear encoder computing a sparse code
which turns out to be as similar as possible to the sparse coding that one
obtains by re-training the neural network. Importantly, the linearity of SCNN
and the choice of the error function allow one to achieve reduced running time
in the learning phase. The proposed architecture is evaluated on the basis of
two standard machine learning tasks. Its performances are compared with those
of recently proposed non-linear auto-associative neural networks. The overall
results suggest that linear encoders can be profitably used to obtain sparse
data representations in the context of machine learning problems, provided that
an appropriate error function is used during the learning phase
Density Evolution for Asymmetric Memoryless Channels
Density evolution is one of the most powerful analytical tools for
low-density parity-check (LDPC) codes and graph codes with message passing
decoding algorithms. With channel symmetry as one of its fundamental
assumptions, density evolution (DE) has been widely and successfully applied to
different channels, including binary erasure channels, binary symmetric
channels, binary additive white Gaussian noise channels, etc. This paper
generalizes density evolution for non-symmetric memoryless channels, which in
turn broadens the applications to general memoryless channels, e.g. z-channels,
composite white Gaussian noise channels, etc. The central theorem underpinning
this generalization is the convergence to perfect projection for any fixed size
supporting tree. A new iterative formula of the same complexity is then
presented and the necessary theorems for the performance concentration theorems
are developed. Several properties of the new density evolution method are
explored, including stability results for general asymmetric memoryless
channels. Simulations, code optimizations, and possible new applications
suggested by this new density evolution method are also provided. This result
is also used to prove the typicality of linear LDPC codes among the coset code
ensemble when the minimum check node degree is sufficiently large. It is shown
that the convergence to perfect projection is essential to the belief
propagation algorithm even when only symmetric channels are considered. Hence
the proof of the convergence to perfect projection serves also as a completion
of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor
Towards practical minimum-entropy universal decoding
Minimum-entropy decoding is a universal decoding algorithm used in decoding block compression of discrete memoryless sources as well as block transmission of information across discrete memoryless channels. Extensions can also be applied for multiterminal decoding problems, such as the Slepian-Wolf source coding problem. The 'method of types' has been used to show that there exist linear codes for which minimum-entropy decoders achieve the same error exponent as maximum-likelihood decoders. Since minimum-entropy decoding is NP-hard in general, minimum-entropy decoders have existed primarily in the theory literature. We introduce practical approximation algorithms for minimum-entropy decoding. Our approach, which relies on ideas from linear programming, exploits two key observations. First, the 'method of types' shows that that the number of distinct types grows polynomially in n. Second, recent results in the optimization literature have illustrated polytope projection algorithms with complexity that is a function of the number of vertices of the projected polytope. Combining these two ideas, we leverage recent results on linear programming relaxations for error correcting codes to construct polynomial complexity algorithms for this setting. In the binary case, we explicitly demonstrate linear code constructions that admit provably good performance
MacWilliams Identities for Terminated Convolutional Codes
Shearer and McEliece [1977] showed that there is no MacWilliams identity for
the free distance spectra of orthogonal linear convolutional codes. We show
that on the other hand there does exist a MacWilliams identity between the
generating functions of the weight distributions per unit time of a linear
convolutional code C and its orthogonal code C^\perp, and that this
distribution is as useful as the free distance spectrum for estimating code
performance. These observations are similar to those made recently by
Bocharova, Hug, Johannesson and Kudryashov; however, we focus on terminating by
tail-biting rather than by truncation.Comment: 5 pages; accepted for 2010 IEEE International Symposium on
Information Theory, Austin, TX, June 13-1
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