30 research outputs found
Modern Coding Theory: The Statistical Mechanics and Computer Science Point of View
These are the notes for a set of lectures delivered by the two authors at the
Les Houches Summer School on `Complex Systems' in July 2006. They provide an
introduction to the basic concepts in modern (probabilistic) coding theory,
highlighting connections with statistical mechanics. We also stress common
concepts with other disciplines dealing with similar problems that can be
generically referred to as `large graphical models'.
While most of the lectures are devoted to the classical channel coding
problem over simple memoryless channels, we present a discussion of more
complex channel models. We conclude with an overview of the main open
challenges in the field.Comment: Lectures at Les Houches Summer School on `Complex Systems', July
2006, 44 pages, 25 ps figure
Polar Codes: Robustness of the Successive Cancellation Decoder with Respect to Quantization
Polar codes provably achieve the capacity of a wide array of channels under
successive decoding. This assumes infinite precision arithmetic. Given the
successive nature of the decoding algorithm, one might worry about the
sensitivity of the performance to the precision of the computation.
We show that even very coarsely quantized decoding algorithms lead to
excellent performance. More concretely, we show that under successive decoding
with an alphabet of cardinality only three, the decoder still has a threshold
and this threshold is a sizable fraction of capacity. More generally, we show
that if we are willing to transmit at a rate below capacity, then we
need only bits of precision, where is a universal
constant.Comment: In ISIT 201
Why We Can Not Surpass Capacity: The Matching Condition
We show that iterative coding systems can not surpass capacity using only
quantities which naturally appear in density evolution. Although the result in
itself is trivial, the method which we apply shows that in order to achieve
capacity the various components in an iterative coding system have to be
perfectly matched. This generalizes the perfect matching condition which was
previously known for the case of transmission over the binary erasure channel
to the general class of binary-input memoryless output-symmetric channels.
Potential applications of this perfect matching condition are the construction
of capacity-achieving degree distributions and the determination of the number
required iterations as a function of the multiplicative gap to capacity.Comment: 10 pages, 27 ps figures. Forty-third Allerton Conference on
Communication, Control and Computing, invited pape
Linear Programming Decoding of Spatially Coupled Codes
For a given family of spatially coupled codes, we prove that the LP threshold
on the BSC of the graph cover ensemble is the same as the LP threshold on the
BSC of the derived spatially coupled ensemble. This result is in contrast with
the fact that the BP threshold of the derived spatially coupled ensemble is
believed to be larger than the BP threshold of the graph cover ensemble as
noted by the work of Kudekar et al. (2011, 2012). To prove this, we establish
some properties related to the dual witness for LP decoding which was
introduced by Feldman et al. (2007) and simplified by Daskalakis et al. (2008).
More precisely, we prove that the existence of a dual witness which was
previously known to be sufficient for LP decoding success is also necessary and
is equivalent to the existence of certain acyclic hyperflows. We also derive a
sublinear (in the block length) upper bound on the weight of any edge in such
hyperflows, both for regular LPDC codes and for spatially coupled codes and we
prove that the bound is asymptotically tight for regular LDPC codes. Moreover,
we show how to trade crossover probability for "LP excess" on all the variable
nodes, for any binary linear code.Comment: 37 pages; Added tightness construction, expanded abstrac
The Space of Solutions of Coupled XORSAT Formulae
The XOR-satisfiability (XORSAT) problem deals with a system of Boolean
variables and clauses. Each clause is a linear Boolean equation (XOR) of a
subset of the variables. A -clause is a clause involving distinct
variables. In the random -XORSAT problem a formula is created by choosing
-clauses uniformly at random from the set of all possible clauses on
variables. The set of solutions of a random formula exhibits various
geometrical transitions as the ratio varies.
We consider a {\em coupled} -XORSAT ensemble, consisting of a chain of
random XORSAT models that are spatially coupled across a finite window along
the chain direction. We observe that the threshold saturation phenomenon takes
place for this ensemble and we characterize various properties of the space of
solutions of such coupled formulae.Comment: Submitted to ISIT 201
Bayes Complexity of Learners vs Overfitting
We introduce a new notion of complexity of functions and we show that it has
the following properties: (i) it governs a PAC Bayes-like generalization bound,
(ii) for neural networks it relates to natural notions of complexity of
functions (such as the variation), and (iii) it explains the generalization gap
between neural networks and linear schemes. While there is a large set of
papers which describes bounds that have each such property in isolation, and
even some that have two, as far as we know, this is a first notion that
satisfies all three of them. Moreover, in contrast to previous works, our
notion naturally generalizes to neural networks with several layers.
Even though the computation of our complexity is nontrivial in general, an
upper-bound is often easy to derive, even for higher number of layers and
functions with structure, such as period functions. An upper-bound we derive
allows to show a separation in the number of samples needed for good
generalization between 2 and 4-layer neural networks for periodic functions
The Generalized Area Theorem and Some of its Consequences
There is a fundamental relationship between belief propagation and maximum a
posteriori decoding. The case of transmission over the binary erasure channel
was investigated in detail in a companion paper. This paper investigates the
extension to general memoryless channels (paying special attention to the
binary case). An area theorem for transmission over general memoryless channels
is introduced and some of its many consequences are discussed. We show that
this area theorem gives rise to an upper-bound on the maximum a posteriori
threshold for sparse graph codes. In situations where this bound is tight, the
extrinsic soft bit estimates delivered by the belief propagation decoder
coincide with the correct a posteriori probabilities above the maximum a
posteriori threshold. More generally, it is conjectured that the fundamental
relationship between the maximum a posteriori and the belief propagation
decoder which was observed for transmission over the binary erasure channel
carries over to the general case. We finally demonstrate that in order for the
design rate of an ensemble to approach the capacity under belief propagation
decoding the component codes have to be perfectly matched, a statement which is
well known for the special case of transmission over the binary erasure
channel.Comment: 27 pages, 46 ps figure
Lossy Source Coding via Spatially Coupled LDGM Ensembles
We study a new encoding scheme for lossy source compression based on
spatially coupled low-density generator-matrix codes. We develop a
belief-propagation guided-decimation algorithm, and show that this algorithm
allows to approach the optimal distortion of spatially coupled ensembles.
Moreover, using the survey propagation formalism, we also observe that the
optimal distortions of the spatially coupled and individual code ensembles are
the same. Since regular low-density generator-matrix codes are known to achieve
the Shannon rate-distortion bound under optimal encoding as the degrees grow,
our results suggest that spatial coupling can be used to reach the
rate-distortion bound, under a {\it low complexity} belief-propagation
guided-decimation algorithm.
This problem is analogous to the MAX-XORSAT problem in computer science.Comment: Submitted to ISIT 201