11,224 research outputs found
On the VC-Dimension of Binary Codes
We investigate the asymptotic rates of length- binary codes with
VC-dimension at most and minimum distance at least . Two upper
bounds are obtained, one as a simple corollary of a result by Haussler and the
other via a shortening approach combining Sauer-Shelah lemma and the linear
programming bound. Two lower bounds are given using Gilbert-Varshamov type
arguments over constant-weight and Markov-type sets
Multiclass Learning Approaches: A Theoretical Comparison with Implications
We theoretically analyze and compare the following five popular multiclass
classification methods: One vs. All, All Pairs, Tree-based classifiers, Error
Correcting Output Codes (ECOC) with randomly generated code matrices, and
Multiclass SVM. In the first four methods, the classification is based on a
reduction to binary classification. We consider the case where the binary
classifier comes from a class of VC dimension , and in particular from the
class of halfspaces over . We analyze both the estimation error and
the approximation error of these methods. Our analysis reveals interesting
conclusions of practical relevance, regarding the success of the different
approaches under various conditions. Our proof technique employs tools from VC
theory to analyze the \emph{approximation error} of hypothesis classes. This is
in sharp contrast to most, if not all, previous uses of VC theory, which only
deal with estimation error
Kneser-Hecke-operators in coding theory
The Kneser-Hecke-operator is a linear operator defined on the complex vector
space spanned by the equivalence classes of a family of self-dual codes of
fixed length. It maps a linear self-dual code over a finite field to the
formal sum of the equivalence classes of those self-dual codes that intersect
in a codimension 1 subspace. The eigenspaces of this self-adjoint linear
operator may be described in terms of a coding-theory analogue of the Siegel
-operator
Multiplicatively Repeated Non-Binary LDPC Codes
We propose non-binary LDPC codes concatenated with multiplicative repetition
codes. By multiplicatively repeating the (2,3)-regular non-binary LDPC mother
code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9,
1/12,... Surprisingly, such simple low-rate non-binary LDPC codes outperform
the best low-rate binary LDPC codes so far. Moreover, we propose the decoding
algorithm for the proposed codes, which can be decoded with almost the same
computational complexity as that of the mother code.Comment: To appear in IEEE Transactions on Information Theor
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