5,057 research outputs found
Locally Testable Codes and Cayley Graphs
We give two new characterizations of (\F_2-linear) locally testable
error-correcting codes in terms of Cayley graphs over \F_2^h:
\begin{enumerate} \item A locally testable code is equivalent to a Cayley
graph over \F_2^h whose set of generators is significantly larger than
and has no short linear dependencies, but yields a shortest-path metric that
embeds into with constant distortion. This extends and gives a
converse to a result of Khot and Naor (2006), which showed that codes with
large dual distance imply Cayley graphs that have no low-distortion embeddings
into .
\item A locally testable code is equivalent to a Cayley graph over \F_2^h
that has significantly more than eigenvalues near 1, which have no short
linear dependencies among them and which "explain" all of the large
eigenvalues. This extends and gives a converse to a recent construction of
Barak et al. (2012), which showed that locally testable codes imply Cayley
graphs that are small-set expanders but have many large eigenvalues.
\end{enumerate}Comment: 22 page
A study of the classification of low-dimensional data with supervised manifold learning
Supervised manifold learning methods learn data representations by preserving
the geometric structure of data while enhancing the separation between data
samples from different classes. In this work, we propose a theoretical study of
supervised manifold learning for classification. We consider nonlinear
dimensionality reduction algorithms that yield linearly separable embeddings of
training data and present generalization bounds for this type of algorithms. A
necessary condition for satisfactory generalization performance is that the
embedding allow the construction of a sufficiently regular interpolation
function in relation with the separation margin of the embedding. We show that
for supervised embeddings satisfying this condition, the classification error
decays at an exponential rate with the number of training samples. Finally, we
examine the separability of supervised nonlinear embeddings that aim to
preserve the low-dimensional geometric structure of data based on graph
representations. The proposed analysis is supported by experiments on several
real data sets
Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many
statistics and machine learning applications ranging from support vector
machines to Gaussian processes and kernel embeddings of distributions.
Operators acting on such spaces are, for instance, required to embed
conditional probability distributions in order to implement the kernel Bayes
rule and build sequential data models. It was recently shown that transfer
operators such as the Perron-Frobenius or Koopman operator can also be
approximated in a similar fashion using covariance and cross-covariance
operators and that eigenfunctions of these operators can be obtained by solving
associated matrix eigenvalue problems. The goal of this paper is to provide a
solid functional analytic foundation for the eigenvalue decomposition of RKHS
operators and to extend the approach to the singular value decomposition. The
results are illustrated with simple guiding examples
Extremal -invariant eigenvalues of the Laplacian of -invariant metrics
The study of extremal properties of the spectrum often involves restricting
the metrics under consideration. Motivated by the work of Abreu and Freitas in
the case of the sphere endowed with -invariant metrics, we consider
the subsequence of the spectrum of a Riemannian manifold
which corresponds to metrics and functions invariant under the action of a
compact Lie group . If has dimension at least 1, we show that the
functional admits no extremal metric under volume-preserving
-invariant deformations. If, moreover, has dimension at least three,
then the functional is unbounded when restricted to any conformal
class of -invariant metrics of fixed volume. As a special case of this, we
can consider the standard O(n)-action on ; however, if we also require the
metric to be induced by an embedding of in , we get an
optimal upper bound on .Comment: To appear in Mathematische Zeitschrif
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