590 research outputs found
On Recursive Edit Distance Kernels with Application to Time Series Classification
This paper proposes some extensions to the work on kernels dedicated to
string or time series global alignment based on the aggregation of scores
obtained by local alignments. The extensions we propose allow to construct,
from classical recursive definition of elastic distances, recursive edit
distance (or time-warp) kernels that are positive definite if some sufficient
conditions are satisfied. The sufficient conditions we end-up with are original
and weaker than those proposed in earlier works, although a recursive
regularizing term is required to get the proof of the positive definiteness as
a direct consequence of the Haussler's convolution theorem. The classification
experiment we conducted on three classical time warp distances (two of which
being metrics), using Support Vector Machine classifier, leads to conclude
that, when the pairwise distance matrix obtained from the training data is
\textit{far} from definiteness, the positive definite recursive elastic kernels
outperform in general the distance substituting kernels for the classical
elastic distances we have tested.Comment: 14 page
Regularized Regression Problem in hyper-RKHS for Learning Kernels
This paper generalizes the two-stage kernel learning framework, illustrates
its utility for kernel learning and out-of-sample extensions, and proves
{asymptotic} convergence results for the introduced kernel learning model.
Algorithmically, we extend target alignment by hyper-kernels in the two-stage
kernel learning framework. The associated kernel learning task is formulated as
a regression problem in a hyper-reproducing kernel Hilbert space (hyper-RKHS),
i.e., learning on the space of kernels itself. To solve this problem, we
present two regression models with bivariate forms in this space, including
kernel ridge regression (KRR) and support vector regression (SVR) in the
hyper-RKHS. By doing so, it provides significant model flexibility for kernel
learning with outstanding performance in real-world applications. Specifically,
our kernel learning framework is general, that is, the learned underlying
kernel can be positive definite or indefinite, which adapts to various
requirements in kernel learning. Theoretically, we study the convergence
behavior of these learning algorithms in the hyper-RKHS and derive the learning
rates. Different from the traditional approximation analysis in RKHS, our
analyses need to consider the non-trivial independence of pairwise samples and
the characterisation of hyper-RKHS. To the best of our knowledge, this is the
first work in learning theory to study the approximation performance of
regularized regression problem in hyper-RKHS.Comment: 25 pages, 3 figure
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