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In-Depth Information
Chapter 1
Optimization in Reproducing Kernel Hilbert
Spaces of Spike Trains
Ant onio R. C. Paiva, Il Park, and Jos´eC.Prıncipe
Abstract
This chapter presents a framework based on reproducing kernel Hilbert
spaces (RKHS) for optimization with spike trains. To establish the RKHS for opti-
mization we start by introducing kernels for spike trains. It is shown that spike train
kernels can be built from ideas of kernel methods or from the intensity functions
underlying the spike trains. However, the later approach shall be the main focus of
this study. We introduce the memoryless cross-intensity (mCI) kernel as an example
of an inner product of spike trains, which defines the RKHS bottom-up as an in-
ner product of intensity functions. Being defined in terms of the intensity functions,
this approach toward defining spike train kernels has the advantage that points in
the RKHS incorporate a statistical description of the spike trains, and the statisti-
cal model is explicitly stated. Some properties of the mCI kernel and the RKHS it
induces will be given to show that this RKHS has the necessary structure for opti-
mization. The issue of estimation from data is also addressed. We finalize with an
example of optimization in the RKHS by deriving an algorithm for principal com-
ponent analysis (PCA) of spike trains.
