Functional classification in Hilbert spaces

Let X be a random variable taking values in a separable Hilbert space X, with label Y is an element of {0, 1}. We establish universal weak consistency of a nearest neighbor-type classifier based on n independent copies (X-i, Y-i) of the pair (X, Y), extending the classical result of Stone to infinite-dimensional Hilbert spaces. Under a mild condition on the distribution of X, we also prove strong consistency. We reduce the infinite dimension of X by considering only the first d coefficients of a Fourier series expansion of each X-i, and then we perform k-nearest neighbor classification in R-d. Both the dimension and the number of neighbors are automatically selected from the data using a simple data-splitting device. An application of this technique to a signal discrimination problem involving speech recordings is presented.