Title: A Hilbert Space of Stationary Ergodic Processes,
Abstract: Identifying meaningful signal buried in noise is a problem of interest
arising in diverse scenarios of data-driven modeling. We present here a
theoretical framework for exploiting intrinsic geometry in data that resists
noise corruption, and might be identifiable under severe obfuscation. Our
approach is based on uncovering a valid complete inner product on the space of
ergodic stationary finite valued processes, providing the latter with the
structure of a Hilbert space on the real field. This rigorous construction,
based on non-standard generalizations of the notions of sum and scalar
multiplication of finite dimensional probability vectors, allows us to
meaningfully talk about "angles" between data streams and data sources, and,
make precise the notion of orthogonal stochastic processes. In particular, the
relative angles appear to be preserved, and identifiable, under severe noise,
and will be developed in future as the underlying principle for robust
classification, clustering and unsupervised featurization algorithms.