A Hilbert Space of Stationary Ergodic Processes

Identifying meaningful signal buried in noise is a problem of interest arising in diverse scenarios of datadriven 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. I. Preliminary Concepts Definition 1 (Inner Product & Inner Product Spaces). An inner product on a real vector spaceX is a function h ; i : X X ! R, such that the following conditions are satis ed: 8u; v; w 2 X; 2 R; hu; (v + w)i = hu; v + wi = (hu; vi+ hu;wi) (Bi-linearity) 8v;w 2 X; hv;wi = hw; vi (Symmetry) 8u 2 X; hu; ui = 0; where hu; ui = 0) u = 0 (Positive De niteness) A vector space with an inner product is an inner product space. Note that an inner product necessarily induces a norm, which in turn induces a metric [6], [10]. Definition 2 (Complete inner product space or Hilbert Space). A complete inner product space, or a Hilbert space [10], is a Banach space with an inner product, i:e:, every Cauchy sequence in the space converges in the space. Notation 1 (Strictly Positive Probability Vectors). For n 2 N, the space of strictly positive probability vectors is de ned as: P n = ( } 2 R : 8i }i > 0; X