On the representation of sparse stochastic matrices with state embedding

Abstract

Embeddings are adjusted to allow points representing states and observations in Markov models, where conditional probabilities are approximately encoded as the exponential of (negative) distances, jointly scaled by a density factor. It is shown that the goodness of this approximation can be managed, mainly if the embedding dimension is chosen in function of entropies associated to the corresponding Markov model. Therefore, for sparse (low entropy) models, their representation as state embeddings can save memory and allow fully geometric versions of probabilistic algorithms, as the Viterbi, taken as an example in this work. Besides, evidences are also gathered in favor of potentially useful properties that emerge from the geometric representation of Markov models, such as analogies, superstates (aggregation) and semantic fields.