A Simple Explanation of A Spectral Algorithm for Learning Hidden Markov Models

This document is my summary explanation of the algorithm in “A Spectral Algorithm for Learning Hidden Markov Models” (COLT 2009), though there may be some slight notational inconsistencies with the original paper. The exposition and the math here are quite different, so if you don’t like this explanation, try the original paper! The idea is to maintain output predictions in a recursive inference algorithm, instead of the usual method of maintaining hidden state predictions, and to represent the HMM only in terms of the maps necessary to update output predictions given new data. This approach limits the inference computations the algorithm can perform (it can’t answer any queries about the hidden states since it doesn’t explicitly deal with them at all), but it also reduces the complexity of the model parameters that are learned and thus makes learning easier. The learning algorithm uses an SVD and matrix operations, so it avoids the local-optima problems of EM or any other algorithms based on maximizing data likelihood over the usual HMM parameterization. The COLT paper includes error bounds and analysis. Notation. For a vector v in a subspace V ⊆ R and a matrix C with linearly independent columns and range(C) ⊇ V I will use [v] to denote the coordinate vector of v relative to the ordered basis given by the columns of C, and [v] or simply v to denote the coordinate vector of v relative to the standard basis of R. Similarly, for a linear map A : V → V I will use [A]C to denote the matrix of A relative to domain and codomain bases given by the columns of C, and [A] to indicate its matrix relative to standard bases. For a matrix A I will also use [A]ij to denote the (i, j)th entry.