116 research outputs found
A Spectral Algorithm for Learning Hidden Markov Models
Hidden Markov Models (HMMs) are one of the most fundamental and widely used
statistical tools for modeling discrete time series. In general, learning HMMs
from data is computationally hard (under cryptographic assumptions), and
practitioners typically resort to search heuristics which suffer from the usual
local optima issues. We prove that under a natural separation condition (bounds
on the smallest singular value of the HMM parameters), there is an efficient
and provably correct algorithm for learning HMMs. The sample complexity of the
algorithm does not explicitly depend on the number of distinct (discrete)
observations---it implicitly depends on this quantity through spectral
properties of the underlying HMM. This makes the algorithm particularly
applicable to settings with a large number of observations, such as those in
natural language processing where the space of observation is sometimes the
words in a language. The algorithm is also simple, employing only a singular
value decomposition and matrix multiplications.Comment: Published in JCSS Special Issue "Learning Theory 2009
Uniform Chernoff and Dvoretzky-Kiefer-Wolfowitz-type inequalities for Markov chains and related processes
We observe that the technique of Markov contraction can be used to establish
measure concentration for a broad class of non-contracting chains. In
particular, geometric ergodicity provides a simple and versatile framework.
This leads to a short, elementary proof of a general concentration inequality
for Markov and hidden Markov chains (HMM), which supercedes some of the known
results and easily extends to other processes such as Markov trees. As
applications, we give a Dvoretzky-Kiefer-Wolfowitz-type inequality and a
uniform Chernoff bound. All of our bounds are dimension-free and hold for
countably infinite state spaces
A Method of Moments for Mixture Models and Hidden Markov Models
Mixture models are a fundamental tool in applied statistics and machine
learning for treating data taken from multiple subpopulations. The current
practice for estimating the parameters of such models relies on local search
heuristics (e.g., the EM algorithm) which are prone to failure, and existing
consistent methods are unfavorable due to their high computational and sample
complexity which typically scale exponentially with the number of mixture
components. This work develops an efficient method of moments approach to
parameter estimation for a broad class of high-dimensional mixture models with
many components, including multi-view mixtures of Gaussians (such as mixtures
of axis-aligned Gaussians) and hidden Markov models. The new method leads to
rigorous unsupervised learning results for mixture models that were not
achieved by previous works; and, because of its simplicity, it offers a viable
alternative to EM for practical deployment
- …