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.
The second idea is that we can represent the probability of sequences as products of matrix operators, as in the literature on multiplicity automata (see for discussion of this relationship). This idea was re-used in both the Observable Operator Model of @cite_3 and the Predictive State Representations of @cite_0 , both of which are closely related and both of which can model HMMs. In fact, the former work by @cite_3 provides a non-iterative algorithm for learning HMMs, with an asymptotic analysis. However, this algorithm assumed knowing a set of characteristic events', which is a rather strong assumption that effectively reveals some relationship between the hidden states and observations. In our algorithm, this problem is avoided through the first idea.
@cite_0 @cite_3
1528056001 1934019294 2114469907 2053569739
Hidden Markov models (HMMs) have proven to be one of the most widely used tools for learning probabilistic models of time series data. In an HMM, information about the past is conveyed through a single discrete variable—the hidden state. We discuss a generalization of HMMs in which this state is factored into multiple state variables and is therefore represented in a distributed manner. We describe an exact algorithm for inferring the posterior probabilities of the hidden state variables given the observations, and relate it to the forward–backward algorithm for HMMs and to algorithms for more general graphical models. Due to the combinatorial nature of the hidden state representation, this exact algorithm is intractable. As in other intractable systems, approximate inference can be carried out using Gibbs sampling or variational methods. Within the variational framework, we present a structured approximation in which the the state variables are decoupled, yielding a tractable algorithm for learning the parameters of the model. Empirical comparisons suggest that these approximations are efficient and provide accurate alternatives to the exact methods. Finally, we use the structured approximation to model Bach‘s chorales and show that factorial HMMs can capture statistical structure in this data set which an unconstrained HMM cannot. Hidden Markov models (HMMs) are a powerful probabilistic tool for modeling sequential data, and have been applied with success to many text-related tasks, such as part-of-speech tagging, text segmentation and information extraction. In these cases, the observations are usually modeled as multinomial distributions over a discrete vocabulary, and the HMM parameters are set to maximize the likelihood of the observations. This paper presents a new Markovian sequence model, closely related to HMMs, that allows observations to be represented as arbitrary overlapping features (such as word, capitalization, formatting, part-of-speech), and defines the conditional probability of state sequences given observation sequences. It does this by using the maximum entropy framework to fit a set of exponential models that represent the probability of a state given an observation and the previous state. We present positive experimental results on the segmentation of FAQ’s. Hidden Markov modeling (HMM) techniques have been used successfully for connected speech recognition in the last several years. In the traditional HMM algorithms, the probability of duration of a state decreases exponentially with time which is not appropriate for representing the temporal structure of speech. Non-parametric modeling of duration using semi-Markov chains does accomplish the task with a large increase in the computational complexity. Applying a postprocessing state duration penalty after Viterbi decoding adds very little computation but does not affect the forward recognition path. The authors present a way of modeling state durations in HMM using time-dependent state transitions. This inhomogeneous HMM (IHMM) does increase the computation by a small amount but reduces recognition error rates by 14-25 . Also, a suboptimal implementation of this scheme that requires no more computation than the traditional HMM is presented which also has reduced errors by 14-22 on a variety of databases. > We present a novel probabilistic method for topic segmentation on unstructured text. One previous approach to this problem utilizes the hidden Markov model (HMM) method for probabilistically modeling sequence data [7]. The HMM treats a document as mutually independent sets of words generated by a latent topic variable in a time series. We extend this idea by embedding Hofmann's aspect model for text [5] into the segmenting HMM to form an aspect HMM (AHMM). In doing so, we provide an intuitive topical dependency between words and a cohesive segmentation model. We apply this method to segment unbroken streams of New York Times articles as well as noisy transcripts of radio programs on SpeechBot , an online audio archive indexed by an automatic speech recognition engine. We provide experimental comparisons which show that the AHMM outperforms the HMM for this task.