Spectral dimensionality reduction for HMMs

Hidden Markov Models (HMMs) can be accurately approximated using co-occurrence frequencies of pairs and triples of observations by using a fast spectral method in contrast to the usual slow methods like EM or Gibbs sampling. We provide a new spectral method which significantly reduces the number of model parameters that need to be estimated, and generates a sample complexity that does not depend on the size of the observation vocabulary. We present an elementary proof giving bounds on the relative accuracy of probability estimates from our model. (Correlaries show our bounds can be weakened to provide either L1 bounds or KL bounds which provide easier direct comparisons to previous work.) Our theorem uses conditions that are checkable from the data, instead of putting conditions on the unobservable Markov transition matrix

Nonparametric Estimation of Multi-View Latent Variable Models

Spectral methods have greatly advanced the estimation of latent variable models, generating a sequence of novel and efficient algorithms with strong theoretical guarantees. However, current spectral algorithms are largely restricted to mixtures of discrete or Gaussian distributions. In this paper, we propose a kernel method for learning multi-view latent variable models, allowing each mixture component to be nonparametric. The key idea of the method is to embed the joint distribution of a multi-view latent variable into a reproducing kernel Hilbert space, and then the latent parameters are recovered using a robust tensor power method. We establish that the sample complexity for the proposed method is quadratic in the number of latent components and is a low order polynomial in the other relevant parameters. Thus, our non-parametric tensor approach to learning latent variable models enjoys good sample and computational efficiencies. Moreover, the non-parametric tensor power method compares favorably to EM algorithm and other existing spectral algorithms in our experiments

PAC learning of Probabilistic Automaton based on the Method of Moments

International audienceProbabilitic Finite Automata (PFA) are gener-ative graphical models that define distributions with latent variables over finite sequences of symbols, a.k.a. stochastic languages. Traditionally , unsupervised learning of PFA is performed through algorithms that iteratively improves the likelihood like the Expectation-Maximization (EM) algorithm. Recently, learning algorithms based on the so-called Method of Moments (MoM) have been proposed as a much faster alternative that comes with PAC-style guarantees. However, these algorithms do not ensure the learnt automata to model a proper distribution , limiting their applicability and preventing them to serve as an initialization to iterative algorithms. In this paper, we propose a new MoM-based algorithm with PAC-style guarantees that learns automata defining proper distributions. We assess its performances on synthetic problems from the PAutomaC challenge and real datasets extracted from Wikipedia against previous MoM-based algorithms and EM algorithm

Spectral Methods for Natural Language Processing

Many state-of-the-art results in natural language processing (NLP) are achieved with statistical models involving latent variables. Unfortunately, computational problems associated with such models (for instance, finding the optimal parameter values) are typically intractable, forcing practitioners to rely on heuristic methods without strong guarantees. While heuristics are often sufficient for empirical purposes, their de-emphasis on theoretical aspects has certain negative ramifications. First, it can impede the development of rigorous theoretical understanding which can generate new ideas and algorithms. Second, it can lead to black art solutions that are unreliable and difficult to reproduce. In this thesis, we argue that spectral methods---that is, methods that use singular value decomposition or other similar matrix or tensor factorization---can effectively remedy these negative ramifications. To this end, we develop spectral methods for two unsupervised language processing tasks. The first task is learning lexical representations from unannotated text (e.g., hierarchical clustering of a vocabulary). The second task is estimating parameters of latent-variable models used in NLP applications (e.g., for unsupervised part-of-speech tagging). We show that our spectral algorithms have the following advantages over previous methods: 1. The algorithms provide a new theoretical framework that is amenable to rigorous analysis. In particular, they are shown to be statistically consistent. 2. The algorithms are simple to implement, efficient, and scalable to large amounts of data. They also yield results that are competitive with the state-of-the-art

Rich and Scalable Models for Text

Topic models have become essential tools for uncovering hidden structures in big data. However, the most popular topic model algorithm—Latent Dirichlet Allocation (LDA)— and its extensions suffer from sluggish performance on big datasets. Recently, the machine learning community has attacked this problem using spectral learning approaches such as the moment method with tensor decomposition or matrix factorization. The anchor word algorithm by Arora et al. [2013] has emerged as a more efficient approach to solve a large class of topic modeling problems. The anchor word algorithm is high-speed, and it has a provable theoretical guarantee: it will converge to a global solution given enough number of documents. In this thesis, we present a series of spectral models based on the anchor word algorithm to serve a broader class of datasets and to provide more abundant and more flexible modeling capacity. First, we improve the anchor word algorithm by incorporating various rich priors in the form of appropriate regularization terms. Our new regularized anchor word algorithms produce higher topic quality and provide flexibility to incorporate informed priors, creating the ability to discover topics more suited for external knowledge. Second, we enrich the anchor word algorithm with metadata-based word representation for labeled datasets. Our new supervised anchor word algorithm runs very fast and predicts better than supervised topic models such as Supervised LDA on three sentiment datasets. Also, sentiment anchor words, which play a vital role in generating sentiment topics, provide cues to understand sentiment datasets better than unsupervised topic models. Lastly, we examine ALTO, an active learning framework with a static topic overview, and investigate the usability of supervised topic models for active learning. We develop a new, dynamic, active learning framework that combines the concept of informativeness and representativeness of documents using dynamically updating topics from our fast supervised anchor word algorithm. Experiments using three multi-class datasets show that our new framework consistently improves classification accuracy over ALTO