Learning HMMs with Nonparametric Emissions via Spectral Decompositions of Continuous Matrices
Recently, there has been a surge of interest in using spectral methods for estimating latent variable models. However, it is usually assumed that the distribution of the observations conditioned on the latent variables is either discrete or belongs to a parametric family. In this paper, we study the estimation of an m-state hidden Markov model (HMM) with only smoothness assumptions, such as Hölderian conditions, on the emission densities. By leveraging some recent advances in continuous linear algebra and numerical analysis, we develop a computationally efficient spectral algorithm for learning nonparametric HMMs. Our technique is based on computing an SVD on nonparametric estimates of density functions by viewing them as continuous matrices. We derive sample complexity bounds via concentration results for nonparametric density estimation and novel perturbation theory results for continuous matrices. We implement our method using Chebyshev polynomial approximations. Our method is competitive with other baselines on synthetic and real problems and is also very computationally efficient.
The paper proposes a learning approach for HMMs with non-parametric emission distributions that computes the observable representation with the help of kernel density estimates. An approximation of the kernel density estimates with Chebyshev polynomials improves the runtime of the approach and ensures its practical applicability.
Strengths and Weaknesses ======================== Positive aspects: + The paper is well written and has a clear and coherent structure. The discussion of related work is comprehensive. + Non-parametric emission distributions add flexibility to the general HMM framework and reduce bias due to wrong modeling assumptions. Progress in this area should have theoretical and practical impact. + The paper builds upon existing spectral methods for parametric HMMs but introduces novel techniques to extend those approaches to the non-parametric case. + The theoretical bounds (section 5) are interesting, even though most of the results are special cases or straightforward extensions of known results.  Negative aspects: - The restriction to triplets (or a sliding window of length 3) is quite limiting. Is this a fundamental limitation of the approach or is an extension to longer subsequences (without a sliding window) straightforward? - Since the paper mentions the possibility to use Chebyshev polynomials to achieve a speed-up, it would have been interesting to see a runtime comparison at test time. - The presentation is at times too equation-driven and the notation, especially in chapter 3, quite convoluted and hard to follow. An illustrative figure of the key concepts in section 3 would have been helpful. - The experimental evaluation compares the proposed approach to 4 other HMM baselines. Even though NP-SPEC-HMM outperforms those baselines, the experimental evaluation has only toy character (simple length 6 conditional distributions, only one training/test sequence in case of the real datasets).  Minor points ============ * l.22: Only few parametric distributions allow for tractable exact inference in an HMM. * l.183: Much faster approximations than Chebyshev polynomials exist for the evaluation of kernel density estimates, especially in low-dimensional spaces (e.g., based on fast multipole methods). * Figure 1: There is probably a ‘x 10^3’ missing in the plot on the bottom right.  Questions ========= * Eq. (3): Why the restriction to an isotropic bandwidth and a product kernel? Especially a diagonal bandwidth matrix could have been helpful. Would the approximation with Chebyshev polynomials still work? * The paper focuses on learning HMMs with non-parametric emission distributions, but it does not become clear how those emission distributions affect inference. Which of the common inference tasks in a discrete HMM (filtering, smoothing, marginal observation likelihood) can be computed exactly/approximately with an NP-SPEC-HMM? * Is it computationally feasible to use the proposed model in a more realistic application, e.g., action recognition from motion capture sequences?  Conclusion ========== The paper is well written and, from a theoretical perspective, interesting to read. However, the experiments are weak and it remains unclear how practical the proposed model would be in real applications. I’m tending towards accept, but the authors should comment in their rebuttal on the above points.