A central problem in dynamical system modeling is state discovery--that is, finding a compact summary of the past that captures the information needed to predict the future. Predictive State Representations (PSRs) enable clever spectral methods for state discovery; however, while consistent in the limit of infinite data, these methods often suffer from poor performance in the low data regime. In this paper we develop a novel algorithm for incorporating domain knowledge, in the form of an imperfect state representation, as side information to speed spectral learning for PSRs. We prove theoretical results characterizing the relevance of a user-provided state representation, and design spectral algorithms that can take advantage of a relevant representation. Our algorithm utilizes principal angles to extract the relevant components of the representation, and is robust to misspecification. Empirical evaluation on synthetic HMMs, an aircraft identification domain, and a gene splice dataset shows that, even with weak domain knowledge, the algorithm can significantly outperform standard PSR learning.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

In this paper, we propose a learning approach for the Mixture of Hidden Markov Models (MHMM) based on the Method of Moments (MoM). Computational advantages of MoM make MHMM learning amenable for large data sets. It is not possible to directly learn an MHMM with existing learning approaches, mainly due to a permutation ambiguity in the estimation process. We show that it is possible to resolve this ambiguity using the spectral properties of a global transition matrix even in the presence of estimation noise. We demonstrate the validity of our approach on synthetic and real data.

Generalized-linear dynamical models (GLDMs) remain a widely-used framework within neuroscience for modeling time-series data, such as neural spiking activity or categorical decision outcomes. Whereas the standard usage of GLDMs is to model a single data source, certain applications require jointly modeling two generalized-linear time-series sources while also dissociating their shared and private dynamics. Most existing GLDM variants and their associated learning algorithms do not support this capability. Here we address this challenge by developing a multi-step analytical subspace identification algorithm for learning a GLDM that explicitly models shared vs. private dynamics within two generalized-linear time-series. In simulations, we demonstrate our algorithm's ability to dissociate and model the dynamics within two time-series sources while being agnostic to their respective observation distributions.

In this paper, we propose a learning approach for the Mixture of Hidden Markov Models (MHMM) based on the Method of Moments (MoM). Computational advantages of MoM make MHMM learning amenable for large data sets. It is not possible to directly learn an MHMM with existing learning approaches, mainly due to a permutation ambiguity in the estimation process. We show that it is possible to resolve this ambiguity using the spectral properties of a global transition matrix even in the presence of estimation noise. We demonstrate the validity of our approach on synthetic and real data.

Update after author feedback: I thank the authors for their constructive response and hope they can incorporate as many of the promised changes as possible. Based on the feedback and reviewer discussion I now have a better idea of the motivation behind the work. Perhaps a brief description of a concrete example application problem motivating the work would make it easier for others more used to machine learning type learning of dynamical models to appreciate the work too. The experimental evaluation would benefit a lot from explicit comparisons with Bayesian alternatives (e.g. Ruttor et al., NIPS 2013; Svensson et al., AISTATS 2016 and references therein) to properly understand the pros and cons of the different approaches.

In this paper, we propose a learning approach for the Mixture of Hidden Markov Models (MHMM) based on the Method of Moments (MoM). Computational advantages of MoM make MHMM learning amenable for large data sets. It is not possible to directly learn an MHMM with existing learning approaches, mainly due to a permutation ambiguity in the estimation process. We show that it is possible to resolve this ambiguity using the spectral properties of a global transition matrix even in the presence of estimation noise. We demonstrate the validity of our approach on synthetic and real data.

We develop a latent variable model and an efficient spectral algorithm motivated by the recent emergence of very large data sets of chromatin marks from multiple human cell types. A natural model for chromatin data in one cell type is a Hidden Markov Model (HMM); we model the relationship between multiple cell types by connecting their hidden states by a fixed tree of known structure. The main challenge with learning parameters of such models is that iterative methods such as EM are very slow, while naive spectral methods result in time and space complexity exponential in the number of cell types. We exploit properties of the tree structure of the hidden states to provide spectral algorithms that are more computationally efficient for current biological datasets. We provide sample complexity bounds for our algorithm and evaluate it experimentally on biological data from nine human cell types.

SUMMARY: This paper proposes a method to incorporate prior knowledge into the spectral learning algorithm for predictive state representations (PSR). The prior knowledge consists of an imperfect/incomplete state representation which is'refined' and'completed' by the learning algorithm. This contribution addresses one of the main caveats of spectral methods: while these methods are fast and consistent, they tend to perform worse than local methods (e.g. By leveraging domain specific knowledge, the proposed algorithm overcomes this issue. The proposed extension, PSR-f, is relatively straightforward: the belief vector at each time step is the concatenation of the user-specified state representation f with a learned state representation b; the parameters of b are learned in the same fashion as for the classical method by solving linear regression problems constrained to the row space of the concatenation of some Hankel/system matrices (e.g. now mapping [P(T h); P(h)f(h)] to P(oT h) for each B_o).