We give a polynomial-time algorithm for learning latent-state linear dynamical systems without system identification, and without assumptions on the spectral radius of the system's transition matrix. The algorithm extends the recently introduced technique of spectral filtering, previously applied only to systems with a symmetric transition matrix, using a novel convex relaxation to allow for the efficient identification of phases.

A key task in Bayesian machine learning is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). One prevalent example of this is sampling posteriors in parametric distributions, such as latent-variable generative models. However sampling (even very approximately) can be #P-hard.Classical results (going back to [B É85]) on sampling focus on log-concave distributions, and show a natural Markov process called Langevin diffusion mixes in polynomial time. However, all log-concave distributions are uni-modal, while in practice it is very common for the distribution of interest to have multiple modes. In this case, Langevin diffusion suffers from torpid mixing.We address this problem by combining Langevin diffusion with simulated tempering. The result is a Markov chain that mixes more rapidly by transitioning between different temperatures of the distribution. We analyze this Markov chain for a mixture of (strongly) log-concave distributions of the same shape. In particular, our technique applies to the canonical multi-modal distribution: a mixture of gaussians (of equal variance). Our algorithm efficiently samples from these distributions given only access to the gradient of the log-pdf. To the best of our knowledge, this is the first result that proves fast mixing for multimodal distributions in this setting.For the analysis, we introduce novel techniques for proving spectral gaps based on decomposing the action of the generator of the diffusion. Previous approaches rely on decomposing the state space as a partition of sets, while our approach can be thought of as decomposing the stationary measure as a mixture of distributions (a "soft partition").Additional materials for the paper can be found at http://tiny.cc/glr17. Note that the proof and results have been improved and generalized from the precursor at http://www.arxiv.org/ abs/1710.02736. See Section 3.1 for a comparison.

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Folsom, Kent, and Ono used the theory of modular forms modulo to establish remarkable "self-similarity" properties of the partition function and give an overarching explanation of many partition congruences. We generalize their work to analyze powers p r of the partition function as well as Andrews's spt-function. By showing that certain generating functions reside in a small space made up of reductions of modular forms, we set up a general framework for congruences for p r and spt on arithmetic progressions of the form m n + δ modulo powers of . Our work gives a conceptual explanation of the exceptional congruences of p r observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.

Identification of a linear time-invariant dynamical system from partial observations is a fundamental problem in control theory. A natural question is how to do so with non-asymptotic statistical rates depending on the inherent dimensionality (order) d of the system, rather than on the sufficient rollout length or on 1 1−ρ(A) , where ρ(A) is the spectral radius of the dynamics matrix. We develop the first algorithm that given a single trajectory of length T with gaussian observation noise, achieves a near-optimal rate of ‹ Oin H 2 error for the learned system. We also give bounds under process noise and improved bounds for learning a realization of the system. Our algorithm is based on low-rank approximation of Hankel matrices of geometrically increasing sizes.