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Recently, many reinforcement learning techniques were shown to have provable guarantees in the simple case of linear dynamics, especially in problems like linear quadratic regulators. However, in practice, many reinforcement learning problems try to learn a policy directly from rich, high dimensional representations such as images. Even if there is an underlying dynamics that is linear in the correct latent representations (such as position and velocity), the rich representation is likely to be nonlinear and can contain irrelevant features. In this work we study a model where there is a hidden linear subspace in which the dynamics is linear. For such a model we give an efficient algorithm for extracting the linear subspace with linear dynamics. We then extend our idea to extracting a nonlinear mapping, and empirically verify the effectiveness of our approach in simple settings with rich observations.

Word embedding is a powerful tool in natural language processing. In this paper we consider the problem of word embedding composition -given vector representations of two words, compute a vector for the entire phrase. We give a generative model that can capture specific syntactic relations between words. Under our model, we prove that the correlations between three words (measured by their PMI) form a tensor that has an approximate low rank Tucker decomposition. The result of the Tucker decomposition gives the word embeddings as well as a core tensor, which can be used to produce better compositions of the word embeddings. We also complement our theoretical results with experiments that verify our assumptions, and demonstrate the effectiveness of the new composition method.

Following our previous work with Dorff and Lawlor, we extend results for the so-called equitent problem of fixed boundary and fixed volume. We define sufficient conditions, which in ‫ޒ‬ 2 and ‫ޒ‬ 3 are also necessary, for local minima to be piecewise spherical, and we show that these are areaminimizing in their homotopy class. We also give new examples of these surfaces in ‫ޒ‬ 2 and ‫ޒ‬ 3 .