CAREER: Principled Unsupervised Learning via Minimum Volume Polytopic Embedding

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Unsupervised learning problems are in general significantly more difficult than their supervised counterparts in machine learning. This poses considerable challenges in not only machine learning research but also education, as nearly all models are NP-hard (with possibly the sole exception of PCA), and the community has been dwelling on algorithms without optimality guarantees for several decades. This project aims at developing a principled framework of minimum volume polytopic embedding that unifies various unsupervised learning problems such as independent component analysis, dictionary learning, and nonnegative matrix factorization, by treating the problem as embedding the set of data points into a regular polytope such as a simplex, a box, or an orthoplex, while guided by a novel matrix volume criterion. The benefit is two-fold: 1) it provides identifiability guarantee with finite samples, and 2) it hinges on the development of algorithms that could optimally solve these NP-hard problems under mild assumptions. The PI’s prior work has showed strong identifiability guarantees for the former benefit, while this project will focus on resolving the latter one, starting from a Frank-Wolfe algorithmic framework that has shown great empirical success. Furthermore, this project will greatly expand its application domains such as POMDP identification in reinforcement learning, aggregate flexibility in power systems, and deep polytopic word embedding in natural language processing. In terms of the mathematical framework, extensions to handle nonlinearity and deep representation learning are also developed, which have been elusive and are expected to be widely impactful beyond the main focus of theory and algorithm development in this project. Extensive education and outreach plans are laid out to corroborate the research impact and encourage students from all backgrounds to engage in computer science and machine learning research.In this project we propose a novel framework that tries to transform all data points as points in a regular polytope (such as a simplex, a box, or an orthoplex), hence the aim polytopic embedding, while guided by a novel matrix volume optimization criterion. The PI's prior work not only showed strong identifiability guarantees of the latent representation, but also found a wide variety of practical success in applications. Prior success inspires the PI to further investigate this direction, resolve unsettled theoretical challenges, broaden the learning framework, and seek even more application domains. This project will evolve along the following synergistic thrusts: in Thrust 1, a Frank-Wolfe algorithm is designed to solve the NP-hard polytopic embedding problem. Inspired by recent developments in analyzing guaranteed non-convex learning, a promising pathway is laid out to provide provable global optimality guarantees. In Thrust 2, the proposed learning framework will be used to identify an unknown POMDP from only observations with computational guarantees. Research along this thrust will be applied to healthcare recommendations from medical data. In Thrust 3, the problem of aggregate flexibility in power systems is introduced, which provides an interesting dual interpretation of polytopic embedding. Experiments on real data will validate the performance and expand the framework to handle nonlinear constraints. In Thrust 4, we propose a novel word embedding scheme with not only computational guarantee but also semantic interpretation. An extension to deep polytopic embedding framework is also introduced.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.