Manifold Learning

Many algorithms can be improved by first learning the structure of input data, then operating on it in the context of a low-dimensional manifold. The ALL applies these "manifold learning" techniques to a variety of algorithms and applications.

Manifold Alignment

Manifold alignment is the task of finding correspondences between two or more data sets by considering inter-set similarity in manifold space. The ALL has explored a wide variety of approaches to this problem, featuring both linear and nonlinear embeddings with one-shot or multi-step aligment.

While most of our manifold alignment algorithms are based on solving a generalized eigenvector problem, more recently we have explored a regression formulation of the problem. This new class of alignment algorithms enables the use of regularization techniques (like an L1 penalty) to improve model sparsity and generalization ability.

Manifold Warping

Many interesting real-world data sets are inherently sequential: text documents, social network activity, audio/video streams, mobile robot trajectories, and more. Manifold warping extends the above alignment techniques to exploit this structure, preserving both the data's ordering semantics and manifold structure while finding inter-set alignment.