The high dimensionality of modern data introduces significant challenges in descriptive and exploratory data analysis. These challenges gave rise to extensive work on dimensionality reduction and manifold learning aiming to provide low dimensional representations that preserve or uncover intrinsic patterns and structures in the data. In this thesis, we expand the current literature in manifold learning developing two methods called DIG (Dynamical Information Geometry) and GRAE (Geometry Regularized Autoencoders). DIG is a method capable of finding low-dimensional representations of high-frequency multivariate time series data, especially suited for visualization. GRAE is a general framework which splices the well-established machinery from kernel manifold learning methods to recover a sensitive geometry, alongside the parametric structure of autoencoders.
Manifold learning can also be useful to study data collected from different measurement instruments, conditions, or protocols of the same underlying system. In such cases the data is acquired in a multi-domain representation. The last two Chapters of this thesis are devoted to two new methods capable of aligning multi-domain data, leveraging their geometric structure alongside limited common information. First, we present DTA (Diffusion Transport Alignment), a semi-supervised manifold alignment method that exploits prior one-to-one correspondence knowledge between distinct data views and finds an aligned common representation. And finally, we introduce MALI (Manifold Alignment with Label Information). Here we drop the one-to-one prior correspondences assumption, since in many scenarios such information can not be provided, either due to the nature of the experimental design, or it becomes extremely costly. Instead, MALI only needs side-information in the form of discrete labels/classes present in both domains
