Low-dimensional data representation in data analysis

Abstract

Many Data Analysis tasks deal with data which are presented in high-dimensional spaces, and the ‘curse of dimensionality’ phenomena is often an obstacle to the use of many methods, including Neural Network methods, for solving these tasks. To avoid these phenomena, various Representation learning algorithms are used, as a first key step in solutions of these tasks, to transform the original high-dimensional data into their lower-dimensional representations so that as much information as possible is preserved about the original data required for the considered task. The above Representation learning problems are formulated as various Dimensionality Reduction problems (Sample Embedding, Data Manifold embedding, Data Manifold reconstruction and newly proposed Tangent Bundle Manifold Learning) motivated by various Data Analysis tasks. A new geometrically motivated algorithm that solves all the considered Dimensionality Reduction problems is presented.