Mathematical Foundations of Dimensionality Reduction

Dimensionality reduction is a crucial technique in data analysis and machine learning, enabling the simplification of complex high-dimensional datasets while preserving their intrinsic structures.

In this talk we will present the mathematical footings of several prominent dimensionality reduction methods: Principal Component Analysis (PCA), Isomap, Laplace Eigenmaps, …

We will explore the specific optimization objectives and the role of weight assignments within k-neighborhood graphs for each method. By examining the theoretical frameworks and optimization processes, we aim to provide a comprehensive understanding of how these techniques transform metric relationships within data into meaningful lower dimensional representations. Insights into the mathematical principles that drive these algorithms highlight their unique approaches to capturing and preserving data structures.