Introducing user-prescribed constraints in Markov chains for nonlinear dimensionality reduction

TL;DR

This paper presents a novel framework for incorporating user-defined constraints into Markov chains used in kernel-based nonlinear dimensionality reduction, enabling more flexible and constrained data embeddings.

Contribution

It introduces a path entropy maximization method to derive Markov chain transition probabilities with user-specified stationary and dynamical constraints.

Findings

Demonstrates the method's ability to incorporate constraints effectively

Provides examples illustrating improved data embedding control

Offers a systematic approach for constrained Markov chain construction

Abstract

Stochastic kernel based dimensionality reduction approaches have become popular in the last decade. The central component of many of these methods is a symmetric kernel that quantifies the vicinity between pairs of data points and a kernel-induced Markov chain on the data. Typically, the Markov chain is fully specified by the kernel through row normalization. However, in many cases, it is desirable to impose user-specified stationary-state and dynamical constraints on the Markov chain. Unfortunately, no systematic framework exists to impose such user-defined constraints. Here, we introduce a path entropy maximization based approach to derive the transition probabilities of Markov chains using a kernel and additional user-specified constraints. We illustrate the usefulness of these Markov chains with examples.