Introducing user-prescribed constraints in Markov chains for nonlinear dimensionality reduction
This work addresses a specific problem in nonlinear dimensionality reduction for researchers needing to incorporate custom constraints, but it appears incremental as it builds on existing stochastic kernel methods.
The authors tackled the lack of a systematic framework for imposing user-specified constraints on Markov chains in kernel-based dimensionality reduction by introducing a path entropy maximization approach to derive transition probabilities, demonstrating its usefulness with examples.
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.