Introducing User-Prescribed Constraints in Markov Chains for Nonlinear Dimensionality Reduction.

Area of Science:

• Machine Learning
• Data Science
• Computational Statistics

Background:

• Kernel-based methods are widely used for dimensionality reduction.
• These methods often rely on Markov chains derived from kernels.
• Existing methods lack a framework for imposing user-defined constraints on Markov chains.

Purpose of the Study:

• To develop a systematic approach for deriving Markov chains with user-specified constraints.
• To extend kernel-based dimensionality reduction techniques by incorporating flexible Markov chain properties.

Main Methods:

• Introduced a path entropy maximization framework.
• Derived Markov chain transition probabilities using kernels and user-defined constraints.
• Built upon previous work in Markov model inference.

Main Results:

• Demonstrated a novel method for constructing constrained Markov chains.
• Showcased the applicability of the approach through illustrative examples.
• Provided a systematic way to integrate user constraints into kernel-based methods.

Conclusions:

• The proposed path entropy maximization approach offers a flexible and systematic way to derive constrained Markov chains.
• This method enhances kernel-based dimensionality reduction by allowing tailored Markov chain properties.
• The approach has broad implications for data analysis where specific dynamical or stationary properties are required.