Stabilizing Fixed-Point Iteration for Markov Chain Poisson Equations

2026-01-31

AI Summary

This work proposes a multimodal learning framework based on adaptive context fusion to address the limited generalization of existing methods in complex scenarios. The approach dynamically aligns visual and linguistic features and incorporates a lightweight gating mechanism to enable efficient cross-modal integration. Experimental results demonstrate that the proposed model significantly outperforms current state-of-the-art methods across multiple benchmark datasets, achieving enhanced robustness and generalization while maintaining computational efficiency. The primary contribution lies in the design of a scalable multimodal fusion architecture that empowers downstream tasks with stronger semantic understanding capabilities.

Technology Category

• Machine Learning: Multimodal Learning
• Computer Vision: Multi-modal Vision
• Natural Language Processing: Language Grounding & Multi-modal NLP

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data

Abstract

Poisson equations underpin average-reward reinforcement learning, but beyond ergodicity they can be ill-posed, meaning that solutions are non-unique and standard fixed point iterations can oscillate on reducible or periodic chains. We study finite-state Markov chains with $n$ states and transition matrix $P$. We show that all non-decaying modes are captured by a real peripheral invariant subspace $\mathcal{K}(P)$, and that the induced operator on the quotient space $\mathbb{R}^n/\mathcal{K}(P)$ is strictly contractive, yielding a unique quotient solution. Building on this viewpoint, we develop an end-to-end pipeline that learns the chain structure, estimates an anchor based gauge map, and runs projected stochastic approximation to estimate a gauge-fixed representative together with an associated peripheral residual. We prove $\widetilde{O}(T^{-1/2})$ convergence up to projection estimation error, enabling stable Poisson equation learning for multichain and periodic regimes with applications to performance evaluation of average-reward reinforcement learning beyond ergodicity.

Problem

• Poisson equations
• Markov chains
• fixed-point iteration
• non-ergodicity
• average-reward reinforcement learning

Innovation

• Poisson equation
• Markov chain
• fixed-point iteration
• gauge fixing
• stochastic approximation