用平方和形式建模马尔可夫过程，实现精确概率传播与低内存占用。
Learning Markov Processes as Sum-of-Square Forms for Analytical Belief Propagation
- 采用稀疏平方和形式建模条件密度，支持解析式信念传播。
- 在低维空间内存远低于现有方法，且可扩展至12维系统。
- 适合需要精确推理与高维建模的机器学习与控制系统研究者。
利用马尔可夫过程模型的预测能力需将其概率密度函数（信念）通过模型传播。然而，许多现有模型难以实现解析式信念传播，通常依赖近似或采样生成预测。本文提出一种基于稀疏平方和（SoS）形式的功能建模框架，用于有效（条件）密度估计。我们研究了使用SoS形式建模条件密度的理论限制，并提出一种新型函数形式以克服这些限制。所提架构支持基函数与系数的联合学习，同时保持解析式信念传播能力。此外，我们设计了一种训练方法，确保完全满足归一化与非负性约束。实验表明，该方法在低维空间中精度接近最先进水平，内存消耗显著降低；更关键的是，当传统方法在超过2维时失效时，本方法仍可成功扩展至12维系统。
原文摘要 · Abstract (English)
Harnessing the predictive capability of Markov process models requires propagating probability density functions (beliefs) through the model. For many existing models however, belief propagation is analytically infeasible, requiring approximation or sampling to generate predictions. This paper proposes a functional modeling framework leveraging sparse Sum-of-Squares (SoS) forms for valid (conditional) density estimation. We study the theoretical restrictions of modeling conditional densities using the SoS form, and propose a novel functional form for addressing such limitations. The proposed architecture enables generalized simultaneous learning of basis functions and coefficients, while preserving analytical belief propagation. In addition, we propose a training method that allows for exact adherence to the normalization and non-negativity constraints. Our results show that the proposed method achieves accuracy comparable to state-of-the-art approaches while requiring significantly less memory in low-dimensional spaces, and it further scales to 12D systems when existing methods fail beyond 2D.
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