Probabilistic Graphical Models (PGMs) encode conditional dependencies among random variables using a graph -nodes for variables, links for dependencies- and factorize the joint distribution into lower-dimensional components. This makes PGMs well-suited for analyzing complex systems and supporting decision-making. Recent advances in topological signal processing highlight the importance of variables defined on topological spaces in several application domains. In such cases, the underlying topology shapes statistical relationships, limiting the expressiveness of canonical PGMs. To overcome this limitation, we introduce Colored Markov Random Fields (CMRFs), which model both conditional and marginal dependencies among Gaussian edge variables on topological spaces, with a theoretical foundation in Hodge theory. CMRFs extend classical Gaussian Markov Random Fields by including link coloring: connectivity encodes conditional independence, while color encodes marginal independence. We quantify the benefits of CMRFs through a distributed estimation case study over a physical network, comparing it with baselines with different levels of topological prior.

--Despite significant advancements, current large language models (LLMs) and vision-language models (L VLMs) continue to struggle with complex, multi-step, cross-modal common sense reasoning tasks, often exhibiting a lack of "deliberative thinking." They tend to rely on superficial associations rather than deep, chained inference, particularly when integrating visual information with abstract concepts. T o address this, we propose the Coherent Multimodal Reasoning Framework (CMRF), a novel approach that enhances L VLMs' common sense reasoning capabilities through an iterative, self-evaluating inference mechanism. CMRF mimics human problem-solving by decomposing complex queries, generating step-by-step inferences, and self-correcting errors. Coupled with an Adaptive Iterative Refinement strategy, CMRF systematically refines its reasoning paths. Built upon LLaV A-1.6-34B and trained on a novel Multimodal Daily Activity Reasoning (MDAR) dataset, CMRF achieves state-of-the-art performance among open-source L VLMs on challenging benchmarks like VCR, A-OKVQA, and DailyLife-MRC. Extensive ablation studies and human evaluations confirm the critical contributions of each module and the effectiveness of iterative refinement in fostering more coherent and accurate reasoning. The remarkable advancements in large language models (LLMs) [1], [2] and vision-language models (L VLMs) have revolutionized various aspects of artificial intelligence, demonstrating unprecedented capabilities in understanding, generating, and processing information across modalities [3]. These models excel in tasks ranging from complex question answering to creative content generation, largely due to their extensive pre-training on vast amounts of data.

In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intractable because of two problems: (1) it includes the evaluation of the partition function and (2) it is formulated in the form of functional optimization. We avoid Problem (1) by using Bethe approximation. Bethe approximation is an approximation technique equivalent to the loopy belief propagation. Problem (2) can be solved by using orthonormal function expansion. Orthonormal function expansion can reduce a functional optimization problem to a function optimization problem. Our method can provide an analytic form of the solution of the inverse problem within the framework of Bethe approximation.