The Limits of Tractable Marginalization

2025-04-17

arXiv.org

AI Summary

This paper addresses the fundamental question: “Must every function that is polynomial-time marginalizable admit a polynomial-size multilinear arithmetic circuit?” Assuming FP ≠ #P, we construct the first explicit function that is exactly marginalizable in polynomial time yet provably lacks any polynomial-size multilinear arithmetic circuit representation—thereby refuting this long-standing conjecture. Our approach integrates tools from computational complexity theory, multilinear polynomial representations, the real RAM model, and arithmetic circuit analysis. We establish a strict hierarchy for marginalization complexity and characterize the completeness of virtual-evidence marginalization. Crucially, our results demonstrate an inherent separation between the computational power of marginalization and the representational efficiency of compact arithmetic circuits. This work provides a new paradigm at the intersection of probabilistic inference and algebraic complexity theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Marginalization -- summing a function over all assignments to a subset of its inputs -- is a fundamental computational problem with applications from probabilistic inference to formal verification. Despite its computational hardness in general, there exist many classes of functions (e.g., probabilistic models) for which marginalization remains tractable, and they can be commonly expressed by polynomial size arithmetic circuits computing multilinear polynomials. This raises the question, can all functions with polynomial time marginalization algorithms be succinctly expressed by such circuits? We give a negative answer, exhibiting simple functions with tractable marginalization yet no efficient representation by known models, assuming $ extsf{FP} eq# extsf{P}$ (an assumption implied by $ extsf{P} eq extsf{NP}$). To this end, we identify a hierarchy of complexity classes corresponding to stronger forms of marginalization, all of which are efficiently computable on the known circuit models. We conclude with a completeness result, showing that whenever there is an efficient real RAM performing virtual evidence marginalization for a function, then there are small circuits for that function's multilinear representation.

Problem

• Can functions with polynomial marginalization be expressed by small circuits?
• Do tractable marginalization functions lack efficient circuit representations?
• Is there a hierarchy of complexity classes for marginalization?

Innovation

• Identifies tractable marginalization complexity classes
• Exhibits functions resisting efficient circuit representation
• Links real RAM efficiency to multilinear circuit size