On the Relationship Between Monotone and Squared Probabilistic Circuits (2408.00876v2)

Abstract: Probabilistic circuits are a unifying representation of functions as computation graphs of weighted sums and products. Their primary application is in probabilistic modeling, where circuits with non-negative weights (monotone circuits) can be used to represent and learn density/mass functions, with tractable marginal inference. Recently, it was proposed to instead represent densities as the square of the circuit function (squared circuits); this allows the use of negative weights while retaining tractability, and can be exponentially more expressive efficient than monotone circuits. Unfortunately, we show the reverse also holds, meaning that monotone circuits and squared circuits are incomparable in general. This raises the question of whether we can reconcile, and indeed improve upon the two modeling approaches. We answer in the positive by proposing Inception PCs, a novel type of circuit that naturally encompasses both monotone circuits and squared circuits as special cases, and employs complex parameters. Empirically, we validate that Inception PCs can outperform both monotone and squared circuits on a range of tabular and image datasets.

• The paper establishes that monotone and squared circuits are often incomparable, with each encoding certain probabilistic models exponentially more succinctly than the other.
• The paper introduces the novel InceptionPCs, a unified circuit framework that combines the strengths of both circuit types through complex latent variable integration.
• Empirical evaluations on MNIST and FashionMNIST confirm that InceptionPCs achieve improved performance and more compact representations compared to traditional probabilistic circuits.

Overview of the Research on the Relationship Between Monotone and Squared Probabilistic Circuits

The paper confronts a critical issue in probabilistic modeling by examining the expressive power and limitations of two classes of probabilistic circuits: monotone circuits and squared circuits. Probabilistic circuits (PCs) are versatile and effective constructs used to represent and perform inference on probability distributions. They function as directed acyclic graphs composed of weighted sums and product nodes, supporting efficient computation of various probabilistic queries. Within this conceptual framework, monotone circuits are characterized by non-negative weights which offer tractable solutions for several inference tasks. However, recent research proposes the use of squared probabilistic circuits, which enable the inclusion of negative weights, promising a more compact representation of certain probability distributions.

Analytical Insights and Theoretical Contributions

The paper rigorously reexamines monotone and squared structured-decomposable PCs. It establishes that the expressiveness of these two circuit types is often incomparable — meaning each can sometimes encode probabilistic models exponentially more succinctly than the other. This finding is particularly significant as it challenges the conventional wisdom that introducing squared circuits, which allow negative parameterization, universally increases expressivity.

The paper addresses this dichotomy by suggesting a novel circuit class known as InceptionPCs. This newly proposed class capitalizes on the strengths of both monotone and squared PCs through a unified mechanism. InceptionPCs encompass latent variables with complex parameters, unifying and enhancing the modeling capacities of both circuit types.

Empirical Considerations

To validate the theoretical insights, empirical assessments were conducted using image datasets such as MNIST and FashionMNIST. These experiments demonstrated notable improvements in model performance, validating the proposition that InceptionPCs can effectively outperform standalone monotone and squared circuits in practice.

Implications and Future Directions

The implications of this work extend both practically and theoretically. Practically, the introduction of InceptionPCs presents an enhanced tool for probabilistic modeling, expanding the possible applications while maintaining computational feasibility. Theoretically, this research poses a valuable contribution to understanding the fundamental expressive differences between monotone and squared circuits, and how their individual strengths can be leveraged collectively.

Looking forward, the versatility of InceptionPCs suggests numerous intriguing avenues for future research and advancement. An optimization of training procedures, possibly utilizing techniques like expectation-maximization (EM) tailored to their latent variable framework, could significantly enhance efficiency. Reducing computational overhead through improved architectures is another promising direction. Furthermore, exploring analogous applications in different contexts — such as quantum computing frameworks, where circuits generalize tensor networks, or machine learning under uncertain conditions — may yield substantial insights and innovations.

In summary, this work effectively illuminates the complexities of circuit expressiveness and proposes a sophisticated approach that melds the expressive advantages of differing probabilistic circuit types within a single framework. By integrating complex parameters and strategically handling latent variables, InceptionPCs offer a powerful and innovative toolset poised to advance the field of tractable probabilistic modeling.