Optimal Transport for Probabilistic Circuits
Adrian Ciotinga, YooJung Choi
TL;DR
This paper introduces a new Wasserstein-type distance for probabilistic circuits, enabling efficient computation and parameter estimation, which was previously unavailable for such models.
Contribution
It proposes a novel optimal transport framework for PCs, including algorithms for computing the distance, retrieving transport plans, and parameter estimation.
Findings
Efficient algorithms for Wasserstein distance computation between PCs.
Method to retrieve optimal transport plans from linear program solutions.
Empirical evaluation shows effective PC parameter tuning using the proposed distance.
Abstract
We introduce a novel optimal transport framework for probabilistic circuits (PCs). While it has been shown recently that divergences between distributions represented as certain classes of PCs can be computed tractably, to the best of our knowledge, there is no existing approach to compute the Wasserstein distance between probability distributions given by PCs. We propose a Wasserstein-type distance that restricts the coupling measure of the associated optimal transport problem to be a probabilistic circuit. We then develop an algorithm for computing this distance by solving a series of small linear programs and derive the circuit conditions under which this is tractable. Furthermore, we show that we can easily retrieve the optimal transport plan between the PCs from the solutions to these linear programs. Lastly, we study the empirical Wasserstein distance between a PC and a dataset,…
Peer Reviews
Decision·UAI 2025 Poster
The ideas in this paper come through clearly, as the text (if not the math) is well-written. The general idea is natural, and it is not hard to see how a more efficient way of calculating Wasserstein distances between the distributions encoded by PCs could, in principle, be tremendously useful. Propositions 1 and 2 provide important grounding for the given constructions. The authors take the space needed to unpack their ideas. They identify appropriate baselines, and some genuine (if perhaps
Unfortunately, the paper is lacking in technical depth. The idea and its implementation seem straightforward to me, and the results are unsurprising --- so I find the mathematical contribution relatively small (putting aside the substantial low-level issues with it, which I detail below). This could easily be forgiven if the techniques enable something really interesting, but, on the experimental side, the examples are all small-scale synthetic toys. To the extent that this is really a basis o
1- The proofs of the theorems are correct, and the mathematical accuracy is high.
1- Using simple examples to illustrate definitions and results could make the paper easier to read and follow. 2- While the proposed metrics could significantly reduce runtime, they also lead to an increase in error. How much error is considered acceptable? There is no analytical approach or numerical result provided to show the impact of this error. 3- The proposed method performs well with a small set of variables; however, runtime challenges typically arise in large-scale systems with many
This paper introduces, for the first time, a Circuit Wasserstein distance, denoted as $ CW_p $, between compatible probabilistic circuits (PCs). Leveraging the recursive properties of the Wasserstein objective, it computes the optimal parameters for the coupling circuit by solving a small linear program at each sum node. Additionally, the paper presents a method for learning the parameters of PCs by minimizing $ ECW_p $, which is computationally efficient.
This paper addresses only a restricted set of cases within the broader context of probabilistic circuits. Regarding optimal transport, the approach feels somewhat formulaic, lacking a deeper exploration of the essential relationship between optimal transport and probabilistic circuits. Additionally, there is a typo on line 786.
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Taxonomy
TopicsLow-power high-performance VLSI design