Exact Multimodal Inference via CP Decomposition as a Probabilistic Circuit

Keywords: probabilistic circuits, CP decomposition, multimodal learning, missing modality, exact inference, tensor methods

Abstract: We propose CP-PC, a generative multimodal model that represents the joint distribution over discrete modality codes as a CANDECOMP/PARAFAC (CP) tensor decomposition. We show that a CP decomposition over discrete modality codes can be interpreted exactly as a smooth, decomposable probabilistic circuit, yielding closed-form marginals and conditionals for arbitrary observed subsets in time linear in the CP rank and number of observed modalities without variational approximations, sampling, or learned inference networks. Missing-modality prediction reduces to an exact Bayesian posterior that collapses to the learned prior when all modalities are absent, and we prove identifiability under a Kruskal-type rank condition with closed-form EM learning. Experiments on synthetic data confirm factor recovery at the true rank; evaluations on MM-IMDb and CMU-MOSI demonstrate graceful degradation under missingness and calibration behavior consistent with exact Bayesian conditioning under missing evidence in our experiments, while using orders-of-magnitude fewer parameters than discriminative alternatives. On CM-MOSI, NLL degrades with rank in the small-data regime, a limitation we discuss.

Submission Number: 118