Subtractive Mixture Models via Squaring: Representation and Learning
This work addresses the problem of efficient representation and learning of complex distributions for machine learning practitioners, offering a novel approach beyond traditional additive mixtures.
The paper tackles the challenge of learning subtractive mixture models, which can reduce the number of components needed for complex distributions, by using squaring in probabilistic circuits. The result includes a theoretical proof of exponential expressiveness and empirical gains on real-world distribution estimation tasks.
Mixture models are traditionally represented and learned by adding several distributions as components. Allowing mixtures to subtract probability mass or density can drastically reduce the number of components needed to model complex distributions. However, learning such subtractive mixtures while ensuring they still encode a non-negative function is challenging. We investigate how to learn and perform inference on deep subtractive mixtures by squaring them. We do this in the framework of probabilistic circuits, which enable us to represent tensorized mixtures and generalize several other subtractive models. We theoretically prove that the class of squared circuits allowing subtractions can be exponentially more expressive than traditional additive mixtures; and, we empirically show this increased expressiveness on a series of real-world distribution estimation tasks.