Probabilistic Modeling of Structure in Science: Statistical Physics to Recommender Systems.

Jaan Altosaar

Abstract:

Applied machine learning relies on translating the structure of a problem into
a computational model. This arises in applications as diverse as statistical
physics and food recommender systems. The pattern of connectivity in an
undirected graphical model or the fact that datapoints in food recommendation are unordered collections of features can inform the structure of a model. First, consider undirected graphical models from statistical physics like the ubiquitous Ising model. Basic
research in physics requires scalable simulations for comparing the behavior of a model
to its experimental counterpart. The Ising model consists of binary random variables
with local connectivity; interactions between neighboring nodes can lead to long-range
correlations. Modeling these correlations is necessary to capture physical phenomena
such as phase transitions. To mirror the local structure of these models, we use flowbased convolutional generative models that can capture long-range correlations. Combining flow-based models designed for continuous variables with recent work on hierarchical variational approximations enables the modeling of discrete random variables.
Compared to existing variational inference methods, this approach scales to statistical
physics models with millions of correlated random variables and uses 100 times fewer
parameters. Just as computational choices can be made by considering the structure of
an undirected graphical model, model construction itself can be guided by the structure
of individual datapoints. Consider a recommendation task where datapoints consist of
unordered sets, and the objective is to maximize top-K recall, a common recommendation metric. Simple results show that a classifier with zero worst-case error achieves
maximum top-K recall. Further, the unordered structure of the data suggests the use of
a permutation-invariant classifier for statistical and computational efficiency. We evaluate such a classifier on human dietary behavior data, where every meal is an unordered
collection of ingredients, and find that it outperforms probabilistic matrix factorization
methods. Finally, we show that building problem structure into an approximate inference algorithm improves the accuracy of probabilistic modeling methods.