Learning Dynamic Bayesian Networks from Data: Foundations, First Principles and Numerical Comparisons

In this paper, we present a guide to the foundations of learning Dynamic Bayesian Networks (DBNs) from data in the form of multiple samples of trajectories for some length of time. We present the formalism for a generic as well as a set of common types of DBNs for particular variable distributions. We present the analytical form of the models, with a comprehensive discussion on the interdependence between structure and weights in a DBN model and their implications for learning. Next, we give a broad overview of learning methods and describe and categorize them based on the most important statistical features, and how they treat the interplay between learning structure and weights. We give the analytical form of the likelihood and Bayesian score functions, emphasizing the distinction from the static case. We discuss functions used in optimization to enforce structural requirements. We briefly discuss more complex extensions and representations. Finally we present a set of comparisons in different settings for various distinct but representative algorithms across the variants.

Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity

We consider the problem of learning local quantum Hamiltonians given copies of their Gibbs state at a known inverse temperature, following Haah et al. [2108.04842] and Bakshi et al. [arXiv:2310.02243]. Our main technical contribution is a new flat polynomial approximation of the exponential function based on the Chebyshev expansion, which enables the formulation of learning quantum Hamiltonians as a polynomial optimization problem. This, in turn, can benefit from the use of moment/SOS relaxations, whose polynomial bit complexity requires careful analysis [O'Donnell, ITCS 2017]. Finally, we show that learning a $k$-local Hamiltonian, whose dual interaction graph is of bounded degree, runs in polynomial time under mild assumptions.