Several works have shown that perturbation stable instances of the MAP inference problem in Potts models can be solved exactly using a natural linear programming (LP) relaxation. However, most of these works give few (or no) guarantees for the LP sol
utions on instances that do not satisfy the relatively strict perturbation stability definitions. In this work, we go beyond these stability results by showing that the LP approximately recovers the MAP solution of a stable instance even after the instance is corrupted by noise. This noisy stable model realistically fits with practical MAP inference problems: we design an algorithm for finding close stable instances, and show that several real-world instances from computer vision have nearby instances that are perturbation stable. These results suggest a new theoretical explanation for the excellent performance of this LP relaxation in practice.
Modeling the time-series of high-dimensional, longitudinal data is important for predicting patient disease progression. However, existing neural network based approaches that learn representations of patient state, while very flexible, are susceptib
le to overfitting. We propose a deep generative model that makes use of a novel attention-based neural architecture inspired by the physics of how treatments affect disease state. The result is a scalable and accurate model of high-dimensional patient biomarkers as they vary over time. Our proposed model yields significant improvements in generalization and, on real-world clinical data, provides interpretable insights into the dynamics of cancer progression.
Unsupervised learning seeks to uncover patterns in data. However, different kinds of noise may impede the discovery of useful substructure from real-world time-series data. In this work, we focus on mitigating the interference of left-censorship in t
he task of clustering. We provide conditions under which clusters and left-censorship may be identified; motivated by this result, we develop a deep generative, continuous-time model of time-series data that clusters while correcting for censorship time. We demonstrate accurate, stable, and interpretable results on synthetic data that outperform several benchmarks. To showcase the utility of our framework on real-world problems, we study how left-censorship can adversely affect the task of disease phenotyping, resulting in the often incorrect assumption that longitudinal patient data are aligned by disease stage. In reality, patients at the time of diagnosis are at different stages of the disease -- both late and early due to differences in when patients seek medical care and such discrepancy can confound unsupervised learning algorithms. On two clinical datasets, our model corrects for this form of censorship and recovers known clinical subtypes.
We prove that the $alpha$-expansion algorithm for MAP inference always returns a globally optimal assignment for Markov Random Fields with Potts pairwise potentials, with a catch: the returned assignment is only guaranteed to be optimal for an instan
ce within a small perturbation of the original problem instance. In other words, all local minima with respect to expansion moves are global minima to slightly perturb
Gaussian state space models have been used for decades as generative models of sequential data. They admit an intuitive probabilistic interpretation, have a simple functional form, and enjoy widespread adoption. We introduce a unified algorithm to ef
ficiently learn a broad class of linear and non-linear state space models, including variants where the emission and transition distributions are modeled by deep neural networks. Our learning algorithm simultaneously learns a compiled inference network and the generative model, leveraging a structured variational approximation parameterized by recurrent neural networks to mimic the posterior distribution. We apply the learning algorithm to both synthetic and real-world datasets, demonstrating its scalability and versatility. We find that using the structured approximation to the posterior results in models with significantly higher held-out likelihood.
Kalman Filters are one of the most influential models of time-varying phenomena. They admit an intuitive probabilistic interpretation, have a simple functional form, and enjoy widespread adoption in a variety of disciplines. Motivated by recent varia
tional methods for learning deep generative models, we introduce a unified algorithm to efficiently learn a broad spectrum of Kalman filters. Of particular interest is the use of temporal generative models for counterfactual inference. We investigate the efficacy of such models for counterfactual inference, and to that end we introduce the Healing MNIST dataset where long-term structure, noise and actions are applied to sequences of digits. We show the efficacy of our method for modeling this dataset. We further show how our model can be used for counterfactual inference for patients, based on electronic health record data of 8,000 patients over 4.5 years.
We introduce a globally-convergent algorithm for optimizing the tree-reweighted (TRW) variational objective over the marginal polytope. The algorithm is based on the conditional gradient method (Frank-Wolfe) and moves pseudomarginals within the margi
nal polytope through repeated maximum a posteriori (MAP) calls. This modular structure enables us to leverage black-box MAP solvers (both exact and approximate) for variational inference, and obtains more accurate results than tree-reweighted algorithms that optimize over the local consistency relaxation. Theoretically, we bound the sub-optimality for the proposed algorithm despite the TRW objective having unbounded gradients at the boundary of the marginal polytope. Empirically, we demonstrate the increased quality of results found by tightening the relaxation over the marginal polytope as well as the spanning tree polytope on synthetic and real-world instances.
