Quasi-Arithmetic Mixtures, Divergence Minimization, and Bregman Information

Markov Chain Monte Carlo methods for sampling from complex distributions and estimating normalization constants often simulate samples from a sequence of intermediate distributions along an annealing path, which bridges between a tractable initial distribution and a target density of interest. Prior work has constructed annealing paths using quasi-arithmetic means, and interpreted the resulting intermediate densities as minimizing an expected divergence to the endpoints. We provide a comprehensive analysis of this 'centroid' property using Bregman divergences under a monotonic embedding of the density function, thereby associating common divergences such as Amari's and Renyi's ${\alpha}$-divergences, ${(\alpha,\beta)}$-divergences, and the Jensen-Shannon divergence with intermediate densities along an annealing path. Our analysis highlights the interplay between parametric families, quasi-arithmetic means, and divergence functions using the rho-tau Bregman divergence framework of Zhang 2004,2013.Comment: 19 pages + appendix (rewritten + changed title in revision

Action Matching: Learning Stochastic Dynamics from Samples

Learning the continuous dynamics of a system from snapshots of its temporal marginals is a problem which appears throughout natural sciences and machine learning, including in quantum systems, single-cell biological data, and generative modeling. In these settings, we assume access to cross-sectional samples that are uncorrelated over time, rather than full trajectories of samples. In order to better understand the systems under observation, we would like to learn a model of the underlying process that allows us to propagate samples in time and thereby simulate entire individual trajectories. In this work, we propose Action Matching, a method for learning a rich family of dynamics using only independent samples from its time evolution. We derive a tractable training objective, which does not rely on explicit assumptions about the underlying dynamics and does not require back-propagation through differential equations or optimal transport solvers. Inspired by connections with optimal transport, we derive extensions of Action Matching to learn stochastic differential equations and dynamics involving creation and destruction of probability mass. Finally, we showcase applications of Action Matching by achieving competitive performance in a diverse set of experiments from biology, physics, and generative modeling.Comment: Published in ICML 202

A Computational Framework for Solving Wasserstein Lagrangian Flows

The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry ($\textit{kinetic energy}$), and the regularization of density paths ($\textit{potential energy}$). These combinations yield different variational problems ($\textit{Lagrangians}$), encompassing many variations of the optimal transport problem such as the Schr\"odinger bridge, unbalanced optimal transport, and optimal transport with physical constraints, among others. In general, the optimal density path is unknown, and solving these variational problems can be computationally challenging. Leveraging the dual formulation of the Lagrangians, we propose a novel deep learning based framework approaching all of these problems from a unified perspective. Our method does not require simulating or backpropagating through the trajectories of the learned dynamics, and does not need access to optimal couplings. We showcase the versatility of the proposed framework by outperforming previous approaches for the single-cell trajectory inference, where incorporating prior knowledge into the dynamics is crucial for correct predictions

A family history of breast cancer will not predict female early onset breast cancer in a population-based setting

ABSTRACT: BACKGROUND: An increased risk of breast cancer for relatives of breast cancer patients has been demonstrated in many studies, and having a relative diagnosed with breast cancer at an early age is an indication for breast cancer screening. This indication has been derived from estimates based on data from cancer-prone families or from BRCA1/2 mutation families, and might be biased because BRCA1/2 mutations explain only a small proportion of the familial clustering of breast cancer. The aim of the current study was to determine the predictive value of a family history of cancer with regard to early onset of female breast cancer in a population based setting. METHODS: An unselected sample of 1,987 women with and without breast cancer was studied with regard to the age of diagnosis of breast cancer. RESULTS: The risk of early-onset breast cancer was increased when there were: (1) at least 2 cases of female breast cancer in first-degree relatives (yes/no; HR at age 30: 3.09; 95% CI: 128-7.44), (2) at least 2 cases of female breast cancer in first or second-degree relatives under the age of 50 (yes/no; HR at age 30: 3.36; 95% CI: 1.12-10.08), (3) at least 1 case of female breast cancer under the age of 40 in a first- or second-degree relative (yes/no; HR at age 30: 2.06; 95% CI: 0.83-5.12) and (4) any case of bilateral breast cancer (yes/no; HR at age 30: 3.47; 95%: 1.33-9.05). The positive predictive value of having 2 or more of these characteristics was 13% for breast cancer before the age of 70, 11% for breast cancer before the age of 50, and 1% for breast cancer before the age of 30. CONCLUSION: Applying family history related criteria in an unselected population could result in the screening of many women who will not develop breast cancer at an early age