Stochastic control liaisons: Richard Sinkhorn meets gaspard monge on a schr\uf6dinger bridge

In 1931-1932, Erwin Schr\uf6dinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schr\uf6dinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes, Burg, Dempster, and Csisz\ue1r. The problem, known as the Schr\uf6dinger bridge problem (SBP) with "uniform"prior, was more recently recognized as a regularization of the Monge-Kantorovich optimal mass transport (OMT) problem, leading to effective computational schemes for the latter. Specifically, OMT with quadratic cost may be viewed as a zerotemperature limit of the problem posed by Schr\uf6dinger in the early 1930s. The latter amounts to minimization of Helmholtz's free energy over probability distributions that are constrained to possess two given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called Sinkhorn algorithm, which appears as a special case of an algorithm first studied in the more challenging continuous space setting by the French analyst Robert Fortet in 1938-1940 specifically for Schr\uf6dinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation of the optimal solution by iteratively solving the so-called Schr\uf6dinger system. Convergence of the iteration is guaranteed by contraction along the steps in suitable metrics, such as Hilbert's projective metric. In both the continuous as well as the discrete time and space settings, stochastic control provides a reformulation of and a context for the dynamic versions of general Schr\uf6dinger bridge problems and of their zero-temperature limit, the OMT problem. These problems, in turn, naturally lead to steering problems for flows of one-time marginals which represent a new paradigm for controlling uncertainty. The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou-Brenier characterization of the McCann displacement interpolation flow in OMT. The formalism and techniques behind these control problems on flows of probability distributions have attracted significant attention in recent years as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, and network routing as well as in computer and data science. This multifaceted and versatile framework, intertwining SBP and OMT, provides the substrate for the historical and technical overview of the field given in this paper. A key motivation has been to highlight links between the classical early work in both topics and the more recent stochastic control viewpoint, which naturally lends itself to efficient computational schemes and interesting generalizations

Reflected Schr\"odinger Bridge for Constrained Generative Modeling

Diffusion models have become the go-to method for large-scale generative models in real-world applications. These applications often involve data distributions confined within bounded domains, typically requiring ad-hoc thresholding techniques for boundary enforcement. Reflected diffusion models (Lou23) aim to enhance generalizability by generating the data distribution through a backward process governed by reflected Brownian motion. However, reflected diffusion models may not easily adapt to diverse domains without the derivation of proper diffeomorphic mappings and do not guarantee optimal transport properties. To overcome these limitations, we introduce the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive elegant reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and explore natural connections to entropic optimal transport for the study of approximate linear convergence - a valuable insight for practical training. Our algorithm yields robust generative modeling in diverse domains, and its scalability is demonstrated in real-world constrained generative modeling through standard image benchmarks

Multiscale neighborhood-wise decision fusion for redundancy detection in image pairs

SIAM Journal Multiscale Modeling & SimulationTo develop better image change detection algorithms, new models able to capture spatio-temporal regularities and geometries present in an image pair are needed. In this paper, we propose a multiscale formulation for modeling semi-local inter-image interactions and detecting local or regional changes in an image pair. By introducing dissimilarity measures to compare patches and binary local decisions, we design collaborative decision rules that use the total number of detections obtained from the neighboring pixels, for different patch sizes. We study the statistical properties of the non-parametric detection approach that guarantees small probabilities of false alarms. Experimental results on several applications demonstrate that the detection algorithm (with no optical flow computation) performs well at detecting occlusions and meaningful changes for a variety of illumination conditions and signal-to-noise ratios. The number of control parameters of the algorithm is small and the adjustment is intuitive in most cases

A non-rigid registration approach for quantifying myocardial contraction in tagged MRI using generalized information measures.

International audienceWe address the problem of quantitatively assessing myocardial function from tagged MRI sequences. We develop a two-step method comprising (i) a motion estimation step using a novel variational non-rigid registration technique based on generalized information measures, and (ii) a measurement step, yielding local and segmental deformation parameters over the whole myocardium. Experiments on healthy and pathological data demonstrate that this method delivers, within a reasonable computation time and in a fully unsupervised way, reliable measurements for normal subjects and quantitative pathology-specific information. Beyond cardiac MRI, this work redefines the foundations of variational non-rigid registration for information-theoretic similarity criteria with potential interest in multimodal medical imaging