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Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modelling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a natural generalisation as diffusion bridges in a stochastic setting. Simulation of such bridges is key to solve inference and registration problems in shape analysis. We demonstrate how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models thereby substantially expanding the applicability of such models. We exemplify these methods by estimating template shapes from observed shape configurations while simultaneously learning model parameters.

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice to model dynamical systems evolving under the influence of randomness. By formulating the search for a mild solution of an SPDE as a neural fixed-point problem, we introduce the Neural SPDE model to learn solution operators of PDEs with (possibly stochastic) forcing from partially observed data. Our model provides an extension to two classes of physics-inspired neural architectures. On the one hand, it extends Neural CDEs, SDEs, RDEs -- continuous-time analogues of RNNs -- in that it is capable of processing incoming sequential information even when the latter evolves in an infinite dimensional state space. On the other hand, it extends Neural Operators -- generalizations of neural networks to model mappings between spaces of functions -- in that it can be used to learn solution operators $(u_0,\xi) \mapsto u$ of SPDEs depending simultaneously on the initial condition $u_0$ and a realization of the driving noise $\xi$. A Neural SPDE is resolution-invariant, it may be trained using a memory-efficient implicit-differentiation-based backpropagation and, once trained, its evaluation is up to 3 orders of magnitude faster than traditional solvers. Experiments on various semilinear SPDEs, including the 2D stochastic Navier-Stokes equations, demonstrate how Neural SPDEs capable of learning complex spatiotemporal dynamics with better accuracy and using only a modest amount of training data compared to all alternative models.

We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusion coefficient is uniformly elliptic, H\"older continuous and weakly differentiable in the spatial variables while the drift satisfies the Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and R\"ockner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and be fine-tuned to achieve the standard $1/2$-strong convergence rate with a logarithmic factor.

Advection-diffusion equations describe a large family of natural transport processes, e.g., fluid flow, heat transfer, and wind transport. They are also used for optical flow and perfusion imaging computations. We develop a machine learning model, D^2-SONATA, built upon a stochastic advection-diffusion equation, which predicts the velocity and diffusion fields that drive 2D/3D image time-series of transport. In particular, our proposed model incorporates a model of transport atypicality, which isolates abnormal differences between expected normal transport behavior and the observed transport. In a medical context such a normal-abnormal decomposition can be used, for example, to quantify pathologies. Specifically, our model identifies the advection and diffusion contributions from the transport time-series and simultaneously predicts an anomaly value field to provide a decomposition into normal and abnormal advection and diffusion behavior. To achieve improved estimation performance for the velocity and diffusion-tensor fields underlying the advection-diffusion process and for the estimation of the anomaly fields, we create a 2D/3D anomaly-encoded advection-diffusion simulator, which allows for supervised learning. We further apply our model on a brain perfusion dataset from ischemic stroke patients via transfer learning. Extensive comparisons demonstrate that our model successfully distinguishes stroke lesions (abnormal) from normal brain regions, while reconstructing the underlying velocity and diffusion tensor fields.

We study a class of algorithms for solving bilevel optimization problems in both stochastic and deterministic settings when the inner-level objective is strongly convex. Specifically, we consider algorithms based on inexact implicit differentiation and we exploit a warm-start strategy to amortize the estimation of the exact gradient. We then introduce a unified theoretical framework inspired by the study of singularly perturbed systems (Habets, 1974) to analyze such amortized algorithms. By using this framework, our analysis shows these algorithms to match the computational complexity of oracle methods that have access to an unbiased estimate of the gradient, thus outperforming many existing results for bilevel optimization. We illustrate these findings on synthetic experiments and demonstrate the efficiency of these algorithms on hyper-parameter optimization experiments involving several thousands of variables.

Manual annotation of medical images is highly subjective, leading to inevitable and huge annotation biases. Deep learning models may surpass human performance on a variety of tasks, but they may also mimic or amplify these biases. Although we can have multiple annotators and fuse their annotations to reduce stochastic errors, we cannot use this strategy to handle the bias caused by annotators' preferences. In this paper, we highlight the issue of annotator-related biases on medical image segmentation tasks, and propose a Preference-involved Annotation Distribution Learning (PADL) framework to address it from the perspective of disentangling an annotator's preference from stochastic errors using distribution learning so as to produce not only a meta segmentation but also the segmentation possibly made by each annotator. Under this framework, a stochastic error modeling (SEM) module estimates the meta segmentation map and average stochastic error map, and a series of human preference modeling (HPM) modules estimate each annotator's segmentation and the corresponding stochastic error. We evaluated our PADL framework on two medical image benchmarks with different imaging modalities, which have been annotated by multiple medical professionals, and achieved promising performance on all five medical image segmentation tasks.

We present a method for learning latent stochastic differential equations (SDEs) from high-dimensional time series data. Given a high-dimensional time series generated from a lower dimensional latent unknown It\^o process, the proposed method learns the mapping from ambient to latent space, and the underlying SDE coefficients, through a self-supervised learning approach. Using the framework of variational autoencoders, we consider a conditional generative model for the data based on the Euler-Maruyama approximation of SDE solutions. Furthermore, we use recent results on identifiability of latent variable models to show that the proposed model can recover not only the underlying SDE coefficients, but also the original latent variables, up to an isometry, in the limit of infinite data. We validate the method through several simulated video processing tasks, where the underlying SDE is known, and through real world datasets.

We consider the problem of minimizing a convex function that is evolving according to unknown and possibly stochastic dynamics, which may depend jointly on time and on the decision variable itself. Such problems abound in the machine learning and signal processing literature, under the names of concept drift, stochastic tracking, and performative prediction. We provide novel non-asymptotic convergence guarantees for stochastic algorithms with iterate averaging, focusing on bounds valid both in expectation and with high probability. The efficiency estimates we obtain clearly decouple the contributions of optimization error, gradient noise, and time drift. Notably, we show that the tracking efficiency of the proximal stochastic gradient method depends only logarithmically on the initialization quality, when equipped with a step-decay schedule. Numerical experiments illustrate our results.

The parameter estimation of epidemic data-driven models is a crucial task. In some cases, we can formulate a better model by describing uncertainty with appropriate noise terms. However, because of the limited extent and partial information, (in general) this kind of model leads to intractable likelihoods. Here, we illustrate how a stochastic extension of the SEIR model improves the uncertainty quantification of an overestimated MCMC scheme based on its deterministic model to count reported-confirmed COVID-19 cases in Mexico City. Using a particular mechanism to manage missing data, we developed MLE for some parameters of the stochastic model, which improves the description of variance of the actual data.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.