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Assume that an unknown integral operator living in some known subspace is observed indirectly, by evaluating its action on a discrete measure containing a few isolated Dirac masses at an unknown location. Is this information enough to recover the impulse response location and the operator with a sub-pixel accuracy? We study this question and bring to light key geometrical quantities for exact and stable recovery. We also propose an in-depth study of the presence of additive white Gaussian noise. We illustrate the well-foundedness of this theory on the challenging optical imaging problem of blind deconvolution and blind deblurring with non-stationary operators. [ABSTRACT FROM AUTHOR]

We study representations of data from an arbitrary metric space X in the space of univariate Gaussian mixtures equipped with a transport metric (Delon and Desolneux 2020). We prove embedding guarantees for feature maps implemented by small neural networks called probabilistic transformers. Our guarantees are of memorization type: we prove that a probabilistic transformer of depth about n log(n) and width about n2 can bi-Hölder embed any n-point dataset from X with low metric distortion, thus avoiding the curse of dimensionality. We further derive probabilistic bi-Lipschitz guarantees, which trade off the amount of distortion and the probability that a randomly chosen pair of points embeds with that distortion. If the geometry of X is sufficiently regular, we obtain stronger bi-Lipschitz guarantees for all points. As applications, we derive neural embedding guarantees for datasets from Riemannian manifolds, metric trees, and certain types of combinatorial graphs. When instead embedding into multivariate Gaussian mixtures, we show that probabilistic transformers compute bi-Hölder embeddings with arbitrarily small distortion. Our results show that any finite metric dataset, from vertices on a graph to functions a function space, can be faithfully represented in a single representation space, and that the representation can be implemented by a simple transformer architecture. Thus one may only need a modular set of machine learning tools compatible with this one representation space, many of which already exist, for downstream supervised and unsupervised learning from a great variety of data types. [ABSTRACT FROM AUTHOR]

The Graetz problem is a convection-diffusion equation in a pipe invariant along a direction. The contribution of the present work is to propose a mathematical analysis of the Neumann, Robin, and periodic boundary condition on the boundary of a semi-infinite pipe. The solution in the 3D space of the original problem is reduced to eigenproblems in the 2D section of the pipe. The set of solutions is described, its structure depending on the type of boundary condition and on the sign of the total flow of the fluid. This analysis is the cornerstone of numerical methods to solve the Graetz problem in finite pipes, semi-infinite pipes, and exchangers of arbitrary cross-section. Numerical test cases illustrate the capabilities of these methods to provide solutions in various configurations. [ABSTRACT FROM AUTHOR]