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We show that Stolarsky's invariance principle, known for point distributions on the Euclidean spheres, can be extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. A part of the results…
We prove the long-standing inverse scattering theory (IST) of perturbed Kadomtsev Petviashvili multi-line solitons. Our work is the first rigorous IST of a multi-dimensional integrable system when both continuous and discrete scattering…
Stability is a key aspect of data analysis. In many applications, the natural notion of stability is geometric, as illustrated for example in computer vision. Scattering transforms construct deep convolutional representations which are…
The dynamics of steps on crystal surfaces is considered. In general, the meandering of the steps obeys a subdiffusive behaviour. The characteristic asymptotic time laws depend on the microscopic mechanism for detachment and attachment of…
Diffusion-based generative models have recently emerged as powerful solutions for high-quality synthesis in multiple domains. Leveraging the bidirectional Markov chains, diffusion probabilistic models generate samples by inferring the…
In this paper the author studies the problem of the homogenization of a diffusion perturbed by a periodic reflection invariant vector field. The vector field is assumed to have fixed direction but varying amplitude. The existence of a…
This paper improves a previously established test involving only coefficients to decide a priori whether or not non-trivial symmetries of a large class of space-time dependent diffusion processes on the real line exist. When the existence…
We consider $\mathbb{R}^d$-valued diffusion processes of type \begin{align*} dX_t\ =\ b(X_t)dt\, +\, dB_t. \end{align*} Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich ($L^1$…
Sampling from diffusion probabilistic models (DPMs) can be viewed as a piecewise distribution transformation, which generally requires hundreds or thousands of steps of the inverse diffusion trajectory to get a high-quality image. Recent…
Subordinate diffusions are constructed by time changing diffusion processes with an independent L\'{e}vy subordinator. This is a rich family of Markovian jump processes which exhibit a variety of jump behavior and have found many…
Face recognition models embed a face image into a low-dimensional identity vector containing abstract encodings of identity-specific facial features that allow individuals to be distinguished from one another. We tackle the challenging task…
Diffusion-driven patterns appear on curved surfaces in many settings, initiated by unstable modes of an underlying Laplacian operator. On a flat surface or perfect sphere, the patterns are degenerate, reflecting translational/rotational…
The change of the effective dimension of spacetime with the probed scale is a universal phenomenon shared by independent models of quantum gravity. Using tools of probability theory and multifractal geometry, we show how dimensional flow is…
We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL$\cdot$E…
The inverse diffusion curve problem focuses on automatic creation of diffusion curve images that resemble user provided color fields. This problem is challenging since the 1D curves have a nonlinear and global impact on resulting color…
Let $X_t$ be a reversible and positive recurrent diffusion in $R^d$ described by \begin{equation}\nonumber X_t=x+\sigma b(t)+\int_0^tm(X_s)\dif s, \end{equation} where the diffusion coefficient $\sigma$ is a positive-definite matrix and the…
Diffusion processes are instrumental to describe the movement of a continuous quantity in a generic network of interacting agents. Here, we present a probabilistic framework for diffusion in networks and propose to classify agent…
Diffusion models, the most popular generative paradigm so far, can inject conditional information into the generation path to guide the latent towards desired directions. However, existing text-to-image diffusion models often fail to…
Let S be a closed orientable surface of genus at least two, and let C be an arbitrary (complex) projective structure on S. We show that there is a decomposition of S into pairs of pants and cylinders such that the restriction of C to each…
We consider a multidimensional reaction-diffusion equation of either ignition or monostable type, involving periodic heterogeneity, and analyze the dependence of the propagation phenomena on the direction. We prove that the (minimal) speed…