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Recent studies show that transformer-based architectures emulate gradient descent during a forward pass, contributing to in-context learning capabilities - an ability where the model adapts to new tasks based on a sequence of prompt…
Image denoising and demosaicking are the most important early stages in digital camera pipelines. They constitute a severely ill-posed problem that aims at reconstructing a full color image from a noisy color filter array (CFA) image. In…
We prove that the crossing number of a graph decays in a continuous fashion in the following sense. For any epsilon>0 there is a delta>0 such that for a sufficiently large n, every graph G with n vertices and m > n^{1+epsilon} edges, has a…
We study an infinite family of one-parameter deformations, so-called $\alpha$-continued fractions, of interval maps associated to distinct triangle Fuchsian groups. In general for such one-parameter deformations, the function giving the…
We consider gradient descent with constant stepsizes and derive exact worst-case convergence rates on the minimum gradient norm of the iterates. Our analysis covers all possible stepsizes and arbitrary upper/lower bounds on the curvature of…
The grokking phenomenon as reported by Power et al. ( arXiv:2201.02177 ) refers to a regime where a long period of overfitting is followed by a seemingly sudden transition to perfect generalization. In this paper, we attempt to reveal the…
In this article, we prove that, on the diffusive time scale, condensing zero-range processes converge to a dimension-decaying diffusion process on the simplex \[ \Sigma = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in S} x_i = 1\}, \] where…
Artifact detection algorithms are crucial to correcting the output generated by diffusion models. However, because of the variety of artifact forms, existing methods require substantial annotated data for training. This requirement limits…
An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This…
We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also…
We investigate a descent on simple graphs, starting with the complete graph on $n$ vertices and ending up with the cycle graph by removing one edge after another. We obtain quantitative results showing that graphs with large diameter must…
We derive a minimalist but powerful deterministic denoising-diffusion model. While denoising diffusion has shown great success in many domains, its underlying theory remains largely inaccessible to non-expert users. Indeed, an understanding…
We investigate the scaling behaviour of a singular perturbation model within the geometrically linearized theory of elasticity involving data of higher lamination order. We study boundary data which are of staircase type and show rather…
We revisit the classical problem of denoising a one-dimensional scalar-valued function by minimizing the sum of an $L^2$ fidelity term and the total variation, scaled by a regularization parameter. This study focuses on proving that the…
In this article, we prove some total variation inequalities for maximal functions. Our results deal with two possible generalizations of the results contained in Aldaz and P\'erez L\'azaro's work, one of whose considers a variable…
We study the maximum matching problem in fully dynamic graphs: a graph is undergoing both edge insertions and deletions, and the goal is to efficiently maintain a large matching after each edge update. This problem has received considerable…
The advancement of imaging devices and countless images generated everyday pose an increasingly high demand on image denoising, which still remains a challenging task in terms of both effectiveness and efficiency. To improve denoising…
This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set $S \subset {\mathbb R}^d$. The general aim is to identify (via stochastic…
Diffusion models have transformed image synthesis through iterative denoising, by defining trajectories from noise to coherent data. While their capabilities are widely celebrated, a critical challenge remains unaddressed: ensuring…
It has long been recognized that emission of radiation from atoms is not an intrinsic property of individual atoms themselves, but it is largely affected by the characteristics of the photonic environment and by the collective interaction…