相关论文: A study in quantitative equidistribution on the un…
We show the existence of a limiting distribution $\cD_\cC$ of the adequately normalized discrepancy function of a random translation on a torus relative to a strictly convex set $\cC$. Using a correspondence between the small divisors in…
In this paper we investigate the distribution of the set of values of a linear map at integer points on a quadratic surface. In particular, it is shown that subject to certain algebraic conditions, this set is equidistributed. This can be…
We prove that the convex least squares estimator (LSE) attains a $n^{-1/2}$ pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process.…
In large-eddy simulations (LES) a computational-domain translation velocity can be used to improve performance by allowing longer time-step intervals. The continuous equations are Galilean invariant, however, standard…
We study the quantitative convergence of drift-diffusion PDEs that arise as Wasserstein gradient flows of linearly convex functions over the space of probability measures on ${\mathbb R}^d$. In this setting, the objective is in general not…
A computational domain translation velocity is often used in LES simulations to improve computational performance by allowing longer time-step intervals. Even though the equations of motion are Galilean invariant, LES results have been…
In the unit tangent bundle of noncompact finite volume negatively curved Riemannian manifolds, we prove the equidistribution towards the measure of maximal entropy for the geodesic flow of the Lebesgue measure along the divergent geodesic…
A linear flow on the torus $\mathbb{R}^d / \mathbb{Z}^d$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $\mathbb{Q}$. In this paper we combine Fourier analysis and the…
Recently, flow-based generative models have shown superior efficiency compared to diffusion models. In this paper, we study rectified flow models, which constrain transport trajectories to be linear from the base distribution to the data…
Recent research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the $\ell_1$ minimization method for…
We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/\Gamma$ of a semisimple algebraic group $G$. We define two families of algebraic…
Let $A^-$ and $A^+$ be properly immersed closed locally convex subsets of a Riemannian manifold $M$ with pinched negative sectional curvature. When the Bowen-Margulis measure on $T^1M$ is finite and mixing for the geodesic flow, we prove…
We develop an analytical framework for understanding how the generated distribution evolves during diffusion model training. Leveraging a Gaussian-equivalence principle, we solve the full-batch gradient-flow dynamics of linear and…
We study the approximability of general convex sets in $\mathbb{R}^n$ by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution $N(0,I_n)$ and the complexity of an…
Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these…
In this short note, we study the distribution of spreads in a point set $\mathcal{P} \subseteq \mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $\varepsilon > 0$, if $|\mathcal{P}|…
Convex splitting is a powerful technique in quantum information theory used in proving the achievability of numerous information-processing protocols such as quantum state redistribution and quantum network channel coding. In this work, we…
We consider the rate of convergence of the expected loss of empirically optimal vector quantizers. Earlier results show that the mean-squared expected distortion for any fixed distribution supported on a bounded set and satisfying some…
For uniform scalar quantization, the error distribution is approximately a uniform distribution over an interval (which is also a 1-dimensional ball). Nevertheless, for lattice vector quantization, the error distribution is uniform not over…
The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterisation of when one Beta distribution is smaller than another…