相关论文: On the convergence of maximum variance unfolding

This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a…

The high energy physics unfolding problem is an important statistical inverse problem in data analysis at the Large Hadron Collider (LHC) at CERN. The goal of unfolding is to make nonparametric inferences about a particle spectrum from…

The Muon optimizer has demonstrated remarkable empirical success in handling matrix-structured parameters for training neural networks. However, a significant gap remains between its practical performance and theoretical understanding.…

Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon…

Consensus on nonlinear spaces is of use in many control applications. This paper proposes a gradient descent flow algorithm for consensus on hypersurfaces. We show that if an inequality holds, then the system converges for almost all…

Although overparameterized models have achieved remarkable practical success, their theoretical properties, particularly their generalization behavior, remain incompletely understood. The well known double descents phenomenon suggests that…

In this paper, we produce explicit examples of mean curvature flow of (2m-1)-dimensional submanifolds which converge to (2m-2)-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in $\mathbb{C}^{m}$…

Optimization techniques are at the core of many scientific and engineering disciplines. The steepest descent methods play a foundational role in this area. In this paper we studied a generalized steepest descent method on Riemannian…

We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We…

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove…

Three well-studied types of subgraph-restricted matchings are induced matchings, uniquely restricted matchings, and acyclic matchings. While it is hard to determine the maximum size of a matching of each of these types, whether some given…

We analyze recurrent neural networks with diagonal hidden-to-hidden weight matrices, trained with gradient descent in the supervised learning setting, and prove that gradient descent can achieve optimality \emph{without} massive…

In many experimental contexts, it is necessary to statistically remove the impact of instrumental effects in order to physically interpret measurements. This task has been extensively studied in particle physics, where the deconvolution…

We study the asymptotic behavior of a class of methods for sufficient dimension reduction in high-dimension regressions, as the sample size and number of predictors grow in various alignments. It is demonstrated that these methods are…

Many tasks in machine learning and signal processing can be solved by minimizing a convex function of a measure. This includes sparse spikes deconvolution or training a neural network with a single hidden layer. For these problems, we study…

Supermodeling is a modern, model-ensembling paradigm that integrates several self-synchronized imperfect sub-models by controlling a few meta-parameters to generate more accurate predictions of complex systems' dynamics. Continual…

Multivariate extreme-value analysis is concerned with the extremes in a multivariate random sample, that is, points of which at least some components have exceptionally large values. Mathematical theory suggests the use of max-stable models…

In this paper, the maximum spacing method is considered for multivariate observations. Nearest neighbour balls are used as a multidimensional analogue to univariate spacings. A class of information-type measures is used to generalize the…

It has previously been hypothesized, and supported with some experimental evidence, that deeper representations, when well trained, tend to do a better job at disentangling the underlying factors of variation. We study the following related…

The total variation distance is a core statistical distance between probability measures that satisfies the metric axioms, with value always falling in $[0,1]$. This distance plays a fundamental role in machine learning and signal…