Communications on Pure and Applied Mathematics最新文献

We prove $�\epsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya–Tonegawa. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $��\frac{\pi }{2}�$, then it coincides with a $�C^{�1�,�\alpha �}$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than 1.

We study gradient flow on the multi-index regression problem for high-dimensional Gaussian data. Multi-index functions consist of a composition of an unknown low-rank linear projection and an arbitrary unknown, low-dimensional link function. As such, they constitute a natural template for feature learning in neural networks. We consider a two-timescale algorithm, whereby the low-dimensional link function is learnt with a non-parametric model infinitely faster than the subspace parametrizing the low-rank projection. By appropriately exploiting the matrix semigroup structure arising over the subspace correlation matrices, we establish global convergence of the resulting Grassmannian gradient flow dynamics, and provide a quantitative description of its associated “saddle-to-saddle” dynamics. Notably, the timescales associated with each saddle can be explicitly characterized in terms of an appropriate Hermite decomposition of the target link function.

We consider the $��3�d�$ energy critical nonlinear Schrödinger equation with data distributed according to the Gaussian measure with covariance operator $�{�(�1�-�\Delta �)�}^{�-�s�}$, where $�\Delta$ is the Laplace operator and $�s$ is sufficiently large. We prove that the flow sends full measure sets to full measure sets. We also discuss some simple applications. This extends a previous result by Planchon-Visciglia and the second author from $��1�d�$ to higher dimensions.

In this article we prove the orbifold version of the Bogomolov–Gieseker inequality for stable $�Q$-sheaves on log terminal Kähler threefolds.

We prove a simple identity relating the length spectrum of a Riemann surface to that of the same surface with an arbitrary number of additional cusps. Our proof uses the Brownian loop measure introduced by Lawler and Werner. In particular, we express the total mass of Brownian loops in a fixed free homotopy class on any Riemann surface in terms of the length of the geodesic representative for the complete constant curvature metric. This expression also allows us to write the electrical thickness of a compact set in $�C$ separating 0 and $�\infty$, or the Velling–Kirillov Kähler potential, in terms of the Brownian loop measure and the zeta-regularized determinant of Laplacian as a renormalization of the Brownian loop measure with respect to the length spectrum.

We study the convergence of gradient flow for the training of deep neural networks. While residual neural networks (ResNet) are a popular example of very deep architectures, their training constitutes a challenging optimization problem, notably due to the non-convexity and the non-coercivity of the objective. Yet, in applications, such tasks are successfully solved by simple optimization algorithms such as gradient descent. To better understand this phenomenon, we focus here on a “mean-field” model of an infinitely deep and arbitrarily wide ResNet, parameterized by probability measures on the product set of layers and parameters, and with constant marginal on the set of layers. Indeed, in the case of shallow neural networks, mean field models have been proven to benefit from simplified loss landscapes and good theoretical guarantees when trained with gradient flow w.r.t. the Wasserstein metric on the set of probability measures. Motivated by this approach, we propose to train our model with gradient flow w.r.t. the conditional optimal transport (COT) distance: a restriction of the classical Wasserstein distance which enforces our marginal condition. Relying on the theory of gradient flows in metric spaces, we first show the well-posedness of the gradient flow equation and its consistency with the training of ResNets at finite width. Performing a local Polyak–Łojasiewicz analysis, we then show convergence of the gradient flow for well-chosen initializations: if the number of features is finite but sufficiently large and the risk is sufficiently small at initialization, the gradient flow converges to a global minimizer. This is the first result of this type for infinitely deep and arbitrarily wide ResNets. In addition, this work is an opportunity to study in more detail the COT metric, particularly its dynamic formulation. Some of our results in this direction might be interesting on their own.