arXiv research

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles.

We consider in dimension four weakly convergent sequences of approximate biharmonic maps to a Riemannian manifold with bi-tension fields bounded in $L^p$ for $p>\frac43$. We prove an energy identity that accounts for the loss of hessian energies by the sum of hessian energies over finitely many nontrivial biharmonic ma…

We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.

New sampling and identity-testing methods for mixtures of distributions that don't satisfy approximate tensorization of entropy.

We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in $L^p$ for some $p>1$. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb…

For a sequence of coupled fields $\{(φ_n,ψ_n)\}$ from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up pr…

For the class of approximate harmonic maps $u\in W^{1,2}(Σ,N)$ from a closed Riemmanian surface $(Σ,g)$ to a compact Riemannian manifold $(N, h)$, we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps $\{u_n\}:Σ\to N$, with tension fields $τ(u_n)$ bounded in the Morrey spa…

Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla u_n\|_{L^2(M)}+\|τ(u_n)\|_{L^{p}(M)}\big)\leq Λ\quad \text{for some}\,\,p>1, \eeno where $τ(u_n)$ is the tension field of $u_n$, then there hold the so called ene…

We analyze algorithms for approximating a function $f(x) = Φx$ mapping $\Re^d$ to $\Re^d$ using deep linear neural networks, i.e. that learn a function $h$ parameterized by matrices $Θ_1,...,Θ_L$ and defined by $h(x) = Θ_L Θ_{L-1} ... Θ_1 x$. We focus on algorithms that learn through gradient descent on the population …

The Lugannani-Rice formula is a saddlepoint approximation method for estimating the tail probability distribution function, which was originally studied for the sum of independent identically distributed random variables. Because of its tractability, the formula is now widely used in practical financial engineering as …

Market portfolio decomposed into body and tail legs

The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth and width. These bounds improve upon state-of-the-art bounds for certain classes o…

Inference methods are often formulated as variational approximations: these approximations allow easy evaluation of statistics by marginalization or linear response, but these estimates can be inconsistent. We show that by introducing constraints on covariance, one can ensure consistency of linear response with the var…

Study decomposes market portfolio into body and tail legs, revealing systematic differences.

There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimension…

The paper finds a pervasive and severe bias in accounting semi-identity models.

We demonstrate that a very deep ResNet with stacked modules with one neuron per hidden layer and ReLU activation functions can uniformly approximate any Lebesgue integrable function in $d$ dimensions, i.e. $\ell_1(\mathbb{R}^d)$. Because of the identity mapping inherent to ResNets, our network has alternating layers of…

Mimetic initialization improves Transformer training on small datasets.

The paper proposes using path signatures for better inference in time series data.

Neural approximate computing gains enormous energy-efficiency at the cost of tolerable quality-loss. A neural approximator can map the input data to output while a classifier determines whether the input data are safe to approximate with quality guarantee. However, existing works cannot maximize the invocation of the a…

New method estimates animal density using acoustic data, accounting for unknown call identities.

Smooth approximations near singularities of constant mean curvature surfaces are found.

Let $\{u_n\}$ be a sequence of maps from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold $N$ with free boundary on a smooth submanifold $K\subset N$ satisfying \[ \sup_n \ \left(\|\nabla u_n\|_{L^2(M)}+\|τ(u_n)\|_{L^2(M)}\right)\leq Λ, \] where $τ(u_n)$ is the tension field o…

Maximum likelihood (ML) estimation using Newton's method in nonlinear state space models (SSMs) is a challenging problem due to the analytical intractability of the log-likelihood and its gradient and Hessian. We estimate the gradient and Hessian using Fisher's identity in combination with a smoothing algorithm. We exp…

A new diffusion method approximates Schrödinger bridge with improved convergence.

In this paper we consider approximations introduced by Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in [6] about limits of $α$-harmonic maps with uniformly bounded energy. Using a recent energy identity in [7], we obtain an optimal gap theorem for the $α$-harmonic maps…

This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.

The paper addresses the invariance issue in Bayesian neural networks using linearized Laplace approximation.

The paper analyzes ridge regression with random features for non-identically distributed data.

Establishes a correspondence between two mathematical identities.

This paper speeds up mean curvature computation for high-dimensional data.

The paper analyzes risk estimation methods and derives bounds for OCE risk.

This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.

We tackle the Multi-task Batch Reinforcement Learning problem. Given multiple datasets collected from different tasks, we train a multi-task policy to perform well in unseen tasks sampled from the same distribution. The task identities of the unseen tasks are not provided. To perform well, the policy must infer the tas…

Since their introduction a year ago, distributional approaches to reinforcement learning (distributional RL) have produced strong results relative to the standard approach which models expected values (expected RL). However, aside from convergence guarantees, there have been few theoretical results investigating the re…

Quandle homology was defined from rack homology as the quotient by a subcomplex corresponding to the idempotency, for invariance under the type I Reidemeister move. Similar subcomplexes have been considered for various identities of racks and moves on diagrams. We observe common aspects of these identities and subcompl…

The Collective Graphical Model (CGM) models a population of independent and identically distributed individuals when only collective statistics (i.e., counts of individuals) are observed. Exact inference in CGMs is intractable, and previous work has explored Markov Chain Monte Carlo (MCMC) and MAP approximations for le…

The paper derives curvature identities for 5D and 6D Einstein manifolds.

Global Pestov identity proved on frame bundle and related fibrations.

Current approaches in approximate inference for Bayesian neural networks minimise the Kullback-Leibler divergence to approximate the true posterior over the weights. However, this approximation is without knowledge of the final application, and therefore cannot guarantee optimal predictions for a given task. To make mo…

Proves Bochner's identity on graphs using a new auxiliary graph.

Doodles link to commutator identities in a 2-sphere.

Duality restored in gauge theory, gravity, and string theory models.

The paper studies harmonic identity maps on Riemannian manifolds.

We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. We define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals, derive a lower and an upper bound on the fractional Beth…

Efficiently distills pretrained text-to-image models without real data, improving FID and CLIP scores.

User identity linkage is a task of recognizing the identities of the same user across different social networks (SN). Previous works tackle this problem via estimating the pairwise similarity between identities from different SN, predicting the label of identity pairs or selecting the most relevant identity pair based …

The importance of Einstein's geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar …