Explains eigenvalue and generalized eigenvalue problems with examples.
problem Eigenvalue and generalized eigenvalue problems.
method Introduction and examples from machine learning.
result Solutions to eigenvalue and generalized eigenvalue problems.
A deep learning approach generates math word problems in multiple languages.
problem Template-based mechanisms for generating mathematical word problems lack customizability and creativity.
method Character Level Long Short Term Memory Network (LSTM) and POS tags are used to generate and resolve constraints in generated problems.
result The approach generates accurate math word problems in English and Sinhala with over 90% accuracy.
New method solves generalized Minkowski problem for torsional rigidity.
problem Generalized Minkowski problem for torsional rigidity.
method Flow method
result Existence of solutions for general measures.
Solves a generalized dual Minkowski problem for specific values of q.
problem Finding solutions to the generalized dual Minkowski problem for given q and star bodies.
method Variational methods
result Existence of solutions for q<0 and 0≤q≤1, sufficient condition for q>1.
Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.
problem Characterize measures generated by electrostatic p-capacity.
method Solves the discrete logarithmic Minkowski problem for 1 < p < n.
result Solves the discrete logarithmic Minkowski problem for measures in general position.
Paper solves isomorphism problem for specific Baumslag-Solitar groups.
problem Isomorphism problem for small rose non-ascending generalized Baumslag-Solitar groups.
method Analyzed group actions on trees with specific stabilizers.
result Isomorphism problem solvable for the specified groups.
Study investigates OOD generalization methods for mechanics problems.
problem Real-world mechanics problems with unknown test environments and data distribution shifts.
method Investigates OOD generalization methods for regression problems in mechanics.
result OOD generalization methods perform better than traditional ML methods on mechanics-specific regression problems.
Generative models improve inverse problems by providing tailored priors.
problem Analyzing the error in inverse problems solved with generative priors.
method Quantitative error bounds for minimum Wasserstein-2 generative models.
result The error in the posterior due to the generative prior is bounded by the prior's error in Wasserstein-1 distance.
Solves characteristic problem in general relativity for null data.
problem Characteristic Cauchy problem in Einstein vacuum field equations.
method Abstract data formalism and tangential components of the ambient Ricci tensor.
result Formulates and solves the characteristic problem completely abstractly.
The so-called inverse problem of dynamics is about constructing a potential for a given family of curves. We observe that there is a more general way of posing the problem by making use of ideas of another inverse problem, namely the inverse problem of the calculus of variations. We critically review and clarify differ…
The paper solves a specific type of curvature problem on curved surfaces.
problem Dirichlet problem of translating mean curvature equations over domains in Riemannian manifolds.
method Defined a new conformal area functional and generalized solution theory to prove existence.
result Existence of generalized solutions under certain conditions, including smooth solutions for mean convex domains.
The paper explores the geometry of level lines of quasiperiodic functions with many periods.
problem Describing the geometry of level lines of quasi-periodic functions with a large number of periods.
method Generalizes the Novikov problem to the multidimensional case of quasiperiodic functions.
result Arises of open or closed level lines of arbitrarily large sizes.
Solves portfolio optimization with cardinality constraints using column generation.
problem Portfolio optimization with cardinality constraints.
method Column generation method applied to a subset of assets in a master convex quadratic problem, using dual information to propose new assets.
result Solves portfolio optimization problems efficiently with cardinality constraints.
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. Machine learning reduces combinatorial optimization problem dimensions.
problem Reducing the complexity of large combinatorial optimization problems.
method Generalization of a machine learning model for problem reduction on TSP.
result Machine learning can predict which variables are not part of an optimal solution.
Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.
problem Existence and uniqueness of solutions to the Dirichlet problem.
method Formulated using subharmonic functions; generalizes Hitchin's equation for diagonal harmonic metrics on cyclic Higgs bundles.
result Existence and uniqueness of solutions to the Dirichlet problem.
Generalizes Hamiltonian theory for variational problems, applied to first order gravity.
problem Formulating Hamiltonian field theory for variational problems of general nature.
method Introduces a generalized Hamiltonian formalism without requiring a Hamiltonian section.
result Develops a novel multisymplectic Hamiltonian field theory for first order gravity.
Extending a previous paper, we present a generalization in dimension 3 of the traditional Szebehely-type inverse problem. In that traditional setting, the data are curves determined as the intersection of two families of surfaces, and the problem is to find a potential V such that the Lagrangian L = T - V, where T is t…
Diffusion models tackle noisy inverse problems with posterior sampling.
problem Efficiently solving general noisy inverse problems.
method Approximation of posterior sampling for diffusion models.
result Diffusion models can handle various noise statistics and nonlinear problems.
Study of learning-based inverse problems using GANs.
problem Solving inverse problems with learned priors.
method Developed a simple non-convex algorithm for certain GAN architectures.
result The approach achieves linear convergence guarantees and improves upon conventional techniques.
New method solves a generalized Minkowski problem using a curvature flow.
problem Generalized Minkowski problem for smooth measures.
method Flow involving Gauss curvature and support function.
result Existence of solutions for the dual Orlicz-Minkowski problem.
New algorithms solve inverse problems using deep learning, converging faster than traditional methods.
problem Solving inverse problems with deep learning models.
method Simple non-convex algorithm for linear and nonlinear inverse problems, with theoretical and empirical support.
result The proposed algorithms converge faster than conventional techniques for certain inverse problems.
New method uses generative models to improve phase retrieval stability.
problem Improving stability of solutions in phase retrieval problems.
method Unified reconstruction approach using generative models to mitigate overfitting.
result Mitigates overfitting to generative model for varying noise levels.
New research shows deep learning struggles with hard problems due to biased data generation.
problem Deep learning's limitations in solving computationally hard problems.
method Proved that polynomial-time sample generators for NP-hard problems sample from easier sub-problems.
result Machine learning models trained on biased datasets overestimate their accuracy for hard problems.
Study inverse problems for twisted geodesic flows on manifolds.
problem Understanding inverse problems for twisted geodesic flows.
method Generalized ray transforms and tensor tomography.
result New insights into rigidity problems for twisted geodesic flows.
We give an updated extended survey of results related to the celebrated unsolved generalized R. L. Moore problem. In particular, we address the problem of characterizing codimension one manifold factors, i.e. spaces X having the property that X×R is a topological manifold. A main part of the paper i…
Generalizes Aubin's result for Yamabe-type problem on smooth metric measure spaces.
problem Solving Yamabe-type problem on smooth metric measure spaces.
method Generalization of Aubin's result for nonlocally conformally flat manifolds with dimension ≥ 6 and parameter m close to nonnegative integers.
result Generalization of Aubin's result for Yamabe-type problem.
Researchers solve Yamabe problems for specific operators, finding both uniqueness and nonuniqueness.
problem Prescribing scalar, Q-, or σ₂-curvatures in conformal classes.
method Formally self-adjoint, conformally covariant, polydifferential operators.
result Uniqueness results on the sphere, nonuniqueness in general.
Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.
problem Finding a function that attains the Escobar weighted constant.
method Introducing Escobar quotient, infimum, and resolving the problem when the weighted constant is negative.
result Obtained an Aubin type inequality connecting weighted Escobar constant and optimal constant for trace inequality.
A new method uses DMs as priors for imaging problems, offering more accurate reconstructions.
problem Accurate probabilistic imaging for complex inverse problems.
method Markov chain Monte Carlo algorithm using DMs as plug-and-play priors for solving Bayesian inverse problems.
result Offers more accurate reconstructions and posterior estimation compared to existing methods.
Smooth even solutions found for a generalized convex geometry problem.
problem Dual Orlicz-Minkowski problem in convex geometry.
method Geometric flow involving Gauss curvature and normal vectors.
result Existence of smooth even solutions for smooth even measures.
Study the monodromy and center-focus problems for rational maps defined by products of generic lines.
problem Monodromy and center-focus problems for rational maps defined by products of generic lines.
method Analyze the 1-homology group and meromorphic 1-forms to characterize vanishing Abelian integrals.
result Characterize meromorphic 1-forms whose Abelian integrals vanish on cycles around a center singularity.
Although much research has been devoted to extremal problems on non-overlapping domains little is known about all solutions of this problems. We generalized some of this problems on the case of more general systems of points. It was solved using separating transformations and learning functions in detail. Methods used …
Deep learning struggles with out-of-distribution data, so this paper tackles domain generalization.
problem Deep learning models fail with out-of-distribution data.
method Formulates domain generalization as a constrained statistical learning problem, then uses nonconvex duality theory to develop an algorithm with convergence guarantees.
result Improves domain generalization by up to 30 percentage points on various benchmarks.
Paper addresses statistical inference for GANs and minimax problems.
problem Statistical properties of GANs and minimax problems.
method Consistent estimation and confidence sets for GAN parameters.
result Confidence sets for GAN parameters contain the population solutions with desired coverage probability.
We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical stock prices. For a non-linear problem with a general performance criterion, th…
We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on ``generic-case complexity'' we show that if a finitely generated group G has the word problem solvable in subexponential time and has a subgroup of…
Paper improves risk bounds for nonconvex-strongly-concave minimax problems.
problem Achieving sharper risk bounds for nonconvex-strongly-concave minimax problems.
method Using uniform localized convergence to derive high probability generalization error bounds.
result Derives n times faster excess primal risk bounds for popular algorithms.
Improves generalization in learning problems with small parameter method.
problem Improving generalization in learning problems with high-dimensional nonlinear functions.
method Perturbation theory applied to a weakly-controlled gradient system.
result Approximate optimal solutions for improving generalization with small noise.
Learning rate annealing helps even in convex problems, improving generalization.
problem Improving generalization in neural networks, especially convex problems.
method Learning rate annealing schedule (large initial, then small learning rate).
result Gradient descent can reach minima with better generalization using learning rate annealing.
Proposes a faster algorithm for machine learning problems.
problem General minimum conical hull problems in machine learning.
method Sublinear classical algorithm for general minimum conical hull problems.
result Achieves exponential speedup over existing methods.
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
Paper introduces a new learner for generalizing complex tasks.
problem Generalizing to new, complex tasks without prior experience.
method Compositional problem graph and compositional recursive learner.
result Compositional approach can generalize to more complex problems than non-compositional learners.
New method for RL with general utilities using variational policy gradient.
problem Optimizing policies with general concave utility functions in RL.
method Derives Variational Policy Gradient Theorem, develops variational Monte Carlo gradient estimation algorithm.
result Global convergence to optimal policy for general objectives, exponential convergence under strong convexity.
Study problem-dependent rates in statistical learning theory, achieving optimal generalization error bounds.
problem Generalization error in statistical learning theory.
method Uniform localized convergence framework.
result Optimal generalization error bounds for various learning problems.
Study MinMax methods for optimization problems, including optimal transport.
problem Optimization problems, especially optimal transport.
method MinMax framework, regularization, neural networks, approximation theorems.
result Justification of neural networks for solving optimization problems.
We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for L…
Locally, isoperimetric problems on Riemannian surfaces are sub-Riemannian problems in dimension 3. The particular case of Dido problems corresponds to a class of singular contact sub-Riemannian metrics : metrics which have the charateristic vertor field as symmetry. We give a classification of the generic conjugate loc…