We define a new formal Riemannian metric on a conformal class in the context of the -Yamabe problem. Our construction leads to a new variational characterization and a new parabolic flow approach to this problem. Moreover, this variational framework suggests that solutions to this problem are unique in…
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We present a new class of solutions for the inverse problem in the calculus of variations in arbitrary dimension . This is the problem of determining the existence and uniqueness of Lagrangians for systems of second order ordinary differential equations. We also provide a number of new theorems concerning the in…
New algorithm for active exploration in bandit problems.
problem Active exploration in bandit problems.
method Gradient ascent with online lazy mirror ascent sampling rule.
result Asymptotically optimal and computationally efficient.
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.
Survey of Bartnik quasi-local mass and new problems.
problem Foundational properties of Bartnik quasi-local mass.
method Survey and formulation of new problems.
result Formulation of new problems and conjectures.
New cylindrical solutions found for Grushin-type problem.
problem Critical Grushin-type problem on CR sphere.
method Local Pohozaev identities for non-degeneracy, Lyapunov-Schmidt reduction for solutions.
result New type of multi-bubbling cylindrical solutions constructed.
New existence results for prescribed mean curvature problem on balls under pinching conditionsmath.DG
New existence results for curvature problem on balls with specific conditions.
problem Existence of solutions for a prescribed mean curvature problem on a ball.
method Combining critical points at infinity approach with Morse theory.
result New existence results for higher dimensional case under pinching conditions.
New statistical biharmonic maps derived from a variation problem.
problem Variation problem for mappings between statistical manifolds.
method Statistical biharmonic maps derived from the Euler-Lagrange equation.
result Improper affine hyperspheres induce examples of statistical biharmonic maps.
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 geometric methods improve optimization and data science problems.
problem Improving optimization and data science problems.
method Geometric tools for high-dimensional optimization and statistical data science.
result New algorithms and statistical guarantees for optimization and data science.
New control theory for self-path-dependent problems solves unique constraints.
problem Optimal control with self-path-dependent constraints in stochastic systems.
method Introduces new HJB equations for variational inequalities with historical maximum controls.
result Value functions are viscosity solutions to HJB equations under Lipschitz conditions.
Gaussian process is a very promising novel technology that has been applied to both the regression problem and the classification problem. While for the regression problem it yields simple exact solutions, this is not the case for the classification problem, because we encounter intractable integrals. In this paper we …
New local method solves Yamabe problems on compact and non-compact manifolds.
problem Yamabe problems on compact and non-compact manifolds.
method Local method for compact and non-compact manifolds.
result Generalizes Brezis and Nirenberg's nonlinear eigenvalue problem to subsets of manifolds.
Orlicz-Minkowski flowsmath.DG
Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.
problem Anisotropic non-homogeneous Gauss curvature flows and Orlicz-Minkowski problems.
method Long-time existence and behavior analysis, parabolic approximation method, curvature flow.
result Existence and new results for Orlicz-Minkowski problems, including versions.
New geometric mechanism solves four envelope problems.
problem Four basic problems on envelopes created by hyperplane families.
method Simple geometric mechanism of intersections of perpendicular bisectors and normal lines.
result Solves all four basic problems on envelopes at once.
We propose a novel reformulation of the stochastic optimal control problem as an approximate inference problem, demonstrating, that such a interpretation leads to new practical methods for the original problem. In particular we characterise a novel class of iterative solutions to the stochastic optimal control problem …
New algorithms solve complex function optimization problems.
problem Optimizing unknown functions in competitive learning models.
method Proposed F-LCB algorithm based on UCB-type methods for nonlinear optimization.
result Regret upper bounds for the F-LCB algorithm derived from base algorithms' convergence rates.
A new presentation of the -string braid group is studied. Using it, a new solution to the word problem in is obtained which retains most of the desirable features of the Garside-Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to t…
ICON learns differential equation operators from prompts, reducing retraining and improving few-shot learning.
problem Training neural networks to solve differential equations without retraining for new problems.
method In-Context Operator Networks (ICON) that learns operators from prompted data and applies them to new problems.
result ICON can generalize to new operators beyond the training distribution and requires only a few demos.
New methods solve inverse problem for Lagrangians in calculus of variations.
problem Determining Lagrangians for systems of differential equations.
method Exterior differential systems theory (EDS) and generalisation of Jesse Douglas's solution.
result Generalised solutions in arbitrary dimension .
New geometric system from Hessian operators offers solutions to geometric problems.
problem Solving geometric problems using Hessian operators.
method Introducing a new differential-geometric system based on -Hessian operators.
result Deduced an a priori -estimate for solutions to the Dirichlet problem for -Hessian equations.
New geometric approach to optimal control theory using Stokes Theorem.
problem Optimal control theory, specifically the Mayer problem.
method Geometric unfolding based on the Stokes Theorem.
result Derives a new necessary and sufficient condition for optimal solutions.
New theorems prove uniqueness of solutions to geometric PDEs.
problem Proving uniqueness of solutions to geometric PDEs.
method Analyzing nonlinear elliptic PDEs of divergence form.
result Proved several Moser-Bernstein type theorems.
New solutions found for a complex boundary problem.
problem One-phase free boundary problem in geometry.
method New connection with minimal surfaces.
result Disproved a conjecture and constructed homogeneous solutions.
New framework tackles geometric structure existence and classification.
problem Existence and classification of geometric structures.
method Developed a new framework of relative algebroids.
result New framework addresses geometric structure problems.
New BE dimension measure reveals rich RL problems with sample-efficient algorithms.
problem Finding sample-efficient algorithms for complex RL problems.
method Introducing Bellman Eluder (BE) dimension and designing GOLF and OLIVE algorithms.
result GOLF and OLIVE algorithms learn near-optimal policies for low BE dimension problems with polynomial samples.
New proof confirms flat equilateral torus is λ1-maximal.
problem Maximizing the first eigenvalue on flat tori.
method Combining El Soufi-Ilias-Ros's method and Bryant's result.
result Positive answer to Berger's isoperimetric problem.
The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and adaptive control. A new point of view is offered for the constrained optimization prob…
New biharmonic Steklov problem on forms yields eigenvalue estimates.
problem Eigenvalue estimates for differential forms with curvature quantities.
method Introduced a new biharmonic Steklov problem and proved existence of a discrete spectrum.
result Established Kuttler-Sigillito inequalities connecting eigenvalues of differential forms.
New heat equation method solves intertwining problems in CR geometry.
problem Intertwining problems in conformal CR geometry.
method Heat equation and extension problems approach.
result New intertwining formulas derived.
A new method for optimizing black-box problems with constraints.
problem Optimizing black-box systems with multiple performance criteria and constraints.
method Developed a novel constrained Bayesian optimization approach based on the knowledge gradient method.
result A new acquisition function that balances optimality and feasibility.
Knowledge Matters: Importance of Prior Information for Optimization [7], by Gulcehre et. al., sought to establish the limits of current black-box, deep learning techniques by posing problems which are difficult to learn without engineering knowledge into the model or training procedure. In our work, we completely solve…
New eigenvalue problem for free boundary minimal surfaces in a ball.
problem Eigenvalue problem for free boundary minimal surfaces in a unit ball.
method New eigenvalue problem associated with the Jacobi operator.
result Eigenfunctions of the unit normal are associated with eigenvalue -2.
New proof of Wulff-Gage inequality with applications.
problem Proving the Wulff-Gage isoperimetric inequality.
method Provided a new proof of the inequality for origin-symmetric convex bodies.
result Uniqueness of log-Minkowski problem and new proof of log-Minkowski inequality.
A new algorithm screens negligible components to efficiently approximate optimal transport distances.
problem Efficiently approximating the Sinkhorn distance between discrete measures.
method Screening of negligible components in the dual solution of the regularized Sinkhorn problem.
result Screenkhorn algorithm provides provable guarantees with smaller computational complexity.
New method solves Cartan's problem for Lie algebroids and groupoids.
problem Classification of extremal Kähler metrics on surfaces.
method New theory of Lie algebroids and groupoids with structure group and connection.
result Local and complete solutions, symmetries, and moduli spaces determined.
New PAC-Bayes bounds for unbounded loss functions.
problem Generalization bounds for learning problems with unbounded loss functions.
method Introducing HYPE, a new notion for loss range, and deriving a novel PAC-Bayesian generalization bound.
result PAC-Bayes framework extended to unbounded loss functions.
New method uses diffusion models to solve inverse problems.
problem Solving ill-posed inverse problems with powerful priors.
method Formulate posterior sampling as a regularized Wasserstein gradient flow in latent space.
result Demonstrates improved performance on standard benchmarks.
A number of important applied problems in engineering, finance and medicine can be formulated as a problem of anomaly detection. A classical approach to the problem is to describe a normal state using a one-class support vector machine. Then to detect anomalies we quantify a distance from a new observation to the const…
In this paper we introduce a new coherent cumulative risk measure on , the space of càdlàg processes having Laplace transform. This new coherent risk measure turns out to be tractable enough within a class of models where the aggregate claims is driven by a spectrally positive Lévy process. Moreover, w…
We present a novel binary convex reformulation of the sparse regression problem that constitutes a new duality perspective. We devise a new cutting plane method and provide evidence that it can solve to provable optimality the sparse regression problem for sample sizes n and number of regressors p in the 100,000s, that…
New Kelvin transform for anisotropic elliptic problems.
problem Semilinear and quasilinear anisotropic elliptic problems.
method Introducing a new Kelvin-type transform in the anisotropic setting.
result New insights into anisotropic elliptic problems.
Paper classifies critical points in half-space with new distance function.
problem Classifying critical points in half-space with capillary CMC hypersurfaces.
method New shifted distance function for capillary problem in half-space.
result Proves Alexandrov-type theorem for singular capillary CMC hypersurfaces.
We propose a portfolio approach for operational risk quantification based on a class of analytical models from which we derive new results on the correlation problem. In particular, we show that uniform correlation is a robust assumption for measuring capital charges in these models.
New approach removes obstructions in gluing spacelike and null hypersurfaces in Einstein equations.
problem Gluing two solutions of the Einstein equations along a hypersurface.
method Active utilization of nonlinearity, low-frequency linear analysis, high-frequency nonlinear control.
result Removes 10-dimensional obstructions in null and spacelike gluing problems.
New neural methods tackle density estimation and likelihood-free inference.
problem Density estimation and likelihood-free inference.
method Neural network-based methods.
result New methods for density estimation and likelihood-free inference.
New framework uses score-based priors to solve ill-conditioned polynomial equations, improving signal recovery from noisy data.
problem Recovering signals from low-order moments in inverse problems, especially ill-conditioned polynomial equations.
method Integrates score-based diffusion priors with moment-based estimators to regularize and solve nonlinear inverse problems.
result Diffusion priors improve recovery from third-order moments and make super-resolution MTD feasible.
New toolkit for directed distances improves flexibility of OT problems.
problem Optimal transport problems with constraints.
method Directed distances between quantile functions.
result Flexibility in solving OT problems enhanced.