New formulations for Ricci flows without smoothness.
problem Characterize Ricci flows without smooth solutions.
method Weak formulations of super Ricci flows with saturation condition.
result Generalized formulations for singular settings.
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48 results for “Weak Formulation”
New approach finds solutions to games with unbounded controls.
problem Existence of equilibrium in mean-field games with unbounded controls.
method Weak formulation and new existence/stability results for quadratic-growth generalized McKean-Vlasov BSDEs.
result Existence of equilibrium result for non-Markovian mean-field games with unbounded control space.
We propose a weak formulation for the binormal curvature flow of curves in This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence t…
Regression problems assume every instance is annotated (labeled) with a real value, a form of annotation we call \emph{strong guidance}. In order for these annotations to be accurate, they must be the result of a precise experiment or measurement. However, in some cases additional \emph{weak guidance} might be given by…
Centralized systemic risk control in the interbank system: Weak formulation and Gamma-convergencemath.OC
Paper studies central bank's strategy to control systemic risk in interbank system.
problem Minimizing average distance between log-monetary reserves and target levels.
method Weak formulation, Ekeland's variational principle, Gamma-convergence, stochastic Fokker-Planck-Kolmogorov equation.
result Proves convergence of optimal strategies as number of banks increases.
This paper studies the Glosten Milgrom model whose risky asset value admits an arbitrary discrete distribution. Contrast to existing results on insider's models, the insider's optimal strategy in this model, if exists, is not of feedback type. Therefore a weak formulation of equilibrium is proposed. In this weak formul…
Proves inextendibility of weak null singularities from curvature blow-up.
problem Inextendibility of weak null singularities in the context of curvature blow-up.
method Introduces a new strategy to infer -inextendibility from curvature blow-up.
result Expected to contribute to the resolution of strong cosmic censorship conjecture.
Compensated Compactness in Banach Spaces and Weak Rigidity of Isometric Immersions of Manifoldsmath.DG
We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is ob…
Develops weak formulation for spacelike flows in pseudo-Euclidean space.
problem Weak formulation of spacelike mean curvature flow in pseudo-Euclidean space.
method Based on spacelike integer rectifiable varifolds and pseudo-Euclidean first variation.
result Existence and compactness of spacelike Brakke flows with fixed boundary.
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagr…
We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the …
The Parametric Willmore Flowmath.DG
Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.
problem Existence time of the Willmore flow for various initial conditions.
method Established minimal existence time for Willmore flow using geometric data and conservation laws.
result Minimal existence time is a function of geometric data for general weak Lipschitz initial data.
ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spacesstat.ML
ROCK method generalizes MOCK for learning dynamical systems efficiently.
problem Learning dynamical systems from data efficiently.
method Variational formulation in Reproducing Kernel Hilbert Spaces.
result ROCK method is more computationally efficient and performs better on benchmarks.
WSINDy for PDEs robustly identifies models from noisy data.
problem Identifying nonlinear dynamics from noisy partial differential equations data.
method Weak formulation of PDEs, Fourier-based model identification, sequential-thresholding least-squares.
result WSINDy enables robust identification of PDEs in noisy conditions.
NOT learns optimal transport plans, kernel costs improve performance.
problem NOT algorithm learns non-optimal plans with weak quadratic costs.
method Introduced kernel weak quadratic costs to improve NOT's performance.
result Kernel costs provide improved theoretical and practical guarantees.
We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian m…
Unified approach for learning with weak labels across various tasks.
problem Learning with noisy or incomplete labels in diverse machine learning settings.
method Implicit posterior models for joint label inference.
result Unified training objective for various machine learning tasks.
We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension . We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also…
We study a geometric flow where the motion of a set is driven by the mean curvature of its boundary and the normal derivative of its capacity potential. We establish local well-posedness and propose two possible weak formulations that exist after singularities.
We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of -regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for -metrics, an…
New conic quadratic formulations improve outlier detection in regression models.
problem Detecting outliers in regression models with corrupted data.
method Deriving stronger second-order conic relaxations without big-M constraints.
result Proposed formulations are significantly faster than existing methods.
Study on test risk dynamics in learning theory with stochastic gradient flow.
problem Understanding test risk in stochastic gradient flow dynamics.
method Path integral formulation for small learning rates, explicit computation for weak features.
result Explicit corrections due to stochastic term in dynamics, good agreement with simulations.
Paper develops equivariant basic cohomology for Lie groupoids.
problem Equivariant cohomology for Lie groupoids with weak actions.
method Using Kan fibrations and fiber structures, constructing models and comparing with existing theories.
result Equivariant basic cohomology theory for orbifolds and Lie groupoids.
The paper studies surfaces in a bounded domain with orthogonal boundaries and proves curvature estimates.
problem Estimating the area of surfaces with orthogonal boundaries in a bounded domain.
method Weak formulation of orthogonality for curvature varifolds, classification of vanishing curvature varifolds.
result Existence of an orthogonal 2-varifold that minimizes curvature in the integer rectifiable class.
Sampling and Reconstruction of Graph Signals via Weak Submodularity and Semidefinite Relaxationstat.ML
We study the problem of sampling a bandlimited graph signal in the presence of noise, where the objective is to select a node subset of prescribed cardinality that minimizes the signal reconstruction mean squared error (MSE). To that end, we formulate the task at hand as the minimization of MSE subject to binary constr…
A geometric interpretation is given for certain elliptic-hyperbolic systems in the plane. Among several examples, one which reduces in the elliptic region to the equations for harmonic 1-forms on the projective disc is studied in detail. A boundary-value problem for this example is formulated and is shown to possess we…
WNVI solves inverse problems without forward models using neural networks.
problem Solving high-dimensional Bayesian inverse problems based on PDEs.
method WNVI uses weighted residuals and SVI with neural networks to infer state variables and unknowns.
result WNVI is more accurate and efficient than traditional methods and handles ill-posed problems.
New varifold solutions for mean curvature flow converge and are unique.
problem Mean curvature flow and Allen-Cahn equation convergence and uniqueness.
method Evolving varifolds coupled to phase volumes, weak-strong uniqueness principle.
result Limits of Allen-Cahn solutions are varifold solutions, and classical flows are unique.
Paper integrates real data into probabilistic models using Fourier transform.
problem Learning from constrained data sets in high dimensions.
method Functional approach based on weak formulation of Fourier transform of probability measures.
result Estimation of posterior probability measures for QoI and QoI with control parameter.
PLRM synthesizes labels from mismatched sources for better training sets.
problem Creating labeled training sets is a major challenge in machine learning.
method PLRM uses probabilistic modeling to synthesize labels from indirect supervision sources with different output spaces.
result PLRM outperforms baselines by 2%-9% on various tasks.
The paper establishes general results in Lorentzian optimal transport theory.
problem Establishing strong duality and optimality conditions in Lorentzian optimal transport.
method Providing non-trivial assumptions on measures, characterizing optimality, and proving regularity results.
result Regularity results for -convex functions and (weak) Kantorovich potentials do not extend to the Lorentzian setting, but under suitable assumptions, they are locally semconvex.
The paper predicts survival functions using random survival trees and concordance maximization.
problem Predicting conditional survival functions in right-censored data.
method The approach combines regression strategies with random survival trees and maximizes concordance.
result The proposed weighted predictor outperforms the usual survival cobra in terms of concordance.
Paper tackles offline RL with weak assumptions on both function classes and data coverage.
problem Achieve sample-efficient offline RL with weak assumptions on both factors.
method Simple algorithm based on primal-dual formulation of MDPs, with density-ratio function modeling dual variables.
result Polynomial sample complexity achieved under realizability and single-policy concentrability.
We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in . This new formulation of Willmore equation appears to be of divergence form, moreover, the non-linearities are made of jacobians. Additionally to that, if $\bH$ denotes the mean curvature vector of the…
We consider the problem of learning optimal binary classification trees. Literature on the topic has burgeoned in recent years, motivated both by the empirical suboptimality of heuristic approaches and the tremendous improvements in mixed-integer programming (MIP) technology. Yet, existing approaches from the literatur…
These notes discuss several topics in neoclassical economics and alternatives, with an aim of reviewing fundamental issues in modeling economic markets. I start with a brief, non-rigorous summary of the basic Arrow-Debreu model of general equilibrium, as well as its extensions to include time and contingency. I then ar…
Study shows continuous evolution of curves in Fréchet distance.
problem Continuous evolution of curves under curvature flow.
method Curvature flow and level-set flow, analyzed in Fréchet distance.
result Evolution of curves depends continuously on initial curve.
Logitboost is an influential boosting algorithm for classification. In this paper, we develop robust logitboost to provide an explicit formulation of tree-split criterion for building weak learners (regression trees) for logitboost. This formulation leads to a numerically stable implementation of logitboost. We then pr…
The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.
problem Analyzing the limiting behavior of wedge products of weakly convergent differential forms on Riemannian manifolds.
method Formulating and proving compensated compactness theorems for wedge products of differential forms on closed Riemannian manifolds.
result The theorem generalizes the div-curl lemma for vectorfields and applies to critical regularity exponents.
We study learning latent models with multi-instance weak supervision.
problem Learning latent models with multi-instance weak supervision.
method Formulated as multi-instance Partial Label Learning (multi-instance PLL), proposed a necessary and sufficient condition for learnability, derived Rademacher-style error bounds.
result First theoretical study of multi-instance PLL with unknown transition function, aligns with empirical results but highlights scalability issues.
We develop the method of stochastic modified equations (SME), in which stochastic gradient algorithms are approximated in the weak sense by continuous-time stochastic differential equations. We exploit the continuous formulation together with optimal control theory to derive novel adaptive hyper-parameter adjustment po…
A financial market is called "diverse" if no single stock is ever allowed to dominate the entire market in terms of relative capitalization. In the context of the standard Ito-process model initiated by Samuelson (1965) we formulate this property (and the allied, successively weaker notions of "weak diversity" and "asy…
Develops weak PINNs for efficient manifold solutions of hyperbolic equations.
problem Challenges in approximating weak solutions of nonlinear hyperbolic equations on manifolds.
method Introduces a novel weak PINN (wPINN) formulation on manifolds leveraging well-posedness theory.
result Demonstrates efficient approximation of entropy solutions on manifolds with a complexity independent of ambient space dimension.
Establishes equivalence between models of derived stacks.
problem Tackles the equivalence between different models of derived geometry.
method Uses Quillen equivalence to show categories of higher derived stacks are equivalent.
result Shows equivalence among models of derived manifolds, Carchedi-Roytenberg, Behrend-Liao-Xu, and Alexandrov-Kontsevich-Schwarz-Zaboronsky.
Study on self-similar sets on Riemannian manifolds with new separation conditions.
problem Analyzing self-similar sets on Riemannian manifolds with new separation conditions.
method Formulated weak separation and finite type conditions for conformal iterated function systems on Riemannian manifolds.
result Obtained formulas for Hausdorff dimensions of self-similar and graph self-similar sets.
The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we pr…
In \cite{CHMY04}, we studied -mean curvature and the associated -minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized -area and associated (-) minimizers in general di…
Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to a positive power k of the mean curvature improves a certain 'isoperimetric difference' for k >= n-1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above mo…