A new robust metric compares distributions more accurately than existing methods.
problem Sensitivity to outliers and sampling discrepancy in Wasserstein distances.
method Introducing k-RPW, a partial p-Wasserstein distance.
result k-RPW converges faster to true distance and is more robust to outliers.
Generative Adversial Networks (GANs) have made a major impact in computer vision and machine learning as generative models. Wasserstein GANs (WGANs) brought Optimal Transport (OT) theory into GANs, by minimizing the 1-Wasserstein distance between model and data distributions as their objective function. Since then, W…
The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.
problem Estimating Wasserstein distances and their bounds in RCD(K,∞) spaces.
method Similar techniques used to prove inequalities involving p-Wasserstein distances and Laplace eigenfunctions. result Proves a conjectured lower bound on p-Wasserstein distance between positive and negative parts of Laplace eigenfunctions. Optimal transport offers an alternative to maximum likelihood for learning generative autoencoding models. We show that minimizing the p-Wasserstein distance between the generator and the true data distribution is equivalent to the unconstrained min-min optimization of the p-Wasserstein distance between the encoder agg…
Study statistical guarantees for DRO with OT and OT-regularized divergences.
problem Enhancing adversarial robustness in machine learning models.
method Derive concentration inequalities for supervised learning via DRO-based adversarial training.
result First to cover soft-constraint costs and reweighting mechanisms in adversarial training.
New smoothing technique improves Wasserstein distance estimation in high dimensions.
problem Estimating statistical distances between high-dimensional distributions.
method Gaussian smoothing of p-Wasserstein distance and analysis of its asymptotic behavior. result Gaussian-smoothed p-Wasserstein distance converges at rate n−1/2, improving over n−1/d for unsmoothed distances. In this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian and Finsler manifolds and led Lott-Villani and Sturm to define an abstract Ricci curvature…
The paper examines rigidity of metric constructions in Wasserstein spaces.
problem Isometric rigidity of metric constructions in Wasserstein spaces.
method Analyzes spaces like Hilbert, rays, half-cylinders, and spherical suspensions.
result Different spaces exhibit varying levels of isometric rigidity in Wasserstein spaces.
Study minimax rates for density estimation under Huber contamination and Besov IPM losses.
problem Minimax convergence rates of nonparametric density estimation under Huber contamination model with outliers.
method Re-scaled thresholding wavelet series estimator and GAN architectures.
result Achieves minimax optimal convergence rates under Besov IPM losses.
The paper explores multidimensional critic output in GANs, improving convergence and diversity.
problem Underexplored in GANs literature, multidimensional critic output.
method Generalized Wasserstein GAN framework, SRVT block, maximal p-centrality discrepancy.
result High-dimensional critic output improves GAN performance in convergence and diversity.
Study absolute continuity of Wasserstein barycenters on manifolds with singular cost functions.
problem Absolute continuity of Wasserstein barycenters on manifolds with singular cost functions.
method Approximation framework to handle singularity, geometrically transparent.
result Precise analytic condition on cost profile for necessary assumptions.
Paper develops KMS Wasserstein for high-dimensional data reduction.
problem Optimal transport's curse of dimensionality in high-dimensional data.
method Kernel max-sliced (KMS) Wasserstein distance for dimensionality reduction.
result Sharp finite-sample guarantees for KMS p-Wasserstein distance. The paper analyzes insurance contracts under distributional uncertainty using Bregman-Wasserstein divergence.
problem Optimal insurance contracts under distributional ambiguity.
method Utilizes Bregman-Wasserstein ball to characterize ambiguity sets, employs robust optimization.
result Derives optimal indemnity functions in closed form and studies their properties.
New robust method for optimal transportation improves statistical inference.
problem Sensitivity to outliers and undefinedness in optimal transportation methods.
method Robust optimal transportation with a tuning parameter λ, leading to robust Wasserstein distance.
result The robust method provides statistical guarantees and improves machine learning applications.
Paper proposes a new algorithm for clustering financial market regimes.
problem Rapid and automated detection of distinct market regimes.
method Wasserstein k-means algorithm for clustering financial time-series.
result Wasserstein k-means algorithm outperforms traditional clustering methods.
Paper provides GOT convergence guarantees for sub-gamma distributions and dependent samples.
problem Estimating GOT distance under general settings.
method Gaussian-smoothed optimal transport (GOT) framework, sub-gamma distributions, dependent samples, kernel MMD distances.
result Convergence guarantees for GOT distance under more general settings.
DRO-REBEL improves LLM alignment by robustly updating models online.
problem Overfitting and drifting of LLMs during RLHF.
method DRO-REBEL uses type-p Wasserstein, KL, and χ2 ambiguity sets for robust online updates. result DRO-REBEL achieves faster convergence and better performance than prior methods.
The paper improves reinforcement learning by estimating return distributions efficiently.
problem Estimating the complete return distribution in reinforcement learning.
method Distributional policy evaluation using the certainty-equivalence method.
result The method provides sample-efficient estimation of return distributions.
Momentum methods such as Polyak's heavy ball (HB) method, Nesterov's accelerated gradient (AG) as well as accelerated projected gradient (APG) method have been commonly used in machine learning practice, but their performance is quite sensitive to noise in the gradients. We study these methods under a first-order stoch…
An differential field (F;∂1,...,∂m) of characteristic zero, a subgroup H of affine group GL(n,C)∝Cn with respect to its identical representation in Fn and the following two fields of differential rational functions in x=(x1,x2,...,xn)-column vector, $$C< x, \partial >^H=\{f^{\part…
The L2-∂∂-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.
problem Extending the L2-∂∂-Lemma to non-compact Kähler manifolds. method Proving the L2-∂∂-Lemma on complete Kähler manifolds with a gap in the spectrum. result The L2-∂∂-Lemma is generalized to complete Kähler manifolds. Sharp Hölder regularity found for complex Frobenius theorem coordinates.
problem Finding optimal Hölder-Zygmund regularity for complex Frobenius theorem coordinates.
method Analyzing necessary and sufficient conditions for coordinate charts achieving the theorem's structure.
result The optimal Hölder-Zygmund regularity for coordinate charts is shown to be α. In a recent paper~\cite{DDL10} we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov in~\cite{Gro70}. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free map…
Two-dimensional Riemannian manifolds uniquely determined by boundary data.
problem Determining a 2D Riemannian manifold from boundary data.
method Calderón problem approach using Dirichlet-to-Neumann operator.
result A 2D compact connected Riemannian manifold is uniquely determined up to conformal equivalence.
Study cohomologies of complex manifolds with symplectic forms and their stability.
problem Analyzing cohomologies of complex manifolds with symplectic forms.
method Investigate the Hard Lefschetz Condition on Dolbeault cohomology groups using a double complex.
result Stability of the ∂∂Λ-Lemma under small deformations of ω but not under complex structure. Let M be a surface sum of 3-manifolds M1 and M2 along a bounded connected surface F and ∂i be the component of ∂Mi containing F. If Mi has a high distance Heegaard splitting, then any minimal Heegaard splitting of M is the amalgamation of those of M1,M2 and M∗, where $M^i=M…
We present a simple, flexible, and general framework titled Partial Registration Network (PRNet), for partial-to-partial point cloud registration. Inspired by recently-proposed learning-based methods for registration, we use deep networks to tackle non-convexity of the alignment and partial correspondence problems. Whi…
The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.
problem Analyzing maps between pseudo-Hermitian and Kähler manifolds.
method Investigates partial energy functionals and critical maps, proving foliated results for ∂b- and ∂b-harmonic maps. result Generalizes Siu's holomorphicity result to ∂b- and ∂b-harmonic maps. The paper proves Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
problem Proving Hodge decompositions and partial bar partial lemmas for G2 and Calabi-Yau manifolds.
method Defining cohomology spaces analogous to Bott-Chern cohomology and relating them to harmonic forms on the manifolds.
result Geometric interpretation of cohomology classes in terms of submanifolds and gerbes for G2 manifolds.
The paper studies holomorphic Poisson manifolds and their deformations under specific cohomology conditions.
problem Understanding deformations of holomorphic Poisson manifolds under certain cohomological assumptions.
method Investigates properties of Koszul-Brylinski homology and Dolbeault cohomology, proving formality of a DGLA.
result The DGLA is shown to be formal, and Maurer-Cartan elements induce complex structure deformations.
The paper characterizes when the ∂∂-lemma holds for twistor spaces.
problem Characterizing the ∂∂-lemma for twistor spaces. method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.
Let M be a compact hypersurface with boundary ∂M=∂D1∪∂D2, ∂D1⊂Π1, ∂D2⊂Π2, Π1 and Π2 two parallel hyperplanes in Rn+1 (n≥2). Suppose that M is contained in the slab determined by these hyperplanes and that the mean cu…
The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.
problem Deformations of Calabi-Yau manifolds under co-polarised conditions.
method Analyzes local deformations of Calabi-Yau ∂∂ˉ-manifolds using Gauduchon metrics and constructs a new hp-HS form. result Proves the p-SKT h-∂∂ˉ-property is deformation open. Paper tackles distribution matching by partially matching distributions, achieving robust results.
problem Robustly aligning two probability distributions.
method Developed a partial Wasserstein adversarial network (PWAN) to efficiently approximate the partial Wasserstein-1 (PW) discrepancy.
result The PWAN effectively produces highly robust matching results, outperforming state-of-the-art methods.
We give a simple proof of a result on the ∂∂ˉ-lemma property under a blow-up transformation by Deligne--Griffiths--Morgan--Sullivan's criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of…
Differential structure on partial isometries over Grassmannian constructed.
problem No specific problem stated; abstract focuses on method and result.
method Construction of differential structure on partial isometries over restricted Grassmannian.
result Set of partial isometries over restricted Grassmannian becomes a Banach Lie groupoid.
Ribbon cobordism forms a partial order in 3-manifolds.
problem Understanding partial orders in 3-manifolds.
method Utilizing recent methods from Ian Agol's work on knot concordance.
result Ribbon rational homology cobordism forms a partial order.
On a compact complex manifold X, we prove a Frölicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if X satisfies the ∂∂-Lemma.
Abstract study of HKT manifolds, proving Hodge theory and formality properties.
problem Analyzing Hodge theory and formality in HKT manifolds.
method Study of Dolbeault operators and Laplacians, proving relations and properties.
result Formality of differential graded algebra for certain HKT manifolds.
On a compact ∂∂ˉ-manifold X, one has the Hodge decomposition: the de Rham cohomology groups split into subspaces of pure-type classes as HdRk(X)=⊕p+q=kHp,q(X), where the Hp,q(X) are canonically isomorphic to the Dolbeault cohomology groups H∂ˉp,q(X). F…
Study on 4-dimensional almost-Hermitian manifolds, proving ∂-harmonic forms invariant under certain metrics.
problem Proving ∂-harmonic forms are topological invariants for specific metrics on 4-dimensional almost-Hermitian manifolds. method Analyzing ∂-Laplacian and using globally conformally Kähler and strictly locally conformally Kähler metrics. result Dimension of ∂-harmonic (1,1)-forms is a topological invariant, answering Kodaira and Spencer's problem. We consider partial matchings, which are finite graphs consisting of edges and vertices of degree zero or one. We consider transformations between two states of partial matchings. We introduce a method of presenting a transformation between partial matchings. We introduce the notion of the lattice presentation of a par…
It is shown that any smooth strictly convex global solution of det(∂ξi∂ξj∂2u)=exp{−∑i=1ndi∂ξi∂u−d0}, where d0, d1,...,dn are constants, must be a quadratic polynomial. This extends a well-known theorem of Jö…
New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. 3-manifold curvature comparison with rotationally symmetric bodies.
problem Comparing scalar curvature of 3-manifolds with rotationally symmetric boundaries.
method Inspired by Gromov, comparing mean curvatures and induced metrics.
result Flatness of 3-manifolds under certain curvature conditions.
Study shows partial hyperbolicity leads to Anosov dynamics in 3-manifolds.
problem Understanding dynamics in hyperbolic 3-manifolds and Seifert manifolds.
method Classification of partially hyperbolic diffeomorphisms and pseudo-Anosov dynamics.
result Complete classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds and Seifert manifolds.
We fully describe the horofunction boundary ∂hL2 with the word metric associated with the generating set {t,at} (i.e the metric arising in the Diestel-Leader graph DL(2,2)). The visual boundary ∂∞L2 with this metric is a subset of ∂hL2. Although $\partial_\infty L_2…
Paper categorifies a polynomial related to ribbon graphs.
problem Enumerating partial duals of ribbon graphs.
method Using an extended Frobenius algebra in unoriented topological quantum field theory.
result A categorification of the partial-dual genus polynomial.