Uniformly K-stable toric varieties are asymptotically Chow stable if their Futaki-Ono invariant vanishes.
problem Determining asymptotic Chow stability of uniformly K-stable toric varieties.
method Detailed study of triangulations of moment polytope neighborhoods and analysis of Futaki-Ono invariant.
result Every uniformly K-stable polarized smooth toric variety with vanishing Futaki-Ono invariant is asymptotically Chow polystable.
Samplets and multiwavelets constructed from scattered data converge to specific densities in the limit.
problem Constructing data-adapted multiresolution analyses and multiwavelets with flexible vanishing moments.
method Probabilistic framework for samplet construction; convergence to multiwavelets with broken polynomial densities.
result Samplet construction converges to multiwavelets in the infinite data limit.
New proof of Sobolev inequality with constraints on sphere.
problem Improving Sobolev inequality on sphere with constraints.
method Careful study of extremal problem on sphere.
result Explicit determination of constant in second order moments case.
New invariants detect Fano varieties' K-stability.
problem Detecting K-stability in Fano varieties.
method Introducing valuative invariants based on higher moments.
result Can detect K-stability of Fano varieties.
New flow connects symplectic maps to hyperKähler geometry.
problem Understanding symplectic maps and their geometry.
method Established a correspondence between symplectic diffeomorphisms and hyperKähler moment maps.
result Introduced a new flow, the modified moment map flow.
We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and a…
The paper examines linking numbers in grid models and finds polynomial moments.
problem Analyzing linking numbers in grid models.
method Examined linking numbers as a random variable on isotopy classes of 2-component links, computed moments and limits.
result The uth moment of the linking number is a polynomial in the grid size with degree d≤u, and all odd moments vanish. Maxout networks study gradients and propose initialization strategies.
problem Complexity in input-output Jacobian distribution complicates stable parameter initialization.
method Obtained bounds on moments of gradients and formulated initialization strategies.
result Parameter initialization strategies improve training of deep maxout networks.
We relate stability properties (i.e. moment exponents) of a stochastic dynamical system on a compact manifold M to the homotopy and integral homology groups of M. In the special case of gradient Brownian systems associated to isometric immersions of M in Euclidean space, these moment exponents can be estimated in…
New result on group actions in CAT(0) spaces with vanishing escape rate.
problem Understanding group actions with vanishing escape rate on CAT(0) spaces.
method Equivariant μ-harmonic map proof. result Existence of a flat subspace invariant under the action of Γ. Study of hyperkähler reduction on abelian varieties and toric manifolds.
problem Understanding hyperkähler reduction on specific manifolds.
method Lifts canonical Kähler reduction to hyperkähler, studies on abelian varieties and toric manifolds.
result Obtains decoupling result, variational characterisation, relation to K-stability, and proves existence and uniqueness under suitable assumptions. The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is t…
Heterotic backgrounds described using generalised geometry, preserving minimal supersymmetry.
problem Characterizing heterotic backgrounds preserving minimal supersymmetry in four dimensions.
method Using generalised geometry, characterizing backgrounds by an SU(3)imesSpin(6+n) structure and an involutive subbundle of the generalised tangent bundle. result The analysis of infinitesimal deformations reproduces known cohomologies of massless moduli.
Bayesian framework uses AI-generated data to improve parameter estimation.
problem Parameter estimation in models with unknown or unspecified likelihood.
method Exponentially tilted empirical likelihood with Dirichlet process posterior.
result AI-generated data can provide useful regularization for parameter estimation.
The study proves the finiteness of moments for Gaussian field zeros and critical points.
problem Finiteness of moments for Gaussian field zeros and critical points.
method Definition and study of multijets, construction of p-multijet bundles.
result Linear statistics of Gaussian field zeros have finite p-th moments for p ≥ 1.
A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibration…
In the paper [Probab. Theory Relat. Fields, 100 (1994) 417-428] Xue-Mei Li has shown that the moment stability of an SDE is closely connected with the topology of the underlying manifold. In particular, she gave sufficient condition on SDE on a manifold M under which the fundamental group π1M=0. We prove that in …
Study on convergence rates for optimal transport with regularization.
problem Convergence analysis of divergence-regularized optimal transport.
method Novel methodology using quantization and martingale couplings.
result Sharp rates for various divergences and transport costs.
Study on estimating rank-one tensors in noisy data with heavy tails.
problem Estimating rank-one spiked tensors in the presence of heavy tailed errors.
method Analysis of spectral norm of random tensors with iid entries.
result Signal strength requirements for optimal estimation are similar for heavy tailed and Gaussian noise, but vanish for noise with finite fourth moment.
Last-iterate guarantees for learning in co-coercive games under noisy feedback.
problem Learning in co-coercive games with noisy feedback.
method Vanilla stochastic gradient descent with a new noise model.
result Last-iterate bound of order O(log(t)/t1/3) for co-coercive games. Study of string theory flux compactifications using generalized geometry.
problem Characterizing generic Minkowski flux compactifications in string theory.
method Using E7(7)×R^+ generalized geometry, analyze involutive subbundles and moment maps.
result Counted massless scalar moduli of GMPT solutions using generalised geometry cohomology.
In this paper we show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As an application we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of…
A tractable pseudo-metric for non-parametric distributions via SPD geometry.
problem Computing distances between non-parametric probability distributions is intractable.
method Two-stage framework: projection onto parametric family, embedding into SPD matrices.
result Closed-form pseudo-metric for two-sample hypothesis testing.
The paper examines how the angle between inputs in ReLU networks decreases with depth, impacting training.
problem Depth degeneracy in neural networks, leading to constant function behavior on initialization.
method Combinatorial expansions and Monte Carlo experiments to analyze the angle between inputs in ReLU networks of increasing depth.
result The angle between inputs in ReLU networks decreases exponentially with depth, leading to constant function behavior on initialization.
Improved CR Sobolev inequalities on CR sphere established.
problem Establishing CR Sobolev inequalities on CR sphere.
method Nice commutator identities involving CR intertwining operators.
result Simpler proof of existence and classification of minimizers.
The paper improves CR Sobolev inequalities and classifies minimizers.
problem Higher-order CR Sobolev inequalities on the CR sphere.
method Improvement through vanishing higher order moments of the volume element.
result New direct proof of minimizers' classification and existence of minimizers in C2k(N). We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the ℓ2 norm of such a product applied to any fixed vector is asymptotically Gaussian. The fluctuations we find can be thought of as a…
New method approximates MMD using pseudo-differential operators and singular values.
problem Approximating MMD with pseudo-differential operators and singular values.
method Corresponding pseudo-differential operators to Mercer kernels, approximating p(x,y) with its first r singular values. result The new MMD distance measures the difference of two distributions with respect to r∗ local moments, where r∗ depends on singular values decay rate. Feature noise causes loss discrepancies across groups even with equal data.
problem Loss discrepancies observed in learning procedures across different groups.
method Characterized the effect of feature noise on loss discrepancy in linear regression.
result Feature noise leads to loss discrepancy even when groups have equal data.
Extends probabilistic approach for Kahler-Einstein metrics on Fano manifolds.
problem Constructing Kahler-Einstein metrics on log Fano manifolds with non-discrete automorphism groups.
method Introduces Gibbs polystability and uses moment map constraint to break symmetry.
result Gibbs polystability conjectured to be equivalent to existence of Kahler-Einstein metric.
In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point q0∈M exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on C0∞(M∖{q0}) is essentially self-adjoint. A particular …
A new method calculates fractional moments using the moment-generating function.
problem Computing fractional moments from probability densities.
method Integral framework based on moment-generating function.
result Exact integral expressions for various types of moments.
Study compares weak and homotopy moment maps in multisymplectic geometry.
problem Existence and equivariance of moment maps in multisymplectic geometry.
method Comparison of weak and homotopy moment maps.
result Analysis of existence and equivariance phenomena.
For a GJR-GARCH specification with a generic innovation distribution we derive analytic expressions for the first four conditional moments of the forward and aggregated returns and variances. Moment for the most commonly used GARCH models are stated as special cases. We also the limits of these moments as the time hori…
This paper identifies and bounds ICE central moments using PO marginal central moments.
problem Identifying and characterizing treatment effect heterogeneity.
method Using only marginal central moments of potential outcomes, the paper identifies and bounds central moments of individual causal effects.
result Identification and bounding of central moments of ICE using marginal moments of POs.
We tackle causal inference under conditional moment restrictions using importance weighting.
problem Challenges in causal inference under conditional moment restrictions, especially in high-dimensional settings.
method Transform conditional moment restrictions to unconditional moment restrictions through importance weighting.
result Successfully estimate nonparametric functions defined under conditional moment restrictions.
We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…
Revisits Lee's Moment Formula, relaxing moment assumptions for implied volatility.
problem Implied volatility constraints under finite log-moments.
method Analyzes stock price martingale with finite log-moments, derives new bounds and proof.
result New bounds on implied volatility growth, relaxes moment assumptions.
Enhanced Adam uses higher-order moments for better performance.
problem Improving the performance of Adam optimization algorithm.
method Proposes HAdam, an extension of Adam using higher-order moments of the stochastic gradient.
result Higher-order moments of the stochastic gradient can lead to better performance than vanilla Adam.
Developed moment estimators for affine stochastic volatility models.
problem Estimating parameters of affine stochastic volatility models.
method Introduced recursive equations for moments and proposed moment estimators.
result Established a central limit theorem and derived asymptotic covariance matrix.
Stiefel-Whitney classes of moment-angle manifolds are trivial.
problem Analyzing the topological properties of moment-angle manifolds.
method Proving triviality of Stiefel-Whitney classes for moment-angle manifolds, including partial quotients.
result Stiefel-Whitney classes of moment-angle manifolds are trivial.
A new method for estimating causal parameters from observables reduces the need for finite moment conditions.
problem Estimating causal parameters from observational data with unknown or infinite moment conditions.
method Variational Method of Moments (VMM) for a general class of estimators, including kernel and neural net-based methods.
result VMM estimators are consistent, asymptotically normal, and semiparametrically efficient.
Introduces generalized moment maps for almost Hermitian settings.
problem Extending classical moment map theory to almost Hermitian settings.
method Introduces momentumly closed forms and proves a variant of the Darboux-Weinstein theorem.
result Establishes convexity property and constructs reduction space for generalized moment maps.
Proposes Moment Exchange to use moments in image recognition models, improving generalization.
problem Discarding moments in image recognition models reduces stability and training time.
method Moment Exchange: replaces moments of learned features with another image's moments and interpolates labels.
result Improves generalization of recognition models across multiple datasets.
New KCM tests improve specification testing via RKHS.
problem Improving specification tests for econometric models.
method Kernel conditional moment (KCM) tests based on RKHS.
result KCM tests have better finite-sample performance than existing tests.
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
A new method of moments estimator goes beyond data reweighting.
problem Estimation of moment restrictions and conditional moment restrictions.
method Kernel Method of Moments (KMM) based on maximum mean discrepancy.
result KMM achieves competitive performance on conditional moment restriction tasks.
Moment Pooling reduces latent space dimensions in machine learning models.
problem High-dimensional latent spaces in machine learning models are hard to interpret.
method Moment Pooling extends Deep Sets networks to arbitrary multivariate moments.
result Latent dimensions as small as 1 can achieve similar performance to higher dimensions.