New insights into identifying mixtures of product distributions using Hadamard extensions.
problem Identifying mixtures of product distributions on binary variables.
method Analysis of Hadamard extensions of matrix products.
result Conditions for full column rank of Hadamard extensions.
Cyclic projections in Hadamard spaces can be irregular, unlike in Hilbert spaces.
problem Understanding the behavior of cyclic projections in Hadamard spaces compared to Hilbert spaces.
method Constructing an example of convex subsets in a Hadamard space.
result Cyclic product of projections is not asymptotically regular in Hadamard spaces.
In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds N×R, submanifolds with bounded φ-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p…
The paper provides formulas for Hadamard coefficients using Green's operators.
problem Calculating Hadamard coefficients from Green's operators.
method Various methods including resolvents, powers of Green's operators, and product with the real line.
result Formulas for Hadamard coefficients in terms of Green's operators.
Characterizes higher rank model geometries using antipodal sets.
problem Identifying higher rank model geometries among Hadamard spaces.
method Using antipodal sets at infinity to characterize model geometries.
result Characterizes Riemannian symmetric spaces, Euclidean buildings, and products as higher rank model geometries.
Stable Hadamard Memory improves reinforcement learning by efficiently managing memory.
problem Memory models struggle in partially observable reinforcement learning environments.
method Introduces a novel memory model using the Hadamard product for efficient memory management and updates.
result Significantly outperforms state-of-the-art memory-based methods on challenging benchmarks.
New theorem extends Hadamard's to transversely affine geometry.
problem Transversely affine foliations and their properties.
method Introduced and investigated novel transversely affine foliations, extending Hadamard's theorem.
result Transversely affine version of Hadamard's theorem for foliations.
Efficient algorithm for Hadamard decomposition of matrices.
problem Decomposing matrices into low-rank factors efficiently.
method Alternating optimization with SVD-inspired initialization and momentum.
result Significantly improved performance compared to existing methods.
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
KReTTaH uses tensor trains and Hadamard overparameterization for fast, interpretable multi-way data imputation.
problem Multi-way data imputation for high-dimensional functional MRI and dynamic graph recovery.
method Reformulates imputation as RKHS regression with TT-constrained coefficients and Hadamard overparameterization. Optimizes TT coefficients and kernel matrices on Riemannian manifolds.
result Consistently outperforms state-of-the-art methods in modeling accuracy.
KReTTaH uses tensor trains and Hadamard overparameterization for fast, interpretable multi-way data imputation.
problem Multi-way data imputation in high-dimensional spaces.
method Reformulates imputation as RKHS regression with TT-constrained coefficients, optimized on manifold frameworks.
result Consistently outperforms state-of-the-art methods in accuracy.
The paper studies isoperimetric inequalities on warped product manifolds.
problem Investigating isoperimetric inequalities on warped product manifolds.
method Exploiting the interplay between isoperimetric inequalities and warped product structures, deriving necessary and sufficient conditions.
result Established a quantitative lower bound for the first nonzero Dirichlet eigenvalue of geodesic balls centered at the pole.
Study of equivariant Poisson 2-algebra bundles over configuration spaces.
problem Understanding Poisson structures on equivariant vector bundles over configuration spaces.
method Construction of induced-equivariance functor, Hadamard and Cauchy tensor products, symmetric 2-monoidal structure, free commutative 2-algebra, compatible Poisson bracket.
result Construction of free commutative 2-algebra and Poisson bracket on equivariant Poisson 2-algebra bundles.
Coupled nonlinear integrable systems are generated from usual zero curvature equation. The relevant Maurer-Cartan forms are constructed by combining suitably chosen matrices (nilpotent, Hadamard, idempotent and k-idempotent) and Lie algebraic elements via Kronecker product. In each case a closure type property among th…
We propose a generalized class of multimodal fusion operators for the task of visual question answering (VQA). We identify generalizations of existing multimodal fusion operators based on the Hadamard product, and show that specific non-trivial instantiations of this generalized fusion operator exhibit superior perform…
Generalizing results due to Brady and Farb we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m_1+m_2-1 in the product X:=X_1^{m_1} times X_2^{m_2} of two Hadamard manifolds X_i^{m_i} of dimension m_i with pinched negative curvature. Combining this result with a Theore…
Paper proposes a method to speed up DNNs by quantizing Winograd/Toom-Cook convolutions.
problem Speeding up convolution computations in DNNs with reduced time consumption and improved accuracy.
method Application of base change technique for quantized Winograd-aware training model.
result 8-bit quantized network achieves nearly the same accuracy as direct quantized convolution with minimal additional operations.
Study dispersive estimates for Schrödinger and wave equations on a cone with specific metric.
problem Pointwise decay estimates for Schrödinger and wave equations on a product cone.
method Modified Hadamard parametrix on Y with ε>π to prove dispersive estimates. result Threshold of conjugate radius ε>π for pointwise dispersive estimates. Paper addresses optimization on Hadamard manifolds, generalizing gradient flow.
problem Optimization of convex functions on Hadamard manifolds.
method Introduces a generalized gradient flow to minimize Q(dfx). result Gradient flow attains infimum in limit for basic manifolds.
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
We continue our study of the space of geodesics of a manifold with linear connection. We obtain sufficient conditions for a product to have a space of geodesics which is a manifold. We investigate the relationship of the space of geodesics of a covering manifold to that of the base space. We obtain sufficient condition…
Harmonic maps between pinched Hadamard surfaces are quasi-conformal.
problem Characterizing harmonic maps between Hadamard surfaces.
method Proving harmonic quasi-isometries are quasi-conformal diffeomorphisms.
result Harmonic quasi-isometries of pinched Hadamard surfaces are injective.
The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.
problem Defining and studying the geometric mean for tensors.
method Generalized geometric mean for tensors using T-product, verified properties, and investigated Riemannian manifold.
result Geometric mean of T-positive definite tensors is a unique solution of algebraic Riccati tensor equations and a midpoint of geodesics.
Study shows spectrum properties for specific Hadamard manifolds.
problem Spectrum properties of Hadamard manifolds.
method Absolute continuity and spectrum determination for two classes of Hadamard manifolds.
result Spectrum properties determined for specific Hadamard manifolds.
Sharp curvature inequality extends Euclidean isoperimetric inequality to Cartan-Hadamard manifolds.
problem Proving the Euclidean isoperimetric inequality on Cartan-Hadamard manifolds with nullity.
method Using the Chern-Gauss-Bonnet theorem to establish a sharp inequality for total curvature.
result The Euclidean isoperimetric inequality extends to Cartan-Hadamard manifolds with nullity.
A new class of harmonic Hadamard manifolds, those spaces called of hypergeometric type, is defined in terms of Gauss hypergeometric equations. Spherical Fourier transform defined on a harmonic Hadamard manifold of hypergeometric type admits an inversion formula. A characterization of harmonic Hadamard manifold being of…
The paper constructs a Poisson algebra bundle for multilocal observables.
problem Representing multilocal observables in classical field theory.
method Working with unordered configuration spaces and using symmetric algebras with respect to two tensor products.
result Obtained a Poisson 2-algebra bundle mimicking Peierls bracket.
Graphical notation simplifies tensor operations and decompositions.
problem Complex tensor operations are difficult to understand and represent.
method Introduces graphical notation to represent tensor operations.
result Simplified representation of tensor operations and decompositions.
Formula for Hadamard coefficients from Green's operators on spacetimes.
problem Calculating Hadamard coefficients from Green's operators on spacetimes.
method Developed formulas for diagonal values and integrals over the diagonal of Hadamard coefficients.
result Formulated analogues of Hadamard expansions and resolvents for Green's operators.
Upper bounds for circumradius in Hadamard surfaces with curvature constraints.
problem Bounding circumradius in Hadamard surfaces with curvature constraints.
method Using curvature constraints to derive upper bounds for circumradius.
result Upper bounds for circumradius in terms of curvature bounds.
Solves Plateau problem for surfaces in pinched curvature manifolds.
problem Asymptotic Plateau problem for immersed surfaces in pinched curvature manifolds.
method Complete solution to asymptotic Plateau problem, providing dynamical stability of hypersurface laminations.
result Achieved complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan--Hadamard manifolds.
Proves Hadamard states for Dirac fields on manifolds with timelike boundaries.
problem Existence of Hadamard states for Dirac fields with MIT boundary conditions.
method Introducing geometric Møller operator to implement unitary isomorphism between spaces of initial data.
result Existence of Hadamard states for Dirac fields with MIT boundary conditions.
Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper we prove that these conditions are mutually equivalent and they hold if and only if the Hadamard manifold is isometri…
Paper computes link determinants using Fourier-Hadamard transforms.
problem Computing determinants of complex link structures.
method Fourier-Hadamard transforms of Boolean functions.
result Determinant of centrally symmetric links with even components equals zero.
McKernel introduces a framework to use kernel approximates in the mini-batch setting with Stochastic Gradient Descent (SGD) as an alternative to Deep Learning. Based on Random Kitchen Sinks [Rahimi and Recht 2007], we provide a C++ library for Large-scale Machine Learning. It contains a CPU optimized implementation of …
The paper proves Liouville-type theorems on Hadamard manifolds.
problem Non-existence of Killing-Yano tensors, Killing tensors, and harmonic symmetric tensors on Hadamard manifolds.
method Proofs use Liouville-type theorems on non-existence of subharmonic and harmonic functions on complete Riemannian manifolds, modified for Hadamard manifolds.
result Proves several Liouville-type theorems on Hadamard manifolds.
The abstract discusses compact quotients of Riemannian products by discrete subgroups, generalizing Inoue-Bombieri surfaces.
problem Compact quotients of Riemannian products by discrete subgroups.
method Study of compact quotients of a Riemannian product Rqimes(N,gN) by discrete subgroups Γ of Sim(Rq)imesIsom(N). result The construction is equivalent to LCP manifolds and provides a Bieberbach-type rigidity result.
In this article, we consider compact surfaces Σ having constant mean curvature H (H-surfaces) whose boundary Γ=∂Σ⊂M0=M×f{0} is transversal to the slice M0 of the warped product M×fR, here M denotes a Hadamard surface…
Maps between Hadamard manifolds are quasi-isometric to harmonic maps.
problem Understanding the relationship between quasi-isometric maps and harmonic maps on Hadamard manifolds.
method Extending a previous result to quotient spaces of Hadamard manifolds by convex cocompact discrete groups.
result Locally quasi-isometric maps to Hadamard manifolds are within bounded distance from a unique harmonic map.
The study extends isoperimetric inequalities to non-positive curvature spaces.
problem Isoperimetric inequalities in spaces of non-positive curvature.
method Analyzes submanifolds and geodesics in Cartan-Hadamard manifolds.
result Extensions of isoperimetric inequalities to non-positive curvature spaces.
Nonexistence of radial optimal functions on certain Cartan-Hadamard manifolds.
problem Proving nonexistence of radial optimal functions for the Sobolev inequality on Cartan-Hadamard manifolds.
method Ad hoc arguments not relying on the Cartan-Hadamard conjecture.
result If the optimal constant in the Sobolev inequality is achieved by a radial function, then the manifold must be isometric to Euclidean space.
Study KKT conditions for multi-objective optimization on Hadamard manifolds.
problem Optimizing multi-objective interval-valued functions on Hadamard manifolds.
method Developed KKT conditions for Pareto optimal solutions under different ordering and convexity notions.
result Results are more general than on Euclidean spaces.
We use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard space, we formulate criterions for the actio…
Proves homological inequality for cycles in Hadamard spaces of asymptotic rank 2.
problem Establishing isoperimetric inequalities in Hadamard spaces of asymptotic rank two.
method Homological inequality for cycles in dimensions at least 2, assuming finite linearly controlled asymptotic dimension.
result Homological inequality for general cycles in Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.
Linear inequality found for certain curved spaces.
problem Finding isoperimetric inequalities for curved spaces.
method Extending known result to homogeneous Hadamard manifolds.
result Linear isoperimetric inequality proven for higher-dimensional cycles.
Paper proves existence of Hadamard states for Maxwell equations.
problem Proving existence of Hadamard states for Maxwell equations on spacetime.
method Introducing Cauchy radiation gauge and new Hodge decomposition.
result Existence of Hadamard states for Maxwell equations proven.
The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
Groups can act on Lp spaces with non-trivial cohomology.
problem Understanding actions of groups on Hadamard spaces.
method Analyzing isometric actions on Hadamard manifolds and Lp-spaces. result Non-trivial Lp-cohomology for certain groups.