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.
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…
We prove two injectivity theorems for the geodesic ray transform on two-dimensional, complete, simply connected Riemannian manifolds with non-positive Gaussian curvature, also known as Cartan-Hadamard manifolds. The first theorem is concerned with bounded non-positive curvature and the second with decaying non-positive…
Study fourth order Schrödinger equation on Cartan-Hadamard manifolds, proving existence, scattering, and blow-up results.
problem Fourth order Schrödinger equation with mixed dispersion on Cartan-Hadamard manifolds.
method Fourier transform for hyperbolic space, weighted Strichartz estimates for rotationally symmetric manifolds, localized virial argument.
result Existence, scattering, and blow-up results for the equation.
We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature de…
Improved SRHT for linear SVM classification with higher accuracy.
problem Inefficient random projection methods for high-dimensional data.
method Importance sampling and deterministic top-r sampling for effective low-dimensional embedding. result Higher classification accuracy on real-life datasets.
The Weyl transform is introduced as a rich framework for data representation. Transform coefficients are connected to the Walsh-Hadamard transform of multiscale autocorrelations, and different forms of dyadic periodicity in a signal are shown to appear as different features in its Weyl coefficients. The Weyl transform …
New algorithms update dynamic graph regression faster than existing methods.
problem Efficiently updating linear regression solutions for dynamic graphs.
method Subsampled randomized Hadamard transform and CountSketch.
result First sublinear update time randomized algorithms for dynamic graph regression.
Extends DCP framework to Hadamard manifolds for geodesically convex functions.
problem Verifying convexity in nonlinear programs on Hadamard manifolds.
method Introduces Disciplined Geodesically Convex Programming (DGCP) framework, defining compositions and transformations for geodesically convex functions.
result Allows verification of geodesic convexity for a broader range of functions, including statistical estimators and matrix-valued optimization.
A new iterative low complexity algorithm has been presented for computing the Walsh-Hadamard transform (WHT) of an N dimensional signal with a K-sparse WHT, where N is a power of two and K=O(Nα), scales sub-linearly in N for some 0<α<1. Assuming a random support model for the non-zero transform domain…
Uniform approximations for RHTs improve kernel approximation and distance estimation.
problem Theoretical guarantees for RHTs in low-dimensional applications.
method Proved uniform convergence of average of function over RHTs entries.
result Improved guarantees for kernel approximation and distance estimation.
Study spherical Fourier transform on hypergeometric type harmonic manifolds.
problem Spherical Fourier transform on harmonic Hadamard manifolds.
method Representation of spherical functions by Gauss hypergeometric functions.
result Inversion formula, convolution rule, and Plancherel theorem are derived.
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.
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.
We give a new algorithm for approximating the Discrete Fourier transform of an approximately sparse signal that has been corrupted by worst-case L0 noise, namely a bounded number of coordinates of the signal have been corrupted arbitrarily. Our techniques generalize to a wide range of linear transformations that are…
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.
Geometrically constructs twist-field correlation functions in CFT.
problem Understanding entanglement entropy in quantum systems.
method Using Cauchy-Hadamard renormalization of Polyakov anomaly integral on surfaces with conical singularities.
result Provides a purely mathematical interpretation of entanglement entropy results.
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.
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.
DFRot improves LLMs by reducing outlier and massive activation effects.
problem Reducing outlier and massive activation effects in rotated LLMs.
method Weighted loss function and orthogonal Procrustes transforms for rotation matrix refinement.
result DFRot achieves dual free (Outlier-Free and Massive Activation-Free) with significant improvements in perplexity.
We propose unitary group convolutions (UGConvs), a building block for CNNs which compose a group convolution with unitary transforms in feature space to learn a richer set of representations than group convolution alone. UGConvs generalize two disparate ideas in CNN architecture, channel shuffling (i.e. ShuffleNet) and…
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.
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.
Uniformly extend maps on Hadamard manifolds with curvature constraints.
problem Extending maps on Hadamard manifolds with curvature constraints.
method Proving uniform extension for contracting maps on Hadamard manifolds with curvature bounds.
result Uniform Lipschitz extension achieved for contracting maps on Hadamard manifolds with curvature constraints.
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.
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…
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.
Paper proposes Walsh-Hadamard Variational Inference for efficient approximate inference in large models.
problem Over-regularization in variational inference for large models.
method Walsh-Hadamard factorization strategies to reduce parameterization, accelerate computations, and increase posterior expressiveness.
result Efficient approximate inference achieved in over-parameterized models.
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.
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.
Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuatio…
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.
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.
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…
Unified methodology for statistical inference in least squares and PCA via randomized sketching.
problem Statistical inference in least squares and PCA problems.
method Randomized sketching and projections, asymptotic normality of quadratic forms.
result Unified statistical inference methods for various sketching distributions.
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.
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.
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.
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.
The article proves isometry theorems for specific types of manifolds.
problem Investigating properties of Cartan-Hadamard manifolds and related solitons.
method Analyzing steady, gradient shrinking, and expanding Ricci solitons.
result Specific manifolds are isometric to Euclidean space under certain conditions.
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. Sharp inequality proved in 3D hyperbolic spaces using flow methods.
problem Proving a sharp Minkowski-type inequality in Cartan-Hadamard 3-spaces.
method Using harmonic mean curvature flow to prove the inequality.
result Sharpened estimates for total mean curvature in hyperbolic 3-space.