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.
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.
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.
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.
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 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.
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.
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.
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.
Proposes a new graph representation method using tensor products.
problem Dynamic graph representation and theoretical properties.
method Bind-and-sum approach in hyperdimensional computing (HDC), tensor product as binding operation.
result Memory vs. size analysis of graph representation size scaling.
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 sample and time complexity for identifying mixtures of product distributions.
problem Identifying a mixture of k product distributions from statistics. method Combining robust tensor decomposition and Hadamard extensions to bound the condition number of key matrices.
result Achieved sample complexity and run-time complexity of (1/ζ)O(k) for n≥2k−1. 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.
This paper derives radial fields on manifolds of symmetric positive definite matrices.
problem Lack of an expression for radial fields on manifolds of symmetric positive definite matrices.
method Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
result Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
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.
NCPF model improves traffic data imputation with neural and tensor methods.
problem Pervasive missing data in traffic analysis due to sensor failures and gaps.
method Neural Canonical Polyadic Factorization (NCPF) integrating CP decomposition and deep learning.
result NCPF outperforms state-of-the-art baselines in urban traffic datasets.
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.
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.
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…
The tensor-tensor product (t-product) [M. E. Kilmer and C. D. Martin, 2011] is a natural generalization of matrix multiplication. Based on t-product, many operations on matrix can be extended to tensor cases, including tensor SVD, tensor spectral norm, tensor nuclear norm [C. Lu, et al., 2018] and many others. The line…
The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined W2-curvature tensor, computed curvature tensors and Ricci tensors, and studied curvature flatness. result Mixed Ricci-flat super twisted product semi-Riemannian manifolds can be expressed as super warped product manifolds.
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…
Killing tensors on reducible spaces are reducible, except for special cases.
problem Characterizing Killing tensors on reducible spaces.
method Analyzing Killing tensors on product manifolds and their lifts.
result Killing tensors on product manifolds are reducible, except for specific cases.
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. New method for optimization on Hadamard manifolds with curvature-independent guarantees.
problem Curvature-dependent complexity in geodesic convex optimization.
method Introducing horospherical convexity and developing algorithms for optimization.
result Curvature-independent convergence of subgradient descent and Nesterov's method.
Paper studies pseudo-projective tensors on warped products.
problem Characterizing pseudo-projectively flat warped products.
method Analyzes sequential warped products and pseudo-projective tensors.
result Necessary and sufficient conditions for pseudo-projectively flat sequential warped products.
Explores tensor products in hyperdimensional computing.
problem Understanding tensor products in hyperdimensional computing.
method Generalized results from graph embeddings to vector symbolic architectures and hyperdimensional computing.
result Tensor product is the most general and expressive representation with errorless unbinding and detection.
Introduces tensor product for quiver representations and applies to stable bundles and character varieties.
problem Stability and classification of quiver bundles and their subvarieties.
method Definition of tensor product for quiver representations and application to stability and character varieties.
result Tensor products of polystable quiver bundles are polystable and provide insights into character varieties.
IPMs struggle with hyperbolic spaces due to polynomially growing barrier parameters.
problem IPMs' efficiency is hindered in hyperbolic spaces.
method Analyzing the barrier parameter growth in hyperbolic and Hadamard spaces.
result The barrier parameter grows polynomially with the domain's diameter in hyperbolic spaces.
Some invariant tensors in two Naveira classes of Riemannian product manifolds are considered. These tensors are related with natural connections, i.e. linear connections preserving the Riemannian metric and the product structure.
Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of…
Defines tensor products for A-infinity structures using diagonals.
problem No specific problem stated; focuses on new definitions.
method Uses diagonals of associahedra and multiplihedra to define tensor products.
result Defines tensor products for various A-infinity structures.
The study extends Jacobi-orthogonality to indefinite scalar product spaces.
problem Generalizing Jacobi-orthogonality to indefinite scalar product spaces.
method Comparing principles, investigating tensor relations, proving properties.
result Every quasi-Clifford tensor is Jacobi-orthogonal; certain tensors are Jacobi-dual or Osserman.
Defines semi-symmetric metric connection on super warped products.
problem Computing curvature and Ricci tensors on super warped products.
method Introduced semi-symmetric metric connection and conditions for Einstein spaces.
result Conditions for super warped product spaces to be Einstein with semi-symmetric metric connection.
The paper studies special warped products with a specific connection on super Riemannian manifolds.
problem Investigating curvature and Ricci tensors on super warped product spaces with a semi-symmetric non-metric connection.
method Defined a semi-symmetric non-metric connection, computed curvature and Ricci tensors, and introduced and analyzed two types of super warped product spaces.
result Conditions for two super warped product spaces with a semi-symmetric non-metric connection to be Einstein spaces are provided.
Paper introduces tensor product of quandles for knot classification.
problem Classifying knot invariants of surface-links with 1-handles.
method Introduces tensor product of quandles and applies to surface-links.
result Tensor product of knot quandles/classical quandles can classify surface-link invariants.
Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the p…
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…
We present two formulas for Chern classes of the tensor product of two vector bundles. In the first formula we consider a matrix containing Chern classes of the first bundle and we take a polynomial of this matrix with Chern classes of the second bundle as coefficients. The determinant of this expression equals the Che…
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.