Tensorial Mixture Models combine tractable structure with rich distribution representation.
problem Lack of tractable marginalization in generative models.
method Derived from tensor analysis, TMMs use simple convolutional networks and leverage theoretical analyses.
result Tensorial Mixture Models deliver state-of-the-art accuracies in classification tasks with missing data.
Proposes TRNN for tensorial time series data analysis.
problem Lack of suitable models for tensorial time series data.
method Introduces Tensorial Recurrent Neural Network (TRNN) based on tensor Tucker decomposition.
result TRNN preserves the spatial or longitudinal dimensions of tensorial time series data.
Paper analyzes online tensorial ICA convergence with stochastic approximation.
problem Online tensorial ICA convergence analysis.
method Stochastic approximation for nonconvex optimization.
result Sharp finite-sample error bound of O ~ ( d / T ) \tilde{O}(\sqrt{d/T}) O ~ ( d / T ) . A new tensorial metric describes geometry in 4D space.
problem Understanding the structure of hypercomplex space.
method Developed a new geometry group in R^4 with a tensorial metric.
result Riemannian and Euclidean distances are special cases of the Alpha Group's metric.
The paper defines constraints for commuting endomorphisms in generalized tangent bundles.
problem Identifying constraints for commuting endomorphisms in generalized tangent bundles.
method Using Gröbner basis techniques to construct and study tensors forming ideals.
result Explicit construction and study of tensors forming ideals of commuting endomorphisms.
We give a tensorial description of the Turaev cobracket on any genus 0 compact surface through the standard group-like expansion, where the Bernoulli numbers appear.
Study peels tensor equations on Schwarzschild spacetime.
problem Analyzing the asymptotic behavior of tensorial wave equations on Schwarzschild spacetime.
method Combining conformal compactification and vector field techniques to estimate tensorial field energies.
result Obtains optimal initial data for peeling at all orders.
New space of tensorial bodies defined, properties and representatives studied.
problem Characterizing convex bodies in tensor norms.
method Introduced a new space of tensorial bodies, defined a Banach-Mazur distance, and proved existence of a compact type compactum.
result Topological representatives for the space of tensorial bodies and the Banach-Mazur type compactum are given.
Developed a new symmetric hyperbolic formulation for Einstein-Yang-Mills system.
problem Future stability of solutions of the Einstein-Yang-Mills system with arbitrary dimension.
method Tensorial symmetric hyperbolic formulation and local well-posedness for Cauchy problem.
result Established local well-posedness for the Cauchy problem of EYM equations in the temporal gauge.
Proves energy estimates for tensorial wave equations, decoupling components for stability proof.
problem Proving stability of ( 1 + 3 ) (1+3) ( 1 + 3 ) -Minkowski space-time with various non-linearities. method Decouples energy estimates for tensorial wave equations, exploiting tensorial structure and Lie derivatives.
result Decoupled energy estimates for tensorial solutions, allowing new stability proofs.
Paper shows invertibility of tensor X-ray transform on certain manifolds.
problem Invertibility of tensor X-ray transform on asymptotically conic manifolds.
method Used 1-cusp pseudodifferential operator algebra and modified solenoidal gauge condition.
result Invertibility of tensor X-ray transform up to natural obstruction.
Tensorial Neural Networks improve neural network compression and performance.
problem Efficiently compressing neural networks while maintaining or improving performance.
method Introducing tensor operations on high-order operands to solve hierarchical nonlinear tensor decomposition using stochastic gradient descent.
result TNNs achieve up to 5% test accuracy improvement on CIFAR10 compared to state-of-the-art compression methods.
TACE unifies scalar and tensorial modeling in Cartesian space for accurate, stable, and efficient atomistic predictions.
problem Complexity and challenges in equivariant atomistic machine learning models.
method Tensor Atomic Cluster Expansion (TACE) in Cartesian space, decomposing local environments into irreducible Cartesian tensors (ICT).
result Universal invariant and equivariant embeddings, enabling explicit control at inference.
Capsule Neural Networks classify graphs from categorical features and relationships.
problem Graph classification in scientific domains, especially with varying graph sizes and features.
method Explicit tensor representations, Capsule Network for classification.
result Capsule Network model performs competitively with state-of-the-art models.
Study finds conditions for operator fields to be in strictly upper triangular form in small dimensions.
problem Jordan-Chevalley decomposition for operator fields in small dimensions.
method Tensorial conditions and proof of conjecture for higher order brackets.
result Proves Tempesta-Tondo conjecture for higher order brackets.
Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor W W W to the setting of smooth metric measure space…
Extended spinor connections associated with composite spin-tensorial bundles are considered. Commutation relationships for covariant and multivariate differentiations and corresponding curvature spin-tensors are derived.
Paper proposes a new algorithm for joint blind source separation using tensorial methods.
problem Joint blind source separation of multi-set data.
method Double coupled canonical polyadic decomposition (DC-CPD) with algebraic approach.
result Deterministic and exact solution for noiseless case, effective initialization for noisy case.
The paper proves volume stability for hyperbolic manifolds and applies it to general relativity.
problem Volume stability of hyperbolic manifolds and its implications in general relativity.
method Sharp volume-stability theorem for closed hyperbolic three-manifolds, tensorial \(C^0\)-convergence.
result Near-equality in the sharp hyperbolic volume bound forces tensorial \(C^0\)-convergence to the hyperbolic metric.
Kosmann-Lie derivatives in the bundle of Weyl spinors are considered. It is shown that the basic spin-tensorial fields of this bundle are constants with respect to these derivatives.
A new geometric method approximates slow invariant manifolds without explicit time-scale separation.
problem Approximating slow invariant manifolds in systems with multiple time-scales.
method Geodesic Stretching and Flow Curvature methods translated into tensorial constructions of Riemannian geometry.
result The method approximates normally attracting invariant manifolds without requiring explicit time-scale separation.
In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to do this with only very weak control on these geometries. Several of these estimates were established in previous works by S. Klainerman and I. Rodn…
Study non-homogeneous operators in 1+0 systems, classifying and analyzing their geometric properties.
problem Classify and analyze geometric properties of non-homogeneous operators in 1+0 systems.
method Complete classification of Casimir functions, tensorial criteria for compatibility, bi-pencils definition.
result Found geometric connections with Nijenhuis geometry, proving compatibility results.
Smooth deformations of a Minkowski type metric in a four-dimensional space-time manifold are considered. Deformations of the basic spin-tensorial fields associated with this metric are calculated and their application to calculating the energy-momentum tensor of a massive spin 1/2 particle is shown.
Defines strongest integrability condition for skew-symmetric endomorphisms.
problem Integrability conditions for skew-symmetric endomorphisms.
method Characterization of the shifted Courant-Nijenhuis torsion.
result Vanishing of the shifted Courant-Nijenhuis torsion as the strongest integrability condition.
Geometrically reformulates wave equation solving method.
problem Solving tensorial wave equations in spacetimes.
method Geometric formulation of the method of descent.
result Representation formula for tensorial wave equation.
Proposes a new kernel technique for tensor data in SVM.
problem Handling tensorial data in machine learning.
method Kernelized support tensor train machine for image classification.
result Tensorizes the standard SVM on its input structure and kernel mapping scheme.
A conformal structure on a manifold M n M^n M n induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of M M M , provided that n ≥ 3 n\ge 3 n ≥ 3 . By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here th…
Introduces a new framization of Hecke algebra type B.
problem No specific problem stated; focuses on algebraic structure.
method Constructs a faithful tensorial representation and two linear bases.
result Derives isotopy invariants for knots and links.
This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl con…
Let ( Q ~ , g ) (\tilde Q,g) ( Q ~ , g ) be a para-quaternionic Hermitian structure on the real vector space V V V . By referring to the tensorial presentation ( V , Q ~ , g ) ≃ ( H 2 ⊗ E 2 n , s l ( H ) , ω H ⊗ ω E ) (V, \tilde{Q},g) \simeq (H^2 \otimes E^{2n}, \mathfrak{sl}(H),ω^H \otimes ω^E) ( V , Q ~ , g ) ≃ ( H 2 ⊗ E 2 n , sl ( H ) , ω H ⊗ ω E ) , we give an explicit description, from an affine and metric point of view, of main classes of subspaces…
Machine learning models accurately predict molecular magnetic anisotropy tensors.
problem Accurately modeling molecular magnetic anisotropy tensors.
method Gaussian-moment neural-network approach for machine learning.
result Achieved accuracy of 0.3--0.4 cm − 1 ^{-1} − 1 for magnetic anisotropy tensor predictions. In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result i…
Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.
problem Stability of Minkowski space-time for perturbations governed by Einstein-Yang-Mills equations.
method Proves exterior energy estimates for tensorial non-linear wave equations in Minkowski space-time.
result Proves exterior stability of Minkowski space-time for Einstein-Yang-Mills equations.
Study fourth-order geometric flow of shape operator for co-dimension one immersions.
problem Analyzing the geometry of isometric immersions in Riemannian manifolds.
method Introduce a moduli flow to decrease curvature variation energy.
result The flow decreases a natural energy measuring curvature variation.
The problem of characterizing conformally Einstein manifolds by tensorial conditions has been tackled recently in papers by M. Listing, and in work by A. R. Gover and P. Nurowski. Their results apply to metrics satisfying a "non-degeneracy" condition on the Weyl tensor \W. We investigate the geometry of the foliations …
Study proves global existence and decay for complex wave equations.
problem Global existence and decay for quasilinear wave equations with weak-null condition.
method Novel decoupling of higher order energy estimates, focusing on tangential components.
result Established global existence and decay for solutions with small data.
A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or ω H ω\mathscr{H} ω H manifolds), as a natural setting where the notion of integrability can be formulated. We prove that the existe…
Enhances mixture models with classifier-defined weights.
problem Density evaluation and sampling in mixture models.
method Introduces Classifier Weighted Mixtures (CWM) with functional weights.
result Improves expressivity in variational estimation without increasing complexity.
New method for estimating mixture models without distributional assumptions.
problem Estimating mixture models without making distributional assumptions.
method Operator-theoretic framework and spectral algorithms.
result Characterization of identifiability in grouped mixture models.
Establishes lower bounds for weighted Ricci curvature and applies to heat flow inequalities.
problem Understanding weighted Ricci curvature and its implications.
method Introduces weighted intermediate Ricci curvature and establishes lower bounds with equivalent characterizations.
result Derives intrinsic-dimensional evolution variational inequalities and Wasserstein contraction estimates for the heat flow.
Paper introduces methods to use mixture models for modal clustering.
problem Clarity between clusters and mixture components in nonparametric mixture modeling.
method Two methods to adopt modal clustering after mixture model fitting.
result Mixture modeling can be used for clustering in a nonparametric sense.
Optimal mixtures of generative models outperform individual models on image datasets.
problem Selecting the best single model from a group of trained generative models.
method Formulated a quadratic optimization problem and proposed the Mixture-UCB algorithm for efficient selection.
result Mixture of generative models achieves better evaluation scores than individual models on benchmark datasets.
New bounds on sample size for identifying mixture models with grouped samples.
problem Identifying mixture models with minimal sample size.
method Generalized identifiability bounds for mixture models with grouped samples.
result Identifiability with ( 2 m − 1 ) / ( k − 1 ) (2m-1)/(k-1) ( 2 m − 1 ) / ( k − 1 ) samples per group, with no improvement possible. We consider unsupervised estimation of mixtures of discrete graphical models, where the class variable corresponding to the mixture components is hidden and each mixture component over the observed variables can have a potentially different Markov graph structure and parameters. We propose a novel approach for estimati…
Spatially constrained Gaussian mixture models reduce covariance complexity.
problem High dimensionality in finite mixture models for spatial data.
method Spatial covariance constraint with only four free parameters.
result Improves clustering of multi-way spatial data and inference of spatial patterns.
DGMM uses deep layers of Gaussian mixtures for flexible data modeling.
problem Efficiently modeling complex data relationships.
method Deep Gaussian Mixture Models (DGMM) with nested mixtures of linear models and factor models.
result DGMM provides a flexible nonlinear model for data description.
Two approaches improve parameter learning in various mixture models.
problem Parameter learning in mixture models.
method Complex-analytic and algebraic-combinatorial methods.
result Improved sample sufficiency for parameter estimation in specific mixture models.