New formula for Bernoulli numbers linking algebra and topology.
problem Explicit formula for Bernoulli numbers and their properties.
method Analytic and algebraic proofs, involving a function in two variables and topological self-intersections.
result A generalized Kronecker formula for Bernoulli numbers with applications in topology.
The study predicts Kronecker coefficients using interpretable machine learning models.
problem Predicting Kronecker coefficients of the symmetric group.
method Employed interpretable machine learning models with input features of triples of partitions and b-loadings.
result Achieved an accuracy of approximately 83% and over 99% with transformer-based models.
We obtain an asymptotic formula for the eigenvalue distribution function of the Laplace-Beltrami operator on the two-dimensional torus in the adiabatic limit given by a Kronecker foliation. Related problems in number theory are discussed.
This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an exp…
Researchers solve the realization of Jordan-Kronecker invariants in Lie algebras.
problem Identifying which Jordan-Kronecker invariants can be realized by Lie algebras.
method Analyzing the Kronecker and Jordan cases, proving impossibility for certain invariants, and describing realizability for others.
result Complete solution for Jordan and Kronecker cases, partial answers for others.
It is shown that in the multivariate case the orders p, of the AR part, and q, of the MA part, are not invariants of the time series. Thus, it is concluded that it only makes sense to define the class of ARMA(p,p)- irreducible models, where p is the biggest of the system's Kronecker indices. This class is shown not to …
Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.
problem Exploring hypersurfaces with constant Gauss-Kronecker curvature.
method Solving ODE for generating curves and analyzing geometric properties.
result Discovery of non-compact rotational hypersurfaces with negative Gauss-Kronecker curvature and finite volume.
In this paper we study some geometrical objects (d-tensors, multi-time semisprays of polymomenta and nonlinear connections) on the dual 1-jet vector bundle J1∗(T,M)→T×M. Some geometrical formulas, which connect the last two geometrical objects, are also derived. Finally, a canonical nonlinear…
Machine learning predicts Kronecker coefficients with high accuracy.
problem Predicting Kronecker coefficients from tensor products of symmetric group representations.
method Training machine learning models (NN, CNN, GBDT) to classify Kronecker coefficients as zero or non-zero.
result Trained models achieve high accuracy (≈0.98) in classifying Kronecker coefficients. TensorSketch solves Kronecker product regression and non-negative regression.
problem Solving regression problems with Kronecker product matrices.
method Extending TensorSketch to other norms for Kronecker product regression.
result Solving Kronecker product regression and non-negative regression in sublinear time.
Bayesian method estimates Kronecker graphical models from autoregressive processes.
problem Estimating Kronecker graphical models from autoregressive Gaussian processes.
method Bayesian approach to estimate Kronecker graphical models.
result Effectiveness demonstrated through numerical experiments and real-world data application.
KoPA approximates matrices using Kronecker products for better flexibility.
problem Matrix approximation and denoising with Kronecker product decomposition.
method Approximate a matrix as a sum of Kronecker products of smaller matrices using extended information criteria for configuration selection.
result KoPA selects the true configuration with high probability under suitable conditions.
Paper surveys Kronecker webs and solves bisymplectic realization problem.
problem Bisymplectic realization of bihamiltonian structures.
method Partial Nijenhuis operator approach.
result Partial solution to bisymplectic realization problem.
EiGLasso speeds up sparse Kronecker-sum covariance estimation.
problem Sparse Kronecker-sum inverse covariance estimation challenges in scalability and parameter identification.
method Newton's method combined with eigendecomposition of sample and feature graphs, approximating Hessian for speed.
result Two to three orders-of-magnitude speed-up on simulated and real-world data.
Scalable Gaussian processes with latent Kronecker structure for large datasets.
problem Limited scalability of Gaussian processes for large datasets.
method Leveraging latent Kronecker structure, projecting kernel matrix onto latent Kronecker product, using iterative linear system solvers and pathwise conditioning.
result Outperforms state-of-the-art sparse and variational GPs on real-world datasets with up to five million examples.
Efficiently trains Kronecker product kernel methods for graph data.
problem Learning from graph data with labeled edges and feature representations.
method Generalizes vec trick approach to non-complete training graphs.
result Order of magnitude improvements in training and prediction time.
In this paper we consider the use of the space vs. time Kronecker product decomposition in the estimation of covariance matrices for spatio-temporal data. This decomposition imposes lower dimensional structure on the estimated covariance matrix, thus reducing the number of samples required for estimation. To allow a sm…
We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space H4 with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and …
Study the geometry of statistical physics hypersurfaces.
problem Understanding the geometry of statistical physics hypersurfaces.
method Calculated induced metric, curvature tensors, and entropy for various statistical hypersurfaces.
result Characterized ideal and non-ideal statistical hypersurfaces, including phase transition singularities.
New method for scalable stochastic neural networks using Kronecker Flow.
problem Scaling stochastic neural networks to high dimensions.
method Kronecker Flow, a scalable parameterization of noise generation.
result Competitive performance on various tasks compared to existing methods.
The present paper discusses that a prescribed Gauss-Kronecker curvature problem on the product of unit spheres.
Efficiently models learning curves using Gaussian processes with latent Kronecker structure.
problem Joint modeling of machine learning model performance across hyper-parameters and training progress.
method Imposes latent Kronecker structure to leverage efficient product kernels and handle missing values.
result Matches the performance of a Transformer on a learning curve prediction task.
We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the total space of a fiber bundle with a torus as the base space and a circle as a typ…
Paper proposes a new method for approximating high-dimensional matrices using Kronecker products.
problem Discovering low-dimensional structure in high-dimensional data.
method Hybrid Kronecker Product Approximation (hKoPA) and estimation procedures.
result The proposed methods provide flexible and effective dimension reduction.
How can we model networks with a mathematically tractable model that allows for rigorous analysis of network properties? Networks exhibit a long list of surprising properties: heavy tails for the degree distribution; small diameters; and densification and shrinking diameters over time. Most present network models eithe…
STARK learns structured dictionaries for tensor data.
problem Representing multidimensional data with structured dictionaries.
method Solves a convex relaxation of a nonconvex rank-1 tensor recovery problem.
result Empirical results show promising performance for tensors of any order.
Detects missing tensor signals in a KS subspace with high probability.
problem Detecting tensor signals with many missing entities in a KS subspace.
method Projecting the signal onto the KS subspace and bounding residual energy.
result Reliable detection is possible if the missing signal cardinality exceeds KS subspace dimensions.
We investigate the structure of 3-dimensional complete minimal hypersurfaces in the unit sphere with Gauss-Kronecker curvature identically zero.
Estimates the dimension of Kronecker product models using Jacobian rank and tropical morphism.
problem Estimating the dimension of Kronecker product models.
method Using Jacobian rank and tropical morphism to describe the limit of the model.
result Combinatorial conditions for the expected dimension and proof for binary restricted Boltzmann machine.
Paper provides conditions for local recovery of tensor data's Kronecker-structured dictionaries.
problem Local recovery of Kronecker-structured dictionaries for tensor data.
method Derives sufficient conditions for local recovery of coordinate dictionaries.
result Sufficient conditions guarantee recovery of individual coordinate dictionaries up to specified error.
Paper explores tradeoffs in classification using tensor subspaces.
problem Supervised classification with sample, computation, and storage complexities.
method Use of tensor subspaces, particularly hierarchical Kronecker structured subspaces.
result Hierarchical Kronecker structured subspaces improve classification tradeoffs.
New method for faster graph parameter inference from large random Kronecker graphs.
problem Efficiently infer graph parameters from large random Kronecker graphs.
method Decompose adjacency matrix into signal and noise components, then use denoising and solving approach.
result Proposed method achieves comparable or better performance than existing methods at lower computational cost.
We investigate complete minimal hypersurfaces in the Euclidean space , with Gauss-Kronecker curvature identically zero. We prove that, if f:M3→R4 is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature b…
This paper studies iteration convergence of Kronecker graphical lasso (KGLasso) algorithms for estimating the covariance of an i.i.d. Gaussian random sample under a sparse Kronecker-product covariance model and MSE convergence rates. The KGlasso model, originally called the transposable regularized covariance model by …
In this paper we present a local description for complete minimal hypersurfaces in S5 with zero Gauss-Kronecker curvature, zero 3-mean curvature and nowhere zero second fundamental form.
New method for matrix completion using Kronecker product approximation.
problem Matrix completion with low Kronecker rank structure.
method Alternative matrix representation using Kronecker product, identification through mean squared error and modified cross-validation.
result Consistency of the method under suitable signal-to-noise ratio conditions.
A parallel algorithm learns efficient Kronecker product dictionaries.
problem Sparse representation of 2D signals like images and hyperspectral data.
method Highly parallelizable algorithm for learning separable dictionaries.
result Competitive sparse representations at lower computational cost.
Shampoo optimizes preconditioners for faster convergence in machine learning.
problem Improving convergence speed in machine learning optimization.
method Explicit connection between Shampoo's Kronecker product approximation and optimal matrix approximations.
result The square of Shampoo's approximation is equivalent to a single power iteration step for optimal Kronecker product approximation.
A new method for optimizing deep neural networks using TKFAC.
problem Optimizing deep neural networks with second-order methods.
method Proposes Trace-restricted Kronecker-factored Approximate Curvature (TKFAC) for Fisher information matrix approximation.
result TKFAC improves performance on deep network architectures compared to state-of-the-art algorithms.
Totally geodesic minimal hypersurfaces in H5 with specific curvature properties.
problem Characterizing minimal hypersurfaces in hyperbolic space with certain curvature conditions.
method Analyzing properties of minimal hypersurfaces in H5 with constant scalar curvature and zero Gauss-Kronecker curvature. result Any complete minimal hypersurface in H5 with constant scalar curvature and zero Gauss-Kronecker curvature is totally geodesic. Study mapping class group action on character varieties, proving Kronecker's Theorem.
problem Topological-dynamical action of mapping class group on character varieties.
method Analyzes Tn-character variety and dense orbit conditions. result Provides a dynamical proof of Kronecker's Theorem.
Study on classification and representation of multidimensional signals using Kronecker-structured models.
problem Performance limits and algorithms for classification and representation of multidimensional signals.
method Analysis of diversity order and classification capacity, development of K-SLD2 algorithm for fast Kronecker-structured learning.
result Agreement between diversity order analysis and empirical classification performance of K-S models.
We give a partial local description of minimal hypersurfaces M3 with identically zero Gauß-Kronecker curvature function in the unit 4-sphere S4(1), without assumption on the compactness of M3.
MCCA extracts shared structure from multiple tensor datasets.
problem Extracting shared structure from multiple tensor datasets.
method Multilinear common component analysis (MCCA) using Kronecker products of mode-wise covariance matrices.
result MCCA constructs a common basis that retains information from multiple tensor datasets.
Kronecker DPPs enable efficient sampling and learning for large DPP problems.
problem Efficient sampling and learning for large Determinantal Point Processes (DPPs).
method Introducing KronDPP, a DPP model with a tensor product kernel matrix, enabling fast exact sampling. Overcoming challenges in learning parameters efficiently.
result Efficient algorithms for learning parameters of KronDPP, overcoming the difficulty of leveraging Kronecker product structure.
Efficient subspace clustering using Kronecker product reduces computational complexity.
problem Efficiency and scalability issues in traditional subspace clustering methods for large datasets.
method Proposes a subspace clustering model based on the Kronecker product to reduce computational complexity.
result Significantly improved efficiency compared to state-of-the-art methods on public datasets.
Stochastic Kronecker graphs supply a parsimonious model for large sparse real world graphs. They can specify the distribution of a large random graph using only three or four parameters. Those parameters have however proved difficult to choose in specific applications. This article looks at method of moments estimators…
The aim of this paper is to complete the local classification of minimal hypersurfaces with vanishing Gauss-Kronecker curvature in a 4-dimensional space form. Moreover, we give a classification of complete minimal hypersurfaces with vanishing Gauss-Kronecker curvature and scalar curvature bounded from below.