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…
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48 results for “tensor-tensor product”
Paper proposes a new method for exact recovery in robust tensor principal component analysis.
problem Exact recovery of low-rank and sparse components in tensors.
method Proposes a new method based on tensor-tensor product and t-SVD to solve a convex optimization problem.
result Exact recovery achieved in a deterministic fashion without randomness assumptions.
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.
In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product). Induced by the t-product, we first rigorously deduce the tensor sp…
We propose a tensor neural network (-NN) framework that offers an exciting new paradigm for designing neural networks with multidimensional (tensor) data. Our network architecture is based on the -product (Kilmer and Martin, 2011), an algebraic formulation to multiply tensors via circulant convolution. In this $t…
ScaledGD algorithm estimates low-rank tensors efficiently from corrupted data.
problem Estimating meaningful information from corrupted tensor data.
method Scaled gradient descent (ScaledGD) algorithm with tailored spectral initializations.
result ScaledGD achieves linear convergence at a constant rate independent of condition number.
This work studies the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is motivated by the recently proposed linear transforms based tensor-tensor product and tensor SVD. We define a new transforms depended tensor rank…
Study spectral learning for odeco tensors, addressing initialization bottlenecks.
problem Recovering orthogonally decomposable tensors under noise.
method Investigates perturbation bounds, non-convex optimization, and initialization strategies.
result Initialization is the main bottleneck for efficient algorithms.
The paper tackles tensor factorization and completion from noisy data.
problem Sparse nonnegative tensor factorization and completion from partial and noisy observations.
method Minimizes the sum of maximum likelihood estimation and tensor norm with nonnegativity constraints.
result Error bounds and minimax lower bounds are established for the proposed model.
Money as a Tensorq-fin.GN
Money analyzed as a multidimensional tensor for better economic policy.
problem Economic complexity and policy responsiveness.
method Tensor analysis of money dynamics.
result Enhanced economic policy design and resilience.
A new tree method for tensor data improves regression accuracy.
problem Efficiently modeling tensor data for regression problems.
method Scalar-output regression tree models for scalar-on-tensor problems, and tensor-on-tensor problems using additive tree ensemble approaches.
result The tensor-input tree (TT) method outperforms tensor-input GP models in efficiency and accuracy.
GETF efficiently decomposes large-scale Boolean tensors.
problem Efficiently factorizing large-scale Boolean tensors.
method Geometric Expansion for all-order Tensor Factorization (GETF).
result GETF significantly improves reconstruction accuracy and efficiency.
Sliding down the stairs: how correlated latent variables accelerate learning with neural networksstat.ML
Neural networks learn faster with correlated latent variables.
problem Efficiently learning from higher-order correlations in neural networks.
method Analytical derivation and simulations of two-layer neural networks.
result Correlations between latent variables speed up learning from higher-order correlations.
Specialized Deep Learning (DL) acceleration stacks, designed for a specific set of frameworks, model architectures, operators, and data types, offer the allure of high performance while sacrificing flexibility. Changes in algorithms, models, operators, or numerical systems threaten the viability of specialized hardware…
Recently, fundamental conditions on the sampling patterns have been obtained for finite completability of low-rank matrices or tensors given the corresponding ranks. In this paper, we consider the scenario where the rank is not given and we aim to approximate the unknown rank based on the location of sampled entries an…
A low-rank tensor model simplifies multi-dimensional Markov chains.
problem Simplifying the dynamics of multi-dimensional Markov chains.
method Low-rank tensor decomposition for multi-dimensional state spaces.
result Our tensor model requires fewer parameters and samples than conventional methods.
Paper proposes a new model for noisy tensor completion.
problem Handling noise in tensor completion.
method Tensor ring nuclear norm (TRNN) and least-squares estimator.
result Effective recovery of noisy incomplete tensor data.
ENTED efficiently decomposes binary and count tensors using nonparametric Gaussian processes.
problem Handling high-dimensional and sparse binary and count data with traditional tensor decompositions.
method ENTED uses nonparametric Gaussian processes and sparse orthogonal variational inference to handle binary and count tensors.
result ENTED outperforms traditional methods in binary and count tensor completion tasks.
New tensor recovery method improves efficiency under strict complementarity.
problem Efficiently recovering low-rank tensors using tensor nuclear norm.
method Developed strict complementarity condition for tensor nuclear norm ball and applied to gradient methods.
result Standard gradient methods achieve linear convergence and nearly linear runtime under strict complementarity.
The paper studies a new type of submanifolds in product spaces.
problem Characterizing and understanding warped product pointwise bi-slant submanifolds.
method Introduced and studied warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds.
result Characterization results and non-trivial examples of these submanifolds.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
Defines products for fibered corners manifolds, generalizing resolutions.
problem Resolving fibered corners manifolds.
method Introduces a category of fibered corners manifolds with products and transverse fiber products, defining the 'ordered product' for wedge metrics.
result The 'ordered product' is a natural product for wedge metrics.
Productivity and credit limits affect aggregate production in non-monotonic ways.
problem Understanding how aggregate production is influenced by individual characteristics and financial constraints.
method Analytical proof of non-monotonic effects of productivity and credit limits on aggregate production in a general equilibrium model.
result Equilibrium aggregate production can be non-monotonic in both individual productivity and credit limit.
Einstein metrics on products are shown to be warped.
problem Characterizing Einstein metrics on conformal products.
method Proving Einstein metrics on conformal products are warped products under natural geometric conditions.
result Einstein metrics on conformal products are proven to be warped products.
Uniform criterion for vanishing products in bounded cohomology.
problem Vanishing of cup products and Massey products in bounded cohomology.
method Uniform vanishing criterion for products in bounded cohomology.
result Reproved and extended previous vanishing results.
The paper studies affine connections on singular warped products and their curvature.
problem Analyzing affine connections on singular warped products.
method Introducing semi-symmetric metric and non-metric Koszul forms, and expressing their curvature in terms of factor manifolds.
result Generalized results for singular multiply warped products.
We refine the intersection product in homology to an equivariant setting, which unifies several known constructions. As an application, we give a common generalisation of the Chas-Sullivan string product on a manifold and the Chataur-Menichi string product on the classifying space by defining a string product on the Bo…
New method compares geometric and standard cup products.
problem Reconciling partially defined and fully defined cochain products.
method Vector field flow through cubulation to compare products.
result Explicit cochain level comparison between intersection and cup products.
The paper examines Einstein doubly warped product manifolds with a semi-symmetric metric connection.
problem Characterizing Einstein doubly warped product manifolds with a semi-symmetric metric connection.
method Deriving curvature formulas and proving necessary and sufficient conditions for a manifold to be a warped product.
result Obtained results for Einstein doubly warped product manifolds and Einstein-like doubly warped product manifolds.
Generalizes warped product submersion to conformal case.
problem Understanding angles preservation in submersions.
method Introduces conformal warped product submersion.
result Fundamental tensors derived for conformal submersion.
The paper explores geometric properties of Riemannian warped product maps and their curvature.
problem Investigating the geometric properties of Riemannian warped product maps.
method The approach involves establishing conditions for geodesics, deriving curvature tensors, and examining various types of maps.
result Derivation of integral formula for scalar curvature of conformal Riemannian warped product maps.
In this article we obtain classification results on the quasi-product production functions in terms of the geometry of their associated graph hypersurfaces, generalizing in a new setting some recent results concerning basic production models. In particular, we obtain several results on the geometry of Spillman-Mitscher…
Conformal product structures on math.DG
Constructing a conformal product structure on using the Reeb foliation.
problem First example of conformal product structure on compact simply connected manifold.
method Reeb foliation
result Conformal product structure on
Super twisted productsmath.DG
The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined -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.
The study examines Einstein-Finsler spaces using Minkowskian products.
problem Characterizing Einstein-Finsler spaces through Minkowskian products.
method Proving conditions for Einstein-Finsler spaces in terms of Minkowskian products.
result If a Minkowskian product Finsler manifold is Einstein, then either the product manifold is Ricci flat or both quotient manifolds are Einstein with same scalar functions.
In this paper, we completely classify homogeneous production functions with an arbitrary number of inputs whose production hypersurfaces are flat. As an immediate consequence, we obtain a complete classification of homogeneous production functions with two inputs whose production surfaces are developable.
Study on warped product Yamabe solitons with constant fiber curvature.
problem Characterizing nontrivial warped product Yamabe gradient solitons.
method Investigation of warped product manifolds, derivation of scalar curvature estimates.
result Nontrivial warped product Yamabe gradient solitons have constant scalar curvature in the fiber.
A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.
Economies grow by upgrading the type of products they produce and export. The technology, capital, institutions and skills needed to make such new products are more easily adapted from some products than others. We study the network of relatedness between products, or product space, finding that most upscale products a…
A new method for analyzing product competition using low-dimensional embeddings.
problem Computational challenges in studying product-level competition for millions of products.
method Product2Vec, a method based on representation learning algorithm Word2Vec.
result The method produces more accurate demand forecasts and price elasticities compared to state-of-the-art models.
The study characterizes submanifolds in product spaces.
problem Limited studies on pseudo-umbilical submanifolds.
method Using projections from product structure, conditions for submanifolds to be invariant, anti-invariant, or semi-invariant are derived.
result Necessary and sufficient conditions for pseudo-umbilical submanifolds in locally product Riemannian manifolds.
Geometry of warped product and CR-warped product submanifolds in Kaehler manifolds: modified versionmath.DG
The warped product of two Riemannian manifolds and is the product manifold equipped with the warped product metric , where is a positive function on . Warped products play very important roles in differential geometry as well as in physic…
Paper reinterprets marginal productivity theory using vectorial products, challenging traditional ethical interpretations.
problem Challenges traditional ethical interpretations of marginal productivity theory.
method Formulates marginal productivity theory using vectorial marginal products, contrasting with traditional scalar approach.
result Vectorial marginal products conflict with traditional distributive shares picture of property.
We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our …
Warped product affects divergences in information geometry.
problem Warped product's impact on divergences in information geometry.
method Study of warped product on information geometry.
result Warped product does not preserve canonical divergences.
Complementary products recommendation is an important problem in e-commerce. Such recommendations increase the average order price and the number of products in baskets. Complementary products are typically inferred from basket data. In this study, we propose the BB2vec model. The BB2vec model learns vector representat…
Study describes conformal product structures on compact Kähler manifolds.
problem Characterizing compact Kähler manifolds with conformal product structures.
method Geometric description of conformal product structures on compact Kähler manifolds.
result Geometric characterization of compact Kähler manifolds with conformal product structures.
Isotropic geometry of graph surfaces associated with product production functions in economicsmath.DG
A production function is a mathematical formalization in economics which denotes the relations between the output generated by a firm, an industry or an economy and the inputs that have been used in obtaining it. In this paper, we study the product production functions of 2 variables in terms of the geometry of their a…