ROD reconstructs conditional mean surfaces from observational data, invariant to term order.
problem Regression estimates from observational data can be biased by multicollinearity and term order.
method Retrospective Orthogonal Design (ROD) reconstructs surfaces on a probability-balanced lattice, preserving observed and completing unsupported cells.
result ROD outperformed polynomial regression across various data-generating processes, achieving high out-of-sample R2. Verma Howe duality connects tensor products of Verma modules to LKB representations.
problem Understanding the relationship between tensor products of Verma modules and LKB representations.
method Established a quantized version of Verma Howe duality and used it to prove the simplicity of LKB representations.
result LKB representations arise from the quantized Verma Howe duality and are shown to be simple modules.
Paper proves Koszul duality for weighted A-infinity algebras.
problem Koszul duality for weighted A-infinity algebras.
method Constructs new box tensor product for weighted A-infinity bimodules and verifies correspondence between maps and bimodules.
result Proves Koszul duality result between weighted A-infinity algebras.
Recurrent Neural Networks (RNNs), which are a powerful scheme for modeling temporal and sequential data need to capture long-term dependencies on datasets and represent them in hidden layers with a powerful model to capture more information from inputs. For modeling long-term dependencies in a dataset, the gating mecha…
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.
Two formulas for Chern classes of tensor products of vector bundles are presented.
problem Calculating Chern classes for tensor products of vector bundles.
method Two formulas using matrices and polynomials to compute Chern classes.
result Determinantal formulas for Chern classes of tensor products.
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.
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.
We prove explicit formulas for Chern classes of tensor products of vector bundles, with coefficients given by certain universal polynomials in the ranks of the two bundles.
Combines RNNs and tensor products for sequential data, outperforming state-of-the-art.
problem Improving symbolic interpretation and systematic generalization in natural language reasoning.
method End-to-end training of a recurrent neural network architecture with tensor product representations.
result Significantly outperforms state-of-the-art models in natural language reasoning tasks.
A new method reduces the complexity of tensor products from cubic to quadratic, improving both speed and accuracy.
problem Efficiently computing high-dimensional tensor products for polynomial kernels.
method Complex-to-Real (CtR) modification of sketches using complex random projections.
result Achieves state-of-the-art performance in accuracy and speed.
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.
A new feature coding method for invariant features using tensor products.
problem Learning invariant features for transformations represented by orthogonal matrices.
method Group-invariant feature vector using tensor-product representations of basic representations.
result Group-invariant feature vector contains sufficient discriminative information for linear classifiers.
Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. T…
Proves elliptic operator images are closed on Hilbert bundles.
problem Closedness of images of elliptic operators on Hilbert bundles.
method Analyzes tensor product of elliptic operators and compares images.
result Establishes closedness of images with respect to natural topology.
It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describi…
The complex analytic methods have found a wide range of applications in the study of multiplicity-free representations. This article discusses, in particular, its applications to the question of restricting highest weight modules with respect to reductive symmetric pairs. We present a number of multiplicity-free branch…
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.
Given a simple algebraic group G, a web is a directed trivalent graph with edges labelled by dominant minuscule weights. There is a natural surjection of webs onto the invariant space of tensor products of minuscule representations. Following the work of Westbury, we produce a set of webs for $\SL_n$ which form a bas…
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.
AMP algorithm for matrix tensor product model provides recovery conditions.
problem Generalization of standard spiked matrix models with multiple pairwise observations.
method Approximate message passing with optimal weighing and combining of estimates.
result Asymptotically exact performance description and necessary/sufficient recovery conditions.
Ejiri's torus in S5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any space forms. Li and Vrancken classified all Willmore surfaces of tensor product in Sn by reducing them into elastic curves in S3, and the Ejiri torus appeared as a special example. I…
Machine learning accelerates Lie algebra computations.
problem Computing tensor products and branching rules of Lie algebras.
method Machine learning for Lie algebra computations.
result Achieves significant speed-ups in Lie algebra computations.
We completely describe Wahlquist-Estabrook prolongation structures (coverings) dependent on u, u_x, u_{xx}, u_{xxx} for the Krichever-Novikov equation u_t=u_{xxx}-3u_{xx}^2/(2u_x)+p(u)/u_x+au_x in the case when the polynomial p(u)=4u^3-g_2u-g_3 has distinct roots. We prove that there is a universal prolongation algebra…
Proposes ANOVA-TPNN for stable interpretation of complex functions.
problem Stability issues in estimating components of functional ANOVA models.
method Introduces ANOVA-TPNN based on tensor product basis expansion.
result ANOVA-TPNN provides stable estimation of components.
Paper finds formulas for mutual information and MMSE in matrix tensor product problems.
problem High-dimensional inference problems involving matrix tensor products.
method Single-letter formulas for mutual information and MMSE, using new techniques.
result Analytical formulas describe leading order terms in mutual information and MMSE.
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed 3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered Floer homology invariants of 3-manifolds with codimension-2 c…
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models ha…
ContraSim learns financial headline similarities for market forecasting.
problem Financial market forecasting accuracy improvement.
method ContraSim framework with Weighted Headline Augmentation and WSSCL.
result Improves financial forecasting accuracy by 7%.
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.
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…
Enhanced Transformer solves math problems better with explicit relation encoding.
problem Improving Transformer models for solving math word problems.
method Integrates Tensor-Product Representations and TP-Attention mechanism.
result Sets new state of the art on the Mathematics Dataset.
Many modern data sets are sampled with error from complex high-dimensional surfaces. Methods such as tensor product splines or Gaussian processes are effective/well suited for characterizing a surface in two or three dimensions but may suffer from difficulties when representing higher dimensional surfaces. Motivated by…
Defines tensor product of profinitely many vector spaces over F2.
problem Defining tensor product of profinitely many copies of a vector space.
method Proposes a definition for finite-dimensional vector spaces over F2 with specific group actions.
result Organizes computations in Heegaard Floer homology.
We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a lin…
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
New projection operators for multipatch spaces with stable properties.
problem Problems with non-matching interfaces in multipatch spaces.
method Construction of commuting projection operators on de Rham sequences of multipatch spaces with local tensor-product parametrization.
result Local and stable projection operators in any Lp norm for shape-regular spline patches with different mappings and local refinements. In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic geometry" of noncompact hypercomplex manifolds. The basic building blocks of the theory …
Study of group orderability using tensor and exterior squares.
problem Circular orderability of groups and torsion elements.
method Analysis of tensor and exterior squares of groups.
result Circularly orderable groups have left-orderable tensor and exterior squares under certain conditions.
In this paper we give an example of a linear group such that its tensor square is not linear. Also, we formulate some sufficient conditions for the linearity of non-abelian tensor products G⊗H and tensor squares G⊗G. Using these results we prove that tensor squares of some groups with one relation a…
We review progress on the generalized Witten conjecture and some of its major ingredients. This conjecture states that certain intersection numbers on the moduli space of higher spin curves assemble into the logarithm of the tau function of a semiclassical limit of the r-th Gelfand-Dickey (or KdV_r) hierarchy. Addition…
HUBERT combines BERT's structure with TPRs to improve NLP task transfer.
problem Improving transferability among NLP tasks.
method Combines BERT's bidirectional Transformer structure with TPRs to learn shared data structures.
result HUBERT outperforms BERT on GLUE and HANS datasets, showing better transferability.
Paper develops polynomial approximations for complex probability densities.
problem Approximating high-dimensional concentrated probability densities.
method Tensor-product spectral polynomials and KR rearrangements.
result Efficient approximation of complex densities using composite maps.
Constructs Koszul dual algebras for star-shaped diagrams in 3-manifolds.
problem Constructing algebraic structures for 3-manifold homology.
method Uses graphical calculus to construct Koszul dual weighted A∞-algebras and dualizing bimodules. result Proves duality of constructed algebras and bimodules.
New method proves non-contrastive self-supervised learning learns useful features.
problem Understanding how non-contrastive self-supervised learning (NS-SL) learns useful features.
method Proved in a linear network, NS-SL learns a desirable projection matrix and reduces sample complexity. Suggested weight decay acts as an implicit threshold.
result DirectCopy, a simpler and more efficient algorithm, outperforms DirectPred on various datasets.
New method adds interactions to interpretable models for large-scale data.
problem Limited model complexity and lack of interactions in interpretable models.
method Factorization method to derive scalable higher-order tensor product spline models.
result Incorporates all higher-order interactions of non-linear feature effects without computational penalties.
HAR regression improves performance on small datasets.
problem Small datasets with complex functions.
method Data-adaptive kernel ridge regression using tensor-product spline basis.
result Achieves n−1/3 convergence rate for right-continuous functions.