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.
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…
Study on Gaussian ensemble of matrix products with mixed moments computed.
problem Understanding the statistical properties of matrix products of Gaussian matrices.
method Analysis of a multi-Wishart ensemble and enumeration of non-crossing pairings.
result Mixed moments of the product matrix are computed and found to be weighted by Fuss-Catalan numbers at large N. 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.
The paper analyzes stability of random matrix products with Markovian noise.
problem Analyzing stability of random matrix products with Markovian noise.
method Using a super-Lyapunov drift condition and controlled growth of matrix-valued functions, the paper provides an exponential stability result for the p-th moment of random matrix product.
result Finite-time p-th moment bounds for linear stochastic approximation and TD learning algorithms.
Revisits neural collaborative filtering vs. matrix factorization, showing dot product superiority.
problem Comparing neural collaborative filtering to matrix factorization in recommendation systems.
method Revisited experiments using MLPs as similarity functions, comparing dot product to MLP outputs.
result Simple dot product outperforms MLP-based learned similarities in practical settings.
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 proposes algorithms for BMF using integer programming.
problem Approximating binary input matrix as product of two smaller binary factors.
method Alternating optimization strategy using integer programming to solve subproblems and combine solutions.
result Proposed algorithms outperform state of the art on medium-scale problems.
Data often comes in the form of an array or matrix. Matrix factorization techniques attempt to recover missing or corrupted entries by assuming that the matrix can be written as the product of two low-rank matrices. In other words, matrix factorization approximates the entries of the matrix by a simple, fixed function-…
Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
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.
New algorithm reduces cold-start costs in multi-armed bandits for many products.
problem High burn-in costs in multi-armed bandits for new products.
method Two-phase bandit algorithm using subsampling and low-rank matrix estimation.
result Reduces burn-in costs and expedites experiment in large product sets.
We propose a statistical model for natural language that begins by considering language as a monoid, then representing it in complex matrices with a compatible translation invariant probability measure. We interpret the probability measure as arising via the Born rule from a translation invariant matrix product state.
Defines cross product for m vectors in n-dimensional spaces.
problem No universal definition for cross product in high-dimensional spaces.
method Defines cross product for m vectors in n-dimensional spaces with any metric matrices.
result Cross product length represents m-dimensional volume, components represent volume directions.
This paper improves compression of large NLP models using doped Kronecker Products.
problem Accuracy loss when compressing large NLP tasks with Kronecker Products.
method Doping Kronecker Products with an overlay matrix to recover accuracy, and a new regularization scheme called co matrix dropout regularization (CMR).
result Compression of a large language model with LSTM layers of size 25 MB by 25x with 1.4% loss in perplexity score.
Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AA^T (in exact arithmetic) from c >= rank(A) columns depend on the right singular vector matrix of A. For a Monte-…
A new matrix concentration inequality for random products of matrices.
problem Understanding the behavior of random matrix products under bounded independent positive semidefinite matrices.
method Developed a non-asymptotic concentration inequality for the product of matrices.
result The inequality provides a bound on the deviation of the matrix product from its expected value.
We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a categorification of MOY polynomial. For the essential colored planar diagrams with additional…
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.
Few attempts have been proposed in order to describe the statistical features and historical evolution of the export bipartite matrix countries/products. An important standpoint is the introduction of a products network, namely a hierarchical forest of products that models the formation and the evolution of commodities…
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.
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…
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.
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.
At the core of any inference procedure in deep neural networks are dot product operations, which are the component that require the highest computational resources. A common approach to reduce the cost of inference is to reduce its memory complexity by lowering the entropy of the weight matrices of the neural network, …
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However, these techniques require tensor decomposition which could lead to the loss of infor…
Paper proposes a new approach to optimal transport for vector and matrix densities.
problem Optimal transport for vector and matrix densities with positivity and action transitivity constraints.
method Gauge-theoretic approach using semi-direct product groups of diffeomorphisms and gauge transformations.
result Bures-type metrics on semi-direct product groups relate to Wasserstein-type metrics on vector and matrix densities via Riemannian submersions.
A new method for higher-order co-occurrences in hypergraphs.
problem Computing higher-order co-occurrences in hypergraphs.
method Face-splitting product or transpose Khatri-Rao product for higher order tuple co-occurrences.
result Demonstrates the utility of the higher order co-occurrence tensor in NLP and hypergraph models.
NG+ method improves deep learning efficiency and accuracy.
problem Efficiency and accuracy in deep learning models.
method Proposes NG+ method using matrix-product natural gradient approach.
result Established global convergence and provided regret bound.
New method estimates large matrices' spectra from small sub-matrices.
problem Estimating large matrices' spectra when full matrix-vector products are not available.
method Free decompression based on free probability theory.
result Estimates eigenspectrum of impalpable matrices from small sub-matrices.
A Kronecker product model is the set of visible marginal probability distributions of an exponential family whose sufficient statistics matrix factorizes as a Kronecker product of two matrices, one for the visible variables and one for the hidden variables. We estimate the dimension of these models by the maximum rank …
Sketch Tomography improves quantum state estimation accuracy.
problem Efficiently estimating quantum states, especially MPS states.
method Hybridizes classical shadow protocol with tensor train ansatz.
result Proven convergence with quadratic sample complexity.
Matrix Product States (MPS), also known as Tensor Train (TT) decomposition in mathematics, has been proposed originally for describing an (especially one-dimensional) quantum system, and recently has found applications in various applications such as compressing high-dimensional data, supervised kernel linear classifie…
Subspace clustering is a useful technique for many computer vision applications in which the intrinsic dimension of high-dimensional data is often smaller than the ambient dimension. Spectral clustering, as one of the main approaches to subspace clustering, often takes on a sparse representation or a low-rank represent…
A new method uses Gram matrix for efficient multivariate functional principal components.
problem Efficiently estimating eigencomponents of multidimensional functional datasets.
method Proposes using inner-product matrix to estimate eigenelements of multivariate and multidimensional functional datasets.
result Established relationship between eigenelements of covariance operator and inner-product matrix.
A new algorithm speeds up matrix multiplication without actual multiplication.
problem Efficiently multiplying matrices in machine learning.
method Learning-based algorithm that uses hashing, averaging, and byte shuffling.
result Often runs 100x faster than exact matrix products and 10x faster than current approximate methods.
The study explores how Matrix Product States can represent boolean and continuous functions.
problem Representing arbitrary boolean and continuous functions using Matrix Product States.
method Developed a construction method for MPS to represent boolean gates and proved density in continuous function space.
result MPS can accurately represent arbitrary boolean functions and continuous functions densely.
The easy access to large data sets has allowed for leveraging methodology in network physics and complexity science to disentangle patterns and processes directly from the data, leading to key insights in the behavior of systems. Here we use to country specific food production data to study binary and weighted topologi…
The paper defines the OU matrix for braid diagrams and finds determinant relationships.
problem Understanding the layeredness of braid diagrams.
method Defining the OU matrix and analyzing its determinant for layered braid diagrams.
result The determinant of the OU matrix for layered braid diagrams is the product of the determinants of the layers.
The graph Laplacian is a standard tool in data science, machine learning, and image processing. The corresponding matrix inherits the complex structure of the underlying network and is in certain applications densely populated. This makes computations, in particular matrix-vector products, with the graph Laplacian a ha…
Deep ReLU networks can approximate matrix-vector products with error bounds.
problem Can deep ReLU networks accurately approximate matrix-vector products?
method Derived error bounds in Lebesgue and Sobolev norms for deep ReLU FNNs.
result Developed deep approximation theory with successful applications.
Study uses Google matrix analysis to show how COVID-19 changed international trade flows.
problem Impact of COVID-19 on international trade patterns.
method Google matrix analysis of World Trade Network (WTN), including PageRank, CheiRank, and reduced Google matrix.
result Significant changes in international trade flows due to the pandemic, affecting export and import balances.
Study on Neural Tangent Kernel of Matrix Product States and their convergence.
problem Understanding the convergence of Neural Tangent Kernel of Matrix Product States.
method Analyzing the Neural Tangent Kernel of Matrix Product States and proving its convergence in the infinite bond dimensional limit.
result The Neural Tangent Kernel of Matrix Product States converges to a constant matrix during training.
New methods estimate covariance for matrix data without assuming fixed size or specific distributions.
problem Estimating covariance for high-dimensional matrix data without distributional assumptions.
method Unified framework for bandable covariance estimation with rank one approximation, robust to heavy-tailed data.
result Proposed estimators are rate-optimal and perform well in simulations and real applications.
This paper introduces matrix product state (MPS) decomposition as a new and systematic method to compress multidimensional data represented by higher-order tensors. It solves two major bottlenecks in tensor compression: computation and compression quality. Regardless of tensor order, MPS compresses tensors to matrices …
Matrix factorization methods are extensively employed to understand complex data. In this paper, we introduce the cross-product penalized component analysis (XCAN), a sparse matrix factorization based on the optimization of a loss function that allows a trade-off between variance maximization and structural preservatio…
Generalizes randomized SVD for better matrix approximations using Gaussian vectors.
problem Computing accurate rank-k approximations of matrices with limited data.
method Extends randomized SVD to multivariate Gaussian vectors, incorporating prior knowledge and using Gaussian processes.
result Demonstrates improved accuracy in approximating matrices and Hilbert-Schmidt operators.
Using the United Nations COMTRADE database \cite{comtrade} we construct the Google matrix G of multiproduct world trade between the UN countries and analyze the properties of trade flows on this network for years 1962 - 2010. This construction, based on Markov chains, treats all countries on equal democratic grounds …