We obtain minimal dimension matrix representations for each indecomposable five-dimensional Lie algebra over and justify in each case that they are minimal. In each case a matrix Lie group is given whose matrix Lie algebra provides the required representation.
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48 results for “representation matrix”
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…
Transfer knowledge from multiple sources to improve matrix completion.
problem Matrix completion with noisy data.
method Aggregating singular subspaces information from multiple sources to solve a two-way PCA problem and transform into a low-dimensional linear regression.
result Guaranteed statistical efficiency in transforming the high-dimensional target matrix completion problem.
We simplify matrix computations for block matrices, especially useful for covariance and correlation matrices.
problem Complex computations for block matrices, especially for covariance and correlation matrices.
method Obtained a canonical representation for block matrices, facilitating computation of various matrix operations.
result Simplified computation of matrix operations for block matrices, particularly useful for covariance and correlation matrices.
This paper clarifies vine copula structures using graph and matrix representations.
problem Ambiguity in vine copula representations in literature.
method Graph and matrix representations to clarify vine structures, including cherry and chordal sequences.
result A unique matrix representation of vine structures when given a perfect elimination ordering.
Generalized matrix-fractional (GMF) functions are a class of matrix support functions introduced by Burke and Hoheisel as a tool for unifying a range of seemingly divergent matrix optimization problems associated with inverse problems, regularization and learning. In this paper we dramatically simplify the support func…
Finding a new mathematical representations for graph, which allows direct comparison between different graph structures, is an open-ended research direction. Having such a representation is the first prerequisite for a variety of machine learning algorithms like classification, clustering, etc., over graph datasets. In…
NIMFA is a Python library for nonnegative matrix factorization.
problem Efficiently factorizing nonnegative matrices for various applications.
method Unified interface, state-of-the-art methods, initialization approaches, quality scoring, supports dense and sparse matrices.
result Unified and efficient implementation of nonnegative matrix factorization methods.
New matrix reveals cluster info in sparse directed graphs.
problem Analyzing cluster information in directed graphs.
method Proposed complex non-backtracking matrix integrating Hermitian adjacency matrix and non-backtracking matrix properties.
result The complex non-backtracking matrix holds cluster information, especially for sparse directed graphs.
Nonnegative Matrix Factorization (NMF) has been a popular representation method for pattern classification problem. It tries to decompose a nonnegative matrix of data samples as the product of a nonnegative basic matrix and a nonnegative coefficient matrix, and the coefficient matrix is used as the new representation. …
Low-dimensional representations of matrix groups and group actions on CAT(0) spaces and manifoldsmath.GT
We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.
On the Optimal Representation Efficiency of Barlow Twins: An Information-Geometric Interpretationcs.LG
This paper analyzes Barlow Twins' representation efficiency using information-geometric methods.
problem Understanding and comparing the efficiency of self-supervised learning methods.
method Introduces an information-geometric framework to quantify representation efficiency and applies it to Barlow Twins.
result Proves that Barlow Twins achieves optimal representation efficiency (η=1).
Study asymptotics of unitary matrix elements in quantum mechanics.
problem Asymptotic behavior of unitary matrix elements in quantum mechanics.
method Uses Berezin-Toeplitz quantization and symplectic geometry.
result Recover asymptotics of Wigner's d-matrix elements for spin representations.
Low-Rank Plus Sparse Matrix Transfer Learning under Growing Representations and Ambient Dimensionscs.LG
The paper tackles transfer learning for growing matrix representations, improving estimation accuracy.
problem Structured matrix estimation under growing ambient dimensions and latent representations.
method Proposes a general transfer framework decomposing target parameters into embedded source components, low-rank innovations, and sparse edits. Develops an anchored alternating projection estimator.
result Establishes deterministic error bounds that separate target noise, representation growth, and source estimation error, yielding improved rates.
A model for grid cells using vectors and matrices for position and motion.
problem Representing self-position and motion in a high-dimensional space.
method Vector-matrix multiplication, magnified local isometry, and global adjacency kernel.
result The model can learn hexagon patterns and correct errors.
This article reviews statistical methods for learning data representations.
problem Learning meaningful representations of data.
method Statistical perspective on unsupervised and supervised representation learning.
result Recent advances in representation learning from a statistical viewpoint.
Paper extends matrix completion to nonlinear algebraic varieties.
problem Matrix completion for nonlinear algebraic varieties.
method Tensorization and Kronecker product approach.
result New method outperforms existing state-of-the-art methods.
New formula simplifies evolution of twist knots and calculates Racah matrices for rectangular representations.
problem Simplifying evolution of twist knots and calculating Racah matrices for rectangular representations.
method Developed a universal formula for triangular evolution matrix applicable to rectangular representations . Used skew characters and Macdonald polynomials.
result Explicit knowledge of twist-family evolution leads to a nearly explicit answer for Racah matrix in arbitrary rectangular representation .
CSA improves recommender systems by learning context-aware feature representations.
problem Limited expressiveness of traditional IMC methods for feature representations.
method Generalizes self-attention mechanism to IMC, learning context-aware feature representations.
result Extensive experiments show CSA's effectiveness on real RS datasets.
New method learns local structure for better data representation.
problem Global structure learning ignores local structure in nonnegative matrix factorization.
method Proposes a new nonnegative matrix factorization method that learns local similarity and clustering.
result The new representation reveals inherent geometric property of the data more effectively.
The paper proposes a model to learn motion perception in V1 using vector and matrix representations.
problem Motion perception in primary visual cortex (V1).
method Coupling vector representations of local contents and matrix representations of local pixel displacements.
result The model can learn Gabor-like filter pairs and infer local motions.
Random matrix theory predicts neural representations generalize well.
problem Understanding why neural representations generalize well in practice.
method Applied random matrix theory to kernel regression and neural networks.
result GCV estimator accurately predicts generalization risk in overparameterized settings.
Matrix Lie Groups as 3-Dimensional Almost Paracontact Almost Paracomplex Riemannian Manifoldsmath.DG
Study connects Lie groups to specific Riemannian manifolds.
problem Understanding Lie groups through Riemannian manifold properties.
method Investigates Lie groups as 3D almost paracontact almost paracomplex Riemannian manifolds.
result Established correspondence between Lie algebra and matrix representation.
FLAMBE tackles RL in low rank MDPs by learning features.
problem Dealing with the curse of dimensionality in RL.
method Develops FLAMBE, a method that engages in exploration and representation learning for RL in low rank transition models.
result FLAMBE efficiently learns features for RL in low rank transition models.
Semi-Non-negative Matrix Factorization is a technique that learns a low-dimensional representation of a dataset that lends itself to a clustering interpretation. It is possible that the mapping between this new representation and our original data matrix contains rather complex hierarchical information with implicit lo…
Quantum theory uses modular group representations to assign invariants to 3-manifolds.
problem Assigning invariants to 3-manifolds via modular group representations.
method Projective representations of the modular group derived from a noncommutative torus.
result Computed traces and determinants of matrices associated with modular group elements.
Efficiently learns sparse low-dimensional Markov chain representations.
problem Learning low-dimensional representations for large-scale Markov chains with sparse structures.
method Formulates as constrained nonnegative matrix factorization and uses gradient descent.
result Proves the effectiveness of the proposed method through convergence analysis.
Linear representations help embed manifolds into matrix spaces.
problem Embedding manifolds into matrix spaces with effective bounds.
method Defining linear representations of -manifolds as maps into matrix spaces, encoding -actions as matrix products.
result Explicit bounds for Mostow-Palais -equivariant embeddings of -manifolds into -modules , showing for compact .
A new multi-view clustering method using deep matrix decomposition and partition alignment.
problem Improving multi-view clustering methods to better utilize data representations and view-specific structures.
method Deep matrix decomposition for partition representations, joint use of partition representations, and alternating optimization.
result Demonstrated effectiveness on six benchmark datasets compared to state-of-the-art methods.
We show that the Lawrence--Krammer representation is unitary. We explicitly present the non-singular matrix representing the sesquilinear pairing invariant under the action. We show that reversing the orientation of a braid is equivalent to the transposition of its Lawrence--Krammer matrix followed by a certain conjuga…
FI-GRL learns graph node representations efficiently and generalizes to unseen nodes.
problem Transductive graph representation learning requires all nodes to be known, limiting generalization.
method FI-GRL uses random projection to preserve graph structure and feature extraction via SVD.
result FI-GRL achieves accurate representations for seen nodes and generalizes to unseen nodes.
Study Lie groups as 4D hypercomplex manifolds with specific metrics.
problem Understanding Lie groups with hypercomplex structures in 4D.
method Investigated Lie groups as almost hypercomplex Hermitian-Norden manifolds, established a correspondence between Lie algebras and matrix representations, and constructed examples.
result Explicit matrix representations of Lie groups with hypercomplex structures in 4D.
Compressed sensing (CS) shows that a signal having a sparse or compressible representation can be recovered from a small set of linear measurements. In classical CS theory, the sampling matrix and representation matrix are assumed to be known exactly in advance. However, uncertainties exist due to sampling distortion, …
New method enforces encoder sparsity in HPF for more interpretable feature selection.
problem Lack of encoder sparsity in HPF leads to lack of column-clustering property.
method Enforces encoder sparsity using a generalized additive model (GAM).
result Gains ability to perform feature selection and relates each representation to original features.
The contribution of reducible connections to the U(N) Chern-Simons invariant of a Seifert manifold can be expressed in some cases in terms of matrix integrals. We show that the U(N) evaluation of the LMO invariant of any rational homology sphere admits a matrix model representation which agrees with the Chern-Simon…
Laplacian matrix helps in reducing data dimensions and clustering.
problem Representing and clustering data using graphs and matrices.
method Using the Laplacian matrix to assign values to nodes based on their connectivity.
result The Laplacian matrix can be used to find a good embedding of data in a low-dimensional space and perform clustering.
SC-InfoNCE improves InfoNCE for feature clustering in contrastive learning.
problem Lack of theoretical understanding of InfoNCE's feature clustering mechanism.
method Introduced a transition probability matrix to model data augmentation dynamics and optimize feature similarity.
result SC-InfoNCE achieves strong performance across diverse domains, aligning feature similarity with downstream data.
Researchers extend knot theory formulas to non-rectangular cases.
problem Applying universal-matrix precursor formulas to non-rectangular knot representations.
method Reformulated previously known formulas for simplest non-rectangular representations [r,1].
result Demonstrated drastic simplification of formulas after reformulation.
Paper presents new matrix formats for deep neural networks that improve inference efficiency.
problem High computational cost of dot product operations in deep neural networks.
method Develops new matrix formats with bounded complexity by entropy of weight matrices.
result Up to x90 energy savings and x5 speed ups in dot product operations.
dCMF learns shared latent representations from multiple matrices, improving predictive modeling.
problem Learning from multiple heterogeneous data sources, especially non-linear interactions.
method Develops a deep-learning based method (dCMF) for unsupervised learning of multiple shared representations.
result dCMF significantly outperforms previous CMF algorithms in integrating heterogeneous data.
Matrix approximation is a common tool in machine learning for building accurate prediction models for recommendation systems, text mining, and computer vision. A prevalent assumption in constructing matrix approximations is that the partially observed matrix is of low-rank. We propose a new matrix approximation model w…
New braid group action defined on projective quantum sl(2) modules.
problem Defining a new braid group action on quantum sl(2) modules.
method Action via R-matrix on tensor powers of simple projective modules.
result The action is faithful for the extended representation.
Solved a specific case of Salter's question on Burau representation.
problem Under what conditions are matrices in the image of the Burau representation of .
method Algorithmically constructed a counterexample to Salter's specific question.
result The central quotient of the Burau image group is not the central quotient of a certain subgroup of the unitary group.
This article gives matrix factorizations for the trivalent diagrams and double line appearing in quantum link invariant. These matrix factorizations reconstruct Khovanov-Rozansky homology. And we show that the Euler characteristic of the matrix factorization for a double loop equals the quantum dimens…
BoostNE learns multiple network embeddings from coarse to fine.
problem Complex node interactions cannot be fully captured by a single low-rank embedding matrix.
method BoostNE proposes a multi-level network embedding framework using gradient boosting.
result BoostNE outperforms existing network embedding methods on various datasets.
A new matrix factorization method that approximates data without requiring nonnegativity or convexity.
problem Approximating data matrices without the constraints of nonnegativity or convexity.
method A multi-objective optimization problem finds conical combinations of templates that approximate a given data matrix.
result The method allows for approximation of data sets without the usual constraints of nonnegativity or convexity.
A new method uses SPDEs to efficiently model random fields on complex domains.
problem Efficient representation of random fields on complex domains for engineering and machine learning.
method Uses SPDEs to develop a scalable framework for statFEM and GP regression.
result Can model anisotropic, non-stationary random fields with arbitrary smoothness.
A new method for learning row and column structures with missing data.
problem Learning row and column structures in data with missing values.
method A three-component unsupervised approach: estimating a complete matrix, computing pairwise distances, and constructing representations.
result Our method outperforms other methods in data visualization and clustering.