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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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…
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.
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.
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).
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.
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.
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.
We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrix , not just its eigenvalues , and provide a universal formula for , applicable to arbitrary rectangular representation . This expression is in terms of s…
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.
In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a pr…
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.
This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The mode…
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…
Graph representation learning aims at transforming graph data into meaningful low-dimensional vectors to facilitate the employment of machine learning and data mining algorithms designed for general data. Most current graph representation learning approaches are transductive, which means that they require all the nodes…
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…
As opposed to manual feature engineering which is tedious and difficult to scale, network representation learning has attracted a surge of research interests as it automates the process of feature learning on graphs. The learned low-dimensional node vector representation is generalizable and eases the knowledge discove…
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.
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…
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.
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…
NIMFA is an open-source Python library that provides a unified interface to nonnegative matrix factorization algorithms. It includes implementations of state-of-the-art factorization methods, initialization approaches, and quality scoring. It supports both dense and sparse matrix representation. NIMFA's component-based…
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…
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, …
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.
Generalization in Representation Models via Random Matrix Theory: Application to Recurrent Networksmath.ST
We analyze generalization in deep learning models using random matrix theory.
problem Understanding the generalization error in deep learning models with random feature representations.
method Applying Random Matrix Theory to derive asymptotic generalization error formulas for various architectures.
result Linear ESNs are equivalent to ridge regression with exponentially time-weighted input covariance, revealing an inductive bias towards recent inputs.
We categorify the coefficients of the Burau representation matrix using elementary geometrical methods. We show the faithfulness of this categorification in the sense that it detects the trivial braid.
Matrix Chernoff bound for Markov chains applied to co-occurrence matrices.
problem Analyzing the behavior of co-occurrence statistics in sequential data.
method Proved a matrix Chernoff-type bound for sums of matrix-valued random variables sampled via a regular Markov chain.
result Achieved exponentially fast convergence rate and sample complexity analysis for co-occurrence matrices.
Recommender systems (RS), which have been an essential part in a wide range of applications, can be formulated as a matrix completion (MC) problem. To boost the performance of MC, matrix completion with side information, called inductive matrix completion (IMC), was further proposed. In real applications, the factorize…
In this article we give an explicit description of the representation matrix of a Heisenberg type action constructed by Blanchet, Habegger, Masbaum and Vogel. We give the matrix in terms of a ribbon graph and its admissible colorings. We show that components of the representation matrix satisfies the {\it external edge…
Proposes a wave-constrained matrix factorization for signal learning.
problem Learning signals constrained by the wave equation.
method Wave-informed matrix factorization with global optimality guarantees.
result Proves global optimality of the proposed model in polynomial time.
New method proves asymptotic normality for matrix sensing problems.
problem Proving asymptotic normality for matrix sensing under general convex losses.
method Riemannian geometry to handle degeneracy of the Hessian due to rotational symmetry.
result Proves as .
The paper introduces a statistical distance matrix for better feature representation and clustering.
problem Lack of detailed distance representation between feature elements.
method Extended traditional statistical distance to a matrix form (statistical distance matrix) and applied hierarchical clustering.
result The statistical distance matrix with clustering (Information Mandala) provides clearer and geometrically arranged feature representations.
This work improves robustness guarantees for neural networks using low rank representations.
problem Certified robustness to adversarial perturbations in neural networks.
method Low rank representations to provide improved robustness guarantees.
result Improved robustness guarantees for perturbations using natural low rank representations.
We propose improved methods to identify stock groups using the correlation matrix of stock price changes. By filtering out the marketwide effect and the random noise, we construct the correlation matrix of stock groups in which nontrivial high correlations between stocks are found. Using the filtered correlation matrix…
A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negative-definite Hermitian.…