Paper improves efficiency in matrix computations for Gaussian processes.
problem Efficiency in matrix computations for Gaussian processes.
method Variance reduction via matrix factorization.
result Factorized estimator can be up to 1,000 times more efficient.
pMMF is a parallel algorithm for matrix computation.
problem Finding multiscale structure and wavelets on matrices.
method pMMF is a parallel algorithm for MMF factorization.
result pMMF scales linearly in the dimension for sparse matrices.
This paper explores how random sampling and coding can speed up approximate matrix multiplication.
problem Efficiently computing large-scale matrix multiplications in distributed systems.
method Proposes two schemes: coding for recovery and random sampling for approximation.
result Investigates tradeoffs between recovery threshold and approximation error.
Efficiently computes option pricing matrix exponentials.
problem Computing matrix exponentials of nested block triangular matrices.
method Incremental computation using scaling and squaring, reusing intermediate quantities.
result Efficiently computes option pricing matrix exponentials.
We provide a proof of backpropagation algorithm in matrix notation.
problem The lack of a full induction proof of backpropagation algorithm in matrix notation.
method We provide a full induction proof of the BP algorithm in matrix notation, situating it in the framework of matrix differential calculus.
result We prove the validity of the backpropagation algorithm in inductive form.
Accelerated RPCholesky speeds up kernel matrix approximations.
problem Efficiently approximating large kernel matrices.
method Accelerated randomly pivoted Cholesky (RPCholesky) with block matrix computations and rejection sampling.
result Approximates kernel matrices up to 40 times faster.
Paper proposes diagnostics for error and variance estimation in randomized matrix computations.
problem Safe use of randomized matrix algorithms in applications.
method Leave-one-out error estimator and jackknife resampling method.
result Provides rapid diagnostics to assess quality of randomized matrix computations.
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.
Bayesian non-linear matrix completion tackles large, sparse data.
problem Predict missing elements in large, sparsely observed matrices.
method Bayesian Gaussian process latent variable models with data-parallel distributed computation.
result Scalable Bayesian non-linear matrix completion outperforms linear methods.
New algorithms for efficient matrix profile computation using various Euclidean distances.
problem Efficiently computing matrix profile for all-pairs-similarity search on time series.
method Proposed AAMP, ACAMP, and extended algorithms for p-norm distance.
result AAMP and ACAMP algorithms outperform existing methods for specific Euclidean distances.
Improved statistical computation through efficient matrix sampling.
problem Reducing computational cost in large-scale statistical methods.
method Accumulative sub-sampling method to improve statistical efficiency.
result Effective matrix size control improves computational efficiency.
A new algorithm speeds up matrix operations in Neural Networks.
problem Time-consuming matrix operations in Neural Networks.
method An algorithm that increases the degree of parallelism of matrix multiplication.
result The algorithm speeds up several matrix operations in Neural Networks.
Randomized HALS for efficient NMF on big data.
problem Challenges in computing nonnegative matrix factorization for big data.
method Randomized hierarchical alternating least squares (HALS) algorithm.
result Efficient nonnegative decomposition for big data applications.
Online algorithm for matrix factorization using Broyden updates.
problem Efficiently compute matrix factorizations with online data.
method Low-rank updates to dictionary matrix, derived from a simple objective function.
result Demonstrated efficiency on real dataset compared to NMF.
Efficiently computes matrix square roots and their inverses for large matrices.
problem Computing matrix square roots and inverses for large matrices efficiently.
method Combines Krylov subspace methods with rational approximation for quadratic-time computation.
result Achieves 4 decimal places of accuracy with fewer than 100 matrix-vector multiplications.
A new method computes link invariants from diagrams.
problem Computing link invariants efficiently.
method Single symmetric matrix from a link diagram.
result Multivariable Alexander polynomial computation.
New method combines simulated annealing and Levy distribution for fast matrix factorization.
problem High complexity and difficulty in parallelizing matrix factorization for large matrices.
method Combining simulated annealing with Levy distribution for matrix factorization.
result Achieves good solutions in acceptable time with low computations.
Parallel SGLD improves MCMC for large matrix factorisation problems.
problem Large-scale matrix factorisation problems.
method Distributed Markov Chain Monte Carlo (MCMC) based on stochastic gradient Langevin dynamics (SGLD).
result PSGLD achieves high performance and superior convergence compared to optimisation methods.
Paper introduces a new method for Gaussian Processes that improves prediction and hyper-parameter optimization.
problem Efficiently predicting unknown functions and optimizing hyper-parameters in Gaussian Processes.
method Sequential randomized low-rank matrix factorization for incremental predictions and hyper-parameter optimization.
result The proposed method outperforms existing approaches in terms of accuracy and computational efficiency.
New algorithms detect large sparse covariance entries quickly.
problem Efficiently detect large sparse entries in covariance matrices.
method Randomized algorithms with sub-quadratic time complexity.
result Detect large sparse entries in O ( n p e x t p o l y l o g p ) O(np ext{ poly log } p) O ( n p e x t p o l y l o g p ) operations. A distributed framework for reducing high-dimensional matrix-variate time series data.
problem Reducing dimensionality of high-dimensional, heterogeneous matrix-variate time series data.
method Data partitioning, distributed two-dimensional tensor PCA, aggregation, final PCA, factor matrix computation.
result Preserves latent matrix structure, improves computational efficiency and information utilization.
A streaming algorithm for matrix completion with noisy data.
problem Matrix completion with limited memory and noisy observed entries.
method A streaming algorithm that estimates the original matrix with minimal memory and computational complexity.
result The algorithm produces an estimate with vanishing mean square error using linear memory and proportional to the number of non-zero entries.
Lower bounds show linear complexity for linear regression.
problem Computational complexity of linear regression.
method Reduction to estimating the least eigenvalue of a random Wishart matrix.
result Θ(d) calls to the oracle are necessary and sufficient for polynomial accuracy.
New method reduces computational cost for nonnegative low rank matrix approximation.
problem Efficiently compute nonnegative low rank matrix approximation for nonnegative matrices.
method Alternating projections onto tangent spaces of fixed rank matrices manifold and nonnegative matrix manifold.
result Sequence converges linearly to optimal solutions, showing better performance in terms of computational time and accuracy.
New algorithm reduces sample and computational complexities for matrix completion.
problem Recovering a rank- r r r matrix from n n n features. method Gradient-based non-convex optimization algorithm with linear convergence rate.
result Linear sample complexity and logarithmic computational complexity.
Localized sketching improves matrix multiplication and ridge regression complexity.
problem Efficiently approximate matrix multiplication and ridge regression with limited data availability.
method Localized sketching matrices for block diagonal structure, reducing sample complexity.
result Localized sketching achieves sample complexity matching global sketching methods.
New algorithms detect anomalies in high-dimensional data using less space than traditional methods.
problem Finding anomalies in high-dimensional data using PCA-based scores efficiently.
method Developed streaming algorithms using linear or sublinear space, and proved matrix perturbation inequalities.
result Proved that certain matrix sketching techniques can approximate PCA-based anomaly scores efficiently.
This paper improves SVD for recommender systems using block-based matrix factorization.
problem Scalability and performance issues in recommender systems.
method Block-based Singular Value Decomposition (BMF) for matrix factorization.
result BMF paired with SVD enhances performance and scalability.
New matrix approximation method speeds up optimization for deep learning.
problem Efficient computation of matrix inverse and square root for high-dimensional optimization.
method Divide matrix into blocks and represent each block by one or two numbers.
result Improved performance of AdaGrad in training deep neural networks compared to diagonal approximation.
New method uses matrix powers for recommendation systems.
problem Predicting unobserved entries in sparse matrices.
method Coordinate descent algorithm to learn embeddings from higher-order matrix powers.
result Outperforms methods using only side information or second-order interactions.
Article presents QR and LQ decomposition algorithms for various matrix sizes and ranks.
problem Solving least squares problems in machine learning and computer vision.
method Developed novel matrix backpropagation algorithms for QR and LQ decompositions of different matrix sizes and ranks.
result Numerical stability and computational efficiency of the proposed methods.
A distributed Bayesian matrix factorization method speeds up inference by 10x with minimal accuracy loss.
problem Scaling Bayesian matrix factorization for large datasets with limited communication.
method Hierarchical decomposition of the joint posterior distribution, coupled subset inferences, and efficient approximate implementation.
result Achieves a speed-up of almost an order of magnitude over full posterior inference with negligible accuracy loss.
Gradient descent algorithms on manifolds solve control and mean computation problems.
problem Control and mean computation on positive definite Hermitian matrices.
method Riemannian and natural gradient algorithms applied to geodesic distance.
result Efficient algorithms for control and mean computation demonstrated.
Algorithm learns a better sketch matrix for low-rank approximations.
problem Efficiently compute low-rank approximations of large matrices.
method Uses a learned sketch matrix instead of random matrix for optimization.
result Learned sketch matrix reduces approximation loss significantly compared to random matrix.
The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem N P {\mathcal{NP}} NP -hard. In this work, we prove that, if the matrix is positive semidefinite and its …
Efficient approximations reduce computation of matrix-based Renyi's entropy.
problem High computational complexity of matrix-based Renyi's entropy.
method Taylor, Chebyshev, and Lanczos approximations to reduce complexity.
result Reduced complexity to significantly less than O ( n 2 ) O(n^2) O ( n 2 ) with negligible accuracy loss. We develop matrix models for Grassmann, flag, and Stiefel manifolds.
problem Creating efficient models for Grassmann, flag, and Stiefel manifolds.
method Orthogonally-equivariant matrix submanifold models derived for each manifold.
result Exhaustive list of orthogonally-equivariant submanifold models for the lowest dimensions.
Improved prediction accuracy in matrix factorization using graph-based priors.
problem Graph side-information may not align with latent-feature relations in matrix completion.
method Identify and remove 'contested' edges using graphical lasso approximation, maintaining linear scalability.
result Improved prediction accuracy with fewer graph edges, demonstrating the often inaccurate nature of graph side-information.
Two algorithms estimate Wasserstein distance matrices from few entries for manifold learning.
problem Estimating Wasserstein distance matrices from limited data for manifold learning.
method Proposes two algorithms: matrix completion and Nyström completion for square Wasserstein matrices.
result Nyström completion can outperform matrix completion with a fixed sample budget and improve classification stability.
New method tackles over-parameterized matrix sensing with FGD, improving statistical and computational complexity.
problem Solving low rank matrix sensing with over-specified factors when rank is unknown.
method Decomposing the factorized matrix into column spaces to capture extra ranks and analyze convergence.
result Convergence to a statistical error of i l d e O ( k d σ 2 / n ) ilde{\mathcal{O}} ({k d σ^2/n}) i l d e O ( k d σ 2 / n ) after i l d e O ( σ r σ n d ) ilde{\mathcal{O}}(\frac{σ_{r}}σ\sqrt{\frac{n}{d}}) i l d e O ( σ σ r d n ) iterations. New method tackles high-dimensional SBL without covariance matrices.
problem Sparse coding problem in high-dimensional settings.
method Parallel solution of multiple linear systems using conjugate gradient algorithm.
result Our method scales better in computation time and memory.
New method improves robust low-rank matrix completion for computer vision.
problem Robust low-rank matrix completion for partially observed data.
method Formulated as a nonsmooth Riemannian optimization problem over Grassmann manifold, solved with an alternating manifold proximal gradient continuation method.
result Demonstrated advantages over existing approaches in background extraction from surveillance videos.
Paper finds a fast method for a matrix norm proximal operator.
problem Optimizing mixed ℓ 1 , ∞ \ell_{1,\infty} ℓ 1 , ∞ matrix norms efficiently. method Closed-form computation using soft-thresholding, iterative algorithm for thresholds.
result Mixed ℓ 1 , ∞ \ell_{1,\infty} ℓ 1 , ∞ prox can be computed in closed form. Dynamic sparseness reduces neural network computation by selectively omitting parts of computations.
problem Reducing the computational and memory footprint of neural networks.
method Combining dynamic sparseness with block-wise matrix-vector multiplications to selectively omit parts of computations.
result The proposed method outperforms static sparseness and achieves similar perplexities at half the computational cost.
Paper proposes a novel method to improve matrix completion with median loss for large datasets.
problem Matrix completion with absolute deviation loss for large-scale data.
method Proposes a refinement step using pseudo data to improve inefficient estimators of median matrix completion.
result Turns inefficient estimators into a rate (near-)optimal matrix completion procedure.
Reservoir computing's success depends on mapping different input time series to separable states.
problem Quantifying the ability of random linear reservoirs to map different input time series.
method Mathematical framework using spectral properties of the connectivity matrix.
result Separation capacity is fully characterized by the spectral properties of the connectivity matrix.
HyperINF improves influence function estimation for large models with better accuracy and efficiency.
problem Inaccurate and computationally expensive influence function estimation for large-scale models.
method HyperINF leverages Schulz's iterative algorithm and GFIM for low-rank approximation of Hessian matrix.
result HyperINF achieves superior accuracy and performance compared to existing methods on LoRA-tuned models.
Incremental method for graph Laplacian eigenpairs improves clustering efficiency.
problem Determining the number of clusters in spectral clustering.
method Incremental computation of graph Laplacian eigenpairs.
result Efficiently computes the K K K -th smallest eigenpair.