Efficient SVD algorithm robust to outliers.
problem Outliers in data matrix affect SVD accuracy and speed.
method Spherically Normalized SVD (SpherSVD) algorithm.
result Significantly faster and more robust than existing methods.
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A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
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48 results for “approximate SVD”
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.
SVD training reduces DNN rank and computation load without SVD per step.
problem High memory and computational load in deep neural networks.
method Explicitly achieves low-rank DNNs during training without SVD per step, using orthogonality regularization and sparsity-inducing regularizers.
result Significantly reduces DNN rank and computation load compared to existing methods.
Truncated Singular Value Decomposition (SVD) calculates the closest rank- approximation of a given input matrix. Selecting the appropriate rank defines a critical model order choice in most applications of SVD. To obtain a principled cut-off criterion for the spectrum, we convert the underlying optimization prob…
Randomized SVD shows phase transitions in noisy data.
problem Noise sensitivity of randomized SVD in large rank matrices.
method Analyzed R-SVD under low-rank signal plus noise model.
result R-SVD exhibits BBP-like phase transition with outliers above detectability threshold.
Sparse Singular Value Decomposition (SVD) models have been proposed for biclustering high dimensional gene expression data to identify block patterns with similar expressions. However, these models do not take into account prior group effects upon variable selection. To this end, we first propose group-sparse SVD model…
K-SVD algorithm has been successfully applied to image denoising tasks dozens of years but the big bottleneck in speed and accuracy still needs attention to break. For the sparse coding stage in K-SVD, which involves constraint, prevailing methods usually seek approximate solutions greedily but are less effe…
Paper proposes a new optimization framework for learning eigenfunctions of operators.
problem Computing eigenvalue decomposition of high-dimensional operators.
method Operator SVD with Neural Networks via Nested Low-Rank Approximation.
result Proposed method efficiently learns top-L singular values and functions in the correct order.
The power of randomized algorithms in numerical methods have led to fast solutions which use the Singular Value Decomposition (SVD) as a core routine. However, given the large data size of modern and the modest runtime of SVD, most practical algorithms would require some form of approximation, such as sketching, when r…
Paper develops a bootstrap method for estimating sketched SVD errors.
problem Lack of tools for accurately estimating sketched SVD errors.
method Develops a fully data-driven bootstrap method for numerical error estimation.
result Allows users to adaptively predict extra work needed for desired error tolerance.
New method approximates high-dimensional probability densities efficiently.
problem Approximating high-dimensional probability densities accurately and efficiently.
method Hierarchical tensor-network approach using randomized SVD and linear equations.
result The method effectively approximates high-dimensional densities with linear complexity.
Spectral embedding based on the Singular Value Decomposition (SVD) is a widely used "preprocessing" step in many learning tasks, typically leading to dimensionality reduction by projecting onto a number of dominant singular vectors and rescaling the coordinate axes (by a predefined function of the singular value). Howe…
New asymmetric kernel methods improve feature learning.
problem Improving feature learning with asymmetric kernels.
method Coupled covariance eigenproblem and Nyström method.
result Empirical evaluations show benefits of KSVD.
Closed-form polynomial approximations replace MLPs in transformers, enabling new interpretability methods.
problem Replacing MLPs with polynomial approximations for transformer models.
method Theoretical derivation of closed-form least-squares approximations of MLPs and GLUs using polynomial functions.
result Polynomial approximations explain over 95% of MLP and GLU outputs' variance, enabling interpretability.
Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.
problem Improving the convergence rate of Muon optimizer.
method Using Newton-Schulz steps for momentum orthogonalization, proving convergence rate and constant factor.
result Muon with Newton-Schulz converges to the same stationary point as SVD-polar, up to a constant factor.
We extend the randomized singular value decomposition (SVD) algorithm \citep{Halko2011finding} to estimate the SVD of a shifted data matrix without explicitly constructing the matrix in the memory. With no loss in the accuracy of the original algorithm, the extended algorithm provides for a more efficient way of matrix…
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.
This paper improves neural network compression by using robust low-rank approximations.
problem Neural network compression sensitivity to outliers.
method Introduces robust low-rank approximations using norms (for ) and provides efficient algorithms.
result Achieves up to 28% compression with minimal accuracy loss compared to existing methods.
Paper improves tensor approximation for streaming data.
problem Challenges in finding accurate low-tubal-rank tensor approximations in streaming settings.
method Extends Frequent Directions for efficient low-tubal-rank tensor approximation.
result The new algorithm achieves arbitrarily small approximation error with linear sketch size growth.
Paper uses autoencoders for efficient reduced-order modeling of eigenvalue problems.
problem Efficiently modeling eigenvalue problems in high dimensions.
method Autoencoder-based reduced-order modeling for eigenvalue problems.
result Autoencoder-based models outperform standard POD-Galerkin methods in neutron diffusion applications.
Study uses random matrix theory to improve tensor approximation accuracy.
problem Improving tensor approximation accuracy in the presence of noise.
method Random matrix theory applied to tensor unfoldings.
result Characterizes spectral behavior of tensor unfoldings and predicts reconstruction performance.
Improved SVD-based Initialization for Nonnegative Matrix Factorization using Low-Rank Correctionmath.NA
Due to the iterative nature of most nonnegative matrix factorization (\textsc{NMF}) algorithms, initialization is a key aspect as it significantly influences both the convergence and the final solution obtained. Many initialization schemes have been proposed for NMF, among which one of the most popular class of methods…
The paper analyzes how random perturbations affect RSVD and its applications.
problem Analyzing the impact of random perturbations on RSVD.
method Derives bounds for distances between exact and approximated singular vectors using RSVD.
result Established nearly-optimal convergence rates and asymptotic normality for RSVD in various inference problems.
New insights into choosing between two data integration methods based on SVD.
problem Choosing between two data integration methods (Stack-SVD and SVD-Stack) for shared latent structure across multiple datasets.
method Derive exact expressions for the asymptotic performance and phase transitions of Stack-SVD and SVD-Stack, and develop optimal weighting schemes.
result Optimally weighted Stack-SVD outperforms optimally weighted SVD-Stack in the asymptotic regime.
We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building …
The paper updates SVD of evolving matrices using projection techniques.
problem Updating the rank-k truncated SVD of evolving matrices.
method Projection viewpoint, building subspaces to approximate singular vectors.
result The proposed algorithm leads to higher accuracy, especially for large singular values.
The Matrix Factorization models, sometimes called the latent factor models, are a family of methods in the recommender system research area to (1) generate the latent factors for the users and the items and (2) predict users' ratings on items based on their latent factors. However, current Matrix Factorization models p…
In this paper we consider the dictionary learning problem for sparse representation. We first show that this problem is NP-hard by polynomial time reduction of the densest cut problem. Then, using successive convex approximation strategies, we propose efficient dictionary learning schemes to solve several practical for…
Efficient algorithm for Hadamard decomposition of matrices.
problem Decomposing matrices into low-rank factors efficiently.
method Alternating optimization with SVD-inspired initialization and momentum.
result Significantly improved performance compared to existing methods.
Ranky solves SVD for large sparse matrices in distributed systems.
problem Rank problem in large sparse matrices for SVD.
method Distributed approach to solve rank problem.
result Recovers SVD with negligible error for large sparse matrices.
Driven by a wide range of applications, many principal subspace estimation problems have been studied individually under different structural constraints. This paper presents a unified framework for the statistical analysis of a general structured principal subspace estimation problem which includes as special cases no…
New method uses SVD entropy to price artworks.
problem Lack of fine measurements in traditional art pricing models.
method SVD entropy of painting images for content measurement.
result SVD entropy positively affects sales price at 1% significance level.
SVD-closed subgroups of the unitary group: generalized principal logarithms and minimizing geodesicsmath.DG
Study of logarithms in SVD-closed subgroups of unitary group.
problem Understanding logarithms in SVD-closed subgroups of unitary groups.
method Analysis of generalized principal logarithms and minimizing geodesics.
result Set of generalized principal logarithms is a disjoint union of diffeomorphic subsets.
New coreset method for near-convex functions.
problem Efficiently approximating loss functions for machine learning.
method Generic framework for computing sensitivities of near-convex functions using f-SVD factorization.
result Significantly improved coresets for various machine learning models.
Deep K-SVD Denoisingcs.LG
This work considers noise removal from images, focusing on the well known K-SVD denoising algorithm. This sparsity-based method was proposed in 2006, and for a short while it was considered as state-of-the-art. However, over the years it has been surpassed by other methods, including the recent deep-learning-based newc…
Efficiently compress pretrained models using RSI for improved predictive accuracy.
problem Efficiently compressing large pretrained models for practical deployment.
method Randomized subspace iteration (RSI) for low-rank approximation of pretrained models.
result RSI achieves near-optimal approximation quality and outperforms RSVD in predictive accuracy.
Globally optimized SVD compression of LLMs via Fermi-function-based rank selection and gauge fixingcs.LG
Physics-inspired methods optimize SVD compression of LLMs.
problem Efficiently compressing large language models (LLMs) using SVD.
method FermiGrad for globally optimal rank selection and PivGa for lossless compression.
result Global optimization of SVD ranks and lossless compression of low-rank factors.
Singular value decomposition (SVD) is the mathematical basis of principal component analysis (PCA). Together, SVD and PCA are one of the most widely used mathematical formalism/decomposition in machine learning, data mining, pattern recognition, artificial intelligence, computer vision, signal processing, etc. In recen…
Block based Singular Value Decomposition approach to matrix factorization for recommender systemscs.LG
With the abundance of data in recent years, interesting challenges are posed in the area of recommender systems. Producing high quality recommendations with scalability and performance is the need of the hour. Singular Value Decomposition(SVD) based recommendation algorithms have been leveraged to produce better result…
We show that given an estimate that is close to a general high-rank positive semi-definite (PSD) matrix in spectral norm (i.e., ), the simple truncated SVD of produces a multiplicative approximation of in Frobenius norm. This observation leads to many inte…
This paper examines the impact of centering in PCA and SVD.
problem The influence of centering in PCA and SVD.
method Detailed exploration of centering in PCA and SVD.
result Two PCA embeddings derived from SVD with and without centering can be aligned under certain conditions.
In this paper, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by usin…
Cross-Layer Subspace Coupling for LLM Compression: A Unifying Framework and Its Empirical Limitscs.LG
Unified SVD compression fails in practical tasks, highlighting the importance of per layer activation reconstruction.
problem The failure of a unified SVD compression method in practical tasks like perplexity and accuracy.
method Unified optimization problem for SVD based compression methods, focusing on cross-layer coupling.
result Downstream metrics like perplexity and accuracy degrade severely compared to standard per layer SVD LLM.
The high-order relations between the content in social media sharing platforms are frequently modeled by a hypergraph. Either hypergraph Laplacian matrix or the adjacency matrix is a big matrix. Randomized algorithms are used for low-rank factorizations in order to approximately decompose and eventually invert such big…
GCNNs gain rotation invariance with more training augmentation, making SVD-Universal more effective.
problem Improving robustness of GCNNs to adversarial attacks.
method SVD-Universal technique applied to GCNNs trained with larger rotations.
result SVD-Universal becomes more effective as GCNNs gain rotation invariance.
We compared the regular Singular Value Decomposition (SVD), truncated SVD, Krylov method and Randomized PCA, in terms of time and space complexity. It is well-known that Krylov method and Randomized PCA only performs well when k << n, i.e. the number of eigenpair needed is far less than that of matrix size. We compared…
New algorithm reduces matrix multiplication time for sparse matrices.
problem Efficiently multiply large sparse matrices with limited space.
method Exploits sparsity to reduce QR decompositions and time complexity.
result Time complexity reduced to $\widetilde{O}\left((
nz(X)+
nz(Y))\ell+n\ell^2
ight)$ in expectation.
New matrix approximation method using RBF components for better memory efficiency.
problem Efficiently approximate any real matrix without being symmetric or positive definite.
method Formulate as an optimization problem with gradient descent methods.
result Significantly reduces memory usage for various matrix types.