Paper tackles fair low-rank approximation and column subset selection.
problem Minimize loss over sub-populations in machine learning.
method Developed algorithms for fair low-rank approximation and fair column subset selection.
result Achieved polynomial time algorithms for fair low-rank approximation.
Low-rank approximation is an effective model compression technique to not only reduce parameter storage requirements, but to also reduce computations. For convolutional neural networks (CNNs), however, well-known low-rank approximation methods, such as Tucker or CP decomposition, result in degraded model accuracy becau…
In this paper, we propose a low-rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noisy-free observations. Sparsity inducing regularization techniques are used within classical algorithms for low-rank approximation in order to exploit the possible sp…
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…
The study assesses low-rank approximations in Gaussian Process regression.
problem Improving Gaussian Process regression efficiency with low-rank approximations.
method Analyzes two low-rank approximations: random Fourier features and Mercer expansion truncation.
result Bounds on the divergence and error between exact and approximate GP models.
New algorithm for weighted low rank approximation with provable guarantees.
problem Weighted low rank approximation (WLRA) is computationally hard.
method Reweights the low rank solution using the weight matrix itself.
result Provably optimal approximation guarantees for WLRA.
The study assesses low-rank approximations in Gaussian Process regression.
problem Improving the efficiency of Gaussian Process regression while maintaining accuracy.
method Analyzes two low-rank approximations: random Fourier features and Mercer expansion truncation, and bounds the divergence and error between exact and approximate models.
result Theoretical bounds on the divergence and error between exact and approximate Gaussian Process models are provided.
We accelerate the power method for strong low-rank approximation using fast sketching.
problem Efficiency bottleneck in power method for large target ranks.
method Developed an algorithmic and theoretical framework for accelerating the power method using fast sketching.
result Simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation.
We develop an efficient algorithm for low-rank approximation with improved approximation guarantees.
problem Optimal low-rank approximation of matrices with ℓ1 norm constraints. method Polynomial time column subset selection-based algorithm achieving ildeO(k1/2)-approximation. result Improved approximation guarantees for ℓ1 low-rank approximation. Stochastic gradient descent on manifolds improves low-rank approximation.
problem Efficiently approximate large matrices with lower rank.
method Stochastic gradient descent on a manifold.
result Algorithm outperforms Euclidean space methods on Netflix Prize data.
New algorithm improves deep learning models' robustness without sacrificing accuracy.
problem Low-rank methods compromise model robustness against adversarial perturbations.
method Robust low-rank training via approximate orthonormal constraints.
result Ensures well-conditioning and better adversarial robustness without sacrificing model accuracy.
Paper develops a new weighted low-rank matrix approximation technique.
problem Matrix completion with missing data.
method Element-wise weighted generalization of low-rank matrix approximation.
result Proposes an algorithm and acceleration techniques for solving the weighted problem.
Unified error analysis for low-rank approximation improves data assimilation performance.
problem Analyzing the error in low-rank approximation methods for data assimilation.
method Unified stochastic analysis framework for Frobenius norm error bounds on centered and non-standard Gaussian matrices.
result Unified bounds provide clearer interpretations and enable better practical choices for covariance matrices.
Matrices of (approximate) low rank are pervasive in data science, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these datasets, there is less attention on explaining why the low rank structure ap…
New method for initializing low-rank neural networks improves performance.
problem Training low-rank neural networks efficiently and accurately.
method Inspired by function approximation, proposes a novel low-rank initialization framework.
result Demonstrates significant gap between spectral and low-rank initialization approaches.
The paper reviews Hankel low-rank methods for time series analysis and forecasting.
problem Developing efficient methods for time series analysis and forecasting.
method Hankel low-rank approximation and completion techniques.
result Discussion of methods and challenges in obtaining optimal solutions.
We provide new approximation guarantees for greedy low rank matrix estimation under standard assumptions of restricted strong convexity and smoothness. Our novel analysis also uncovers previously unknown connections between the low rank estimation and combinatorial optimization, so much so that our bounds are reminisce…
This work tackles sparse coding in DLRA for interpretable multiway data.
problem Sparse coding in DLRA for interpretable multiway data.
method Proposes a new sparse-coding subproblem (MSC) and several algorithms to solve it.
result DLRA extends low-rank approximations, reducing variance and enhancing interpretability.
A new method improves convergence in low-rank approximation.
problem Efficiently solving large-scale numerical linear algebra problems.
method Error-Powered Sketched Inverse Iteration (EPSI) Method.
result Convergence rate improves at least linearly with sketch size.
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 …
We study the ℓ0-Low Rank Approximation Problem, where the goal is, given an m×n matrix A, to output a rank-k matrix A′ for which ∥A′−A∥0 is minimized. Here, for a matrix B, ∥B∥0 denotes the number of its non-zero entries. This NP-hard variant of low rank approximation is natural for pro…
Review of algorithms for linear system approximations.
problem Linear approximation of high-dimensional dynamical systems.
method State-of-the-art algorithms for low-rank DMD.
result Provides additional details for comprehensive understanding.
New methods recover best rank-r approximations from few entries.
problem Recovering best rank-r approximations from limited data entries.
method Two agnostic approaches: spectral truncation and projected gradient descent.
result Projected gradient descent yields superior performance.
Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address this problem and reduce the arithmetic and storage cost. However, we observed tha…
Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding low-rank approximations to Boolean matrices. In this paper we give what is to the best …
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.
A new method reduces high-dimensional filtering to quadratic complexity.
problem High-dimensional dynamical systems inference and simulation.
method Low-rank Kalman filtering using dynamical low-rank integrator.
result The method reproduces exact Kalman filter in low-rank limit.
Efficiently reduces tensor ranks using mean-field approximation.
problem Low-rank approximation of non-negative tensors.
method Mean-field approximation of tensor rank reduction.
result Our algorithm achieves faster and competitive tensor rank reduction.
New method reduces summary points for datasets while maintaining quality.
problem Thinning datasets to reduce summary points while maintaining quality.
method Low-rank analysis of sub-Gaussian thinning.
result Guarantees high-quality compression for any distribution and kernel.
This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we aim for finding an approximated low-rank representation of the (possibly high-rank) matrix [f(Aij)]. To this end, we propose an efficient sketching-based algorithm…
New method finds efficient low-rank neural networks during training.
problem High memory and computational demands of neural networks.
method Restricts weight matrices to a low-rank manifold and updates low-rank factors.
result Significantly reduced time and memory resources required for training and evaluation.
PILNO uses neural operators to solve PDEs efficiently on point clouds.
problem Solving partial differential equations (PDEs) on point cloud data efficiently.
method Physics-informed low-rank neural operator framework combining low-rank kernel approximations and an encoder-decoder architecture.
result PILNO efficiently approximates solution operators of PDEs on point cloud data, satisfying PDE constraints and boundary conditions.
We study the convergence of a variant of distributed gradient descent (DGD) on a distributed low-rank matrix approximation problem wherein some optimization variables are used for consensus (as in classical DGD) and some optimization variables appear only locally at a single node in the network. We term the resulting a…
New ACV method speeds up CV in high dimensions with approximate low-rank data.
problem Accurate model assessment in high-dimensional, large data settings with expensive algorithms.
method Developed a new ACV algorithm that uses low-rank approximations of the Hessian matrix.
result The new method is fast and accurate in the presence of approximate low-rank data.
We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation guarantees. Each iteration of the algorithm involves (approximately) finding the left an…
We simplify SSL by approximating redundant structural components with low-rank factorization.
problem Improving self-supervised learning performance with limited labeled data.
method Low-rank approximation of structural redundancy, introducing ε_s to measure approximation quality.
result The proposed method enhances SSL performance, as shown by theoretical and experimental validations.
PSI-LinUCB improves scalability for large recommender systems.
problem Efficiently training and inferring for large action spaces in recommender systems.
method Represent inverse design matrix as diagonal + low-rank correction, derive stable rank-1 and batched updates, use projector-splitting integrator.
result Demonstrated effectiveness on recommender system datasets, achieving scalable training and inference.
Solves weakly supervised regression using low-rank approximations and manifold regularization.
problem Weakly supervised regression with known, unknown, and uncertain labels.
method Combines manifold regularization and low-rank matrix decomposition for optimization.
result Improves solution quality and stability for large datasets.
LaLoRA prevents forgetting in LoRA fine-tuning.
problem Catastrophic forgetting in fine-tuned models.
method LaLoRA applies Laplace approximation to LoRA weights for regularization.
result Improved learning-forgetting trade-off with controllable regularization strength.
Algorithm compresses large matrices by approximating them as low rank and low precision factors.
problem Efficiently storing and processing large matrices with billions of elements.
method Randomized sketching and quantization of matrix columns to achieve low rank and low precision factorization.
result Achieves compression ratios as low as one bit per matrix coordinate while maintaining or improving performance.
Improves BBVI for high-dimensional Gaussian approximations by using low-rank approximations.
problem Scalability issues with BBVI for high-dimensional multivariate Gaussian approximations.
method Extends BaM framework to handle full covariance matrices by integrating patch step for low-rank parameterization.
result Shows improved efficiency and scalability on synthetic and real-world high-dimensional inference problems.
This paper concerns a fundamental class of convex matrix optimization problems. It presents the first algorithm that uses optimal storage and provably computes a low-rank approximation of a solution. In particular, when all solutions have low rank, the algorithm converges to a solution. This algorithm, SketchyCGM, modi…
We consider the problem of approximately reconstructing a partially-observed, approximately low-rank matrix. This problem has received much attention lately, mostly using the trace-norm as a surrogate to the rank. Here we study low-rank matrix reconstruction using both the trace-norm, as well as the less-studied max-no…
This work studies low-rank approximation of a positive semidefinite matrix from partial entries via nonconvex optimization. We characterized how well local-minimum based low-rank factorization approximates a fixed positive semidefinite matrix without any assumptions on the rank-matching, the condition number or eigensp…
This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non-convex and state-of-the-art algor…
New RL algorithm maximizes CVaR in low-rank MDPs with provable efficiency.
problem Maximizing CVaR in large state spaces with function approximation.
method Upper Confidence Bound (UCB) bonus-driven algorithm for low-rank MDPs.
result Achieves sample complexity of O(H^7 A^2 d^4 / τ^2 ε^2) for ε-optimal CVaR.
This paper proposes a new Nystrom-based clustering algorithm for large-scale data.
problem Spectral clustering's high computational complexity for large-scale data.
method Centroid Minimum Sum of Squared Similarities (CMS3) sampling procedure with eigen spectrum shape heuristic.
result Competitive low-rank approximations in test datasets compared to state-of-the-art methods.
Bayesian framework for sequential learning tasks with low-rank approximations.
problem Balancing knowledge retention and adaptability in sequential neural networks.
method Bayesian framework with diagonal plus low-rank approximations of the precision matrix.
result Unlocking capabilities to encode task relationships and incorporate prior knowledge from later tasks.