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…
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48 results for “Low-Rank Kernels”
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.
A general framework of least squares support vector machine with low rank kernels, referred to as LR-LSSVM, is introduced in this paper. The special structure of low rank kernels with a controlled model size brings sparsity as well as computational efficiency to the proposed model. Meanwhile, a two-step optimization al…
Low-Rank Thinningstat.ML
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.
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.
Although the convolutional neural networks (CNNs) have become popular for various image processing and computer vision task recently, it remains a challenging problem to reduce the storage cost of the parameters for resource-limited platforms. In the previous studies, tensor decomposition (TD) has achieved promising co…
Constructing the adjacency graph is fundamental to graph-based clustering. Graph learning in kernel space has shown impressive performance on a number of benchmark data sets. However, its performance is largely determined by the chosen kernel matrix. To address this issue, the previous multiple kernel learning algorith…
Estimates low-rank distributional matrices from incomplete samples.
problem Matrix completion for distributional entries with limited observed data.
method Kernel mean embeddings, Tucker rank, functional unfolding operators.
result Effective estimator for distributional matrix completion established.
Determinantal point processes (DPPs) are an elegant model for encoding probabilities over subsets, such as shopping baskets, of a ground set, such as an item catalog. They are useful for a number of machine learning tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix…
The paper examines how kernel approximations affect Gaussian process regression in large data applications.
problem Effect of kernel approximations on Gaussian process regression in large data applications.
method Unified framework to analyze Gaussian process regression under computational and epistemic misspecification.
result Theoretical analysis of Gaussian process regression under various misspecifications.
Paper bounds the minimal rank for kernel ridge regression approximations.
problem Efficient memory and computation for kernel ridge regression.
method Lower bound on minimal rank for reliable prediction power.
result Nyström method's computational cost is almost linear in sample size.
Estimates spatio-temporal Hawkes processes using tensor recovery.
problem Estimating influence functions for spatio-temporal Hawkes processes.
method Formulates influence function as a tensor kernel, assumes low-rank structure, solves as convex optimization problem.
result Provides theoretical guarantees and demonstrates efficiency with simulations.
Kernel clustering algorithm improved for large datasets using incomplete Cholesky factorization.
problem Large memory usage in kernel-based clustering for large-scale datasets.
method Approximate the kernel matrix using incomplete Cholesky factorization and apply linear -means clustering.
result The proposed method achieves similar performance to kernel -means clustering but handles large-scale datasets efficiently.
New algorithm tackles low-rank constraints in optimal transport problems.
problem Optimal transport problems with low-rank constraints.
method Explicit factorization of low-rank couplings as a product of sub-coupling factors linked by a common marginal.
result Stationary convergence of the algorithm proved.
Determinantal point processes (DPPs) have garnered attention as an elegant probabilistic model of set diversity. They are useful for a number of subset selection tasks, including product recommendation. DPPs are parametrized by a positive semi-definite kernel matrix. In this work we present a new method for learning th…
We propose a novel class of kernels to alleviate the high computational cost of large-scale nonparametric learning with kernel methods. The proposed kernel is defined based on a hierarchical partitioning of the underlying data domain, where the Nyström method (a globally low-rank approximation) is married with a locall…
Gaussian processes (GP) for machine learning have been studied systematically over the past two decades and they are by now widely used in a number of diverse applications. However, GP kernel design and the associated hyper-parameter optimization are still hard and to a large extend open problems. In this paper, we con…
The paper reveals low-rank structure in neural network gradients, influenced by data and model parameters.
problem Investigating low-rank structure in gradients of neural networks under relaxed assumptions.
method Spiked data model, relaxation of isotropy assumptions, analysis of mean-field and neural-tangent-kernel scalings.
result Gradient of input weights is approximately low rank, dominated by two rank-one terms.
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…
Proposes a framework to extract ordered eigenfunctions from contextual kernels.
problem Lack of exact spectral decomposition in existing methods.
method Modular building blocks for compatibility with contextual kernels and scalability.
result Extracted eigenfunctions provide effective importance scores for feature selection.
DM2L tackles missing labels in multi-label learning by modeling local and global rank structures.
problem Missing labels in multi-label learning.
method DM2L imposes local low-rank structures and global high-rank structures on predictions of instances from the same and different labels, respectively.
result DM2L outperforms state-of-the-art methods in multi-label learning with missing labels.
Efficient and accurate low-rank approximations of multiple data sources are essential in the era of big data. The scaling of kernel-based learning algorithms to large datasets is limited by the O(n^2) computation and storage complexity of the full kernel matrix, which is required by most of the recent kernel learning a…
We consider fast kernel summations in high dimensions: given a large set of points in dimensions (with ) and a pair-potential function (the {\em kernel} function), we compute a weighted sum of all pairwise kernel interactions for each point in the set. Direct summation is equivalent to a (dense) matrix-vec…
This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results obtained in the finite dimensional case for the Frobenius norm. The theorem provide…
New framework estimates eigenvalues of kernel matrices without full matrix construction.
problem Estimating eigenvalues of large kernel matrices efficiently.
method Eigenvalue quantile estimation framework for kernel matrices with quick decay.
result Validates framework with empirical evidence and proves interlacing theorem.
New algorithm improves dynamic mode decomposition for high-dimensional data.
problem Reduced modeling in high-dimensional spaces.
method Low rank constraint optimization and kernel-based computation.
result Gain in approximation accuracy and computational efficiency.
Non-linear reduced modeling of dynamical systems using kernel methods and low-rank approximationstat.ML
Reduced modeling of a computationally demanding dynamical system aims at approximating its trajectories, while optimizing the trade-off between accuracy and computational complexity. In this work, we propose to achieve such an approximation by first embedding the trajectories in a reproducing kernel Hilbert space (RKHS…
GLSKF improves tensor completion by capturing both global and local variations.
problem Tensor completion with missing entries, especially in data with spatial or temporal side information.
method Integrates smoothness-constrained low-rank factorization with a locally correlated residual process.
result GLSKF achieves superior performance and scalability on real-world datasets.
Paper tackles matrix estimation under arbitrary noise, achieving minimax optimality.
problem Noisy low-rank-plus-sparse matrix recovery under arbitrary dependence.
method Incoherent-constrained least-square estimator, novel energy spreading result.
result Achieves minimax optimality in estimating structured Markov transition kernels.
Algorithm POLO learns low-rank MDPs with adversarial changes in full-info feedback.
problem Learning low-rank MDPs with adversarial changes and unknown transition probabilities.
method Policy optimization-based algorithm POLO with regret guarantee.
result POLO achieves sublinear regret guarantee with no dependence on state space size.
In this paper, we present a framework for fitting multivariate Hawkes processes for large-scale problems both in the number of events in the observed history and the number of event types (i.e. dimensions). The proposed Low-Rank Hawkes Process (LRHP) framework introduces a low-rank approximation of the kernel m…
Low-rank tensor regression, a new model class that learns high-order correlation from data, has recently received considerable attention. At the same time, Gaussian processes (GP) are well-studied machine learning models for structure learning. In this paper, we demonstrate interesting connections between the two, espe…
New spectral methods improve matrix estimation in RL with low-rank structure.
problem Estimating matrices with low-rank structure in reinforcement learning.
method Spectral-based matrix estimation approaches.
result Spectral methods efficiently recover singular subspaces and minimize entry-wise error.
Boosting Nyström Methodstat.ML
Boosting Nyström improves accuracy of matrix approximations.
problem Generating low-rank approximations of large matrices efficiently.
method Iteratively generate multiple weak Nyström approximations, combine them to form a strong approximation.
result Boosting Nyström yields more efficient and accurate low-rank approximations.
Kernel-based methods enjoy powerful generalization capabilities in handling a variety of learning tasks. When such methods are provided with sufficient training data, broadly-applicable classes of nonlinear functions can be approximated with desired accuracy. Nevertheless, inherent to the nonparametric nature of kernel…
Scalable Spatiotemporally Varying Coefficient Modelling with Bayesian Kernelized Tensor Regressionstat.ML
BKTR models spatiotemporal data with scalable tensor regression.
problem High computational cost in applying STVC to large-scale spatiotemporal data.
method Summarize STVC coefficients in a tensor, reformulate as low-rank tensor regression, incorporate GP priors for local dependencies.
result BKTR efficiently models large spatiotemporal datasets with reduced parameters and local dependencies.
The paper tackles pure exploration in multi-armed bandits with low rank structure using oblivious sampling.
problem Pure exploration in multi-armed bandits with low rank reward sequences.
method The approach involves separating the exploration strategy from feedback, using oblivious sampling, and incorporating kernel information of reward vectors.
result Efficient algorithms with regret bound for both time-varying and fixed cases, with a lower bound gap of .
New method samples DPPs efficiently without downsampling or low-rank approximations.
problem Efficient sampling from DPPs for diverse selections.
method Directly approximates distribution function of linear statistics using Laplace inversion.
result Scalable sampling for general DPPs, beyond symmetric kernels.
This paper describes a new method for low rank kernel approximation called IKA. The main advantage of IKA is that it produces a function defined as a linear combination of arbitrarily chosen functions. In contrast the approximation produced by Nyström method is a linear combination of kernel evaluations. The pro…
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.
Bayesian Complementary Kernelized Learning models complex spatiotemporal data.
problem Modeling complex, nonstationary, and nonseparable spatiotemporal data.
method Integrates kernelized low-rank tensor factorization and short-range spatiotemporal Gaussian Processes.
result BCKL offers superior performance in providing accurate posterior mean and high-quality uncertainty estimates.
This work tackles scalable sampling for nonsymmetric DPPs.
problem Scalability issue in existing DPP sampling algorithms for nonsymmetric DPPs.
method Developed a linear-time algorithm for kernels with low-rank structure and a sublinear-time rejection sampling algorithm.
result Bounded rejection rate for kernels with structural constraints.
A new algorithm reduces sample complexity for learning Q-functions in reinforcement learning.
problem Efficiently learning Q-functions in reinforcement learning with continuous state and action spaces.
method Developed a simple, iterative learning algorithm that estimates low-rank Q-functions.
result Achieved exponential improvement in sample complexity for low-rank Q-functions.
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.
TensorGuide improves LoRA efficiency and expressivity through joint tensor-train optimization.
problem Limited expressivity and generalization of standard LoRA.
method TensorGuide uses a unified tensor-train structure with controlled Gaussian noise to generate correlated low-rank matrices.
result TensorGuide achieves superior accuracy and scalability with fewer parameters compared to standard LoRA and TT-LoRA.
The smart grid vision entails advanced information technology and data analytics to enhance the efficiency, sustainability, and economics of the power grid infrastructure. Aligned to this end, modern statistical learning tools are leveraged here for electricity market inference. Day-ahead price forecasting is cast as a…
This work improves fair tensor decomposition using a kernel criterion.
problem Learning fair low-rank tensor decompositions with statistical parity.
method Regularizes Canonical Polyadic Decomposition with KHSIC to ensure approximate statistical parity.
result The proposed algorithm achieves better fairness and fit than state-of-the-art FATR.
Paper improves kernel approximations for better statistical learning.
problem Improving kernel approximations for better statistical learning.
method Taylor series approximations of radial kernel functions.
result Establishes upper bounds for eigenfunctions, leading to better approximations.