scholarly journals Approximating Hamiltonian dynamics with the Nyström method

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 234 ◽  
Author(s):  
Alessandro Rudi ◽  
Leonard Wossnig ◽  
Carlo Ciliberto ◽  
Andrea Rocchetto ◽  
Massimiliano Pontil ◽  
...  
Keyword(s):  
Hamiltonian Dynamics ◽  
Randomized Algorithm ◽  
Numerical Linear Algebra ◽  
Low Rank ◽  
Frobenius Norm ◽  
Nyström Method ◽  
Nystrom Method ◽  
Quantum Mechanical Systems ◽  
Symmetric Approximation ◽  
Matrix Exponentials

Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nyström method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.

scholarly journals Improved Guarantees and a Multiple-descent Curve for Column Subset Selection and the Nystrom Method (Extended Abstract)

2021 ◽  
Author(s):  
Michał Dereziński ◽  
Rajiv Khanna ◽  
Michael W. Mahoney
Keyword(s):  
Subset Selection ◽  
Low Rank ◽  
Data Matrix ◽  
Worst Case Analysis ◽  
Nyström Method ◽  
Worst Case ◽  
Nystrom Method ◽  
Column Subset Selection ◽  
Low Rank Approximations ◽  
The Cost

The Column Subset Selection Problem (CSSP) and the Nystrom method are among the leading tools for constructing interpretable low-rank approximations of large datasets by selecting a small but representative set of features or instances. A fundamental question in this area is: what is the cost of this interpretability, i.e., how well can a data subset of size k compete with the best rank k approximation? We develop techniques which exploit spectral properties of the data matrix to obtain improved approximation guarantees which go beyond the standard worst-case analysis. Our approach leads to significantly better bounds for datasets with known rates of singular value decay, e.g., polynomial or exponential decay. Our analysis also reveals an intriguing phenomenon: the cost of interpretability as a function of k may exhibit multiple peaks and valleys, which we call a multiple-descent curve. A lower bound we establish shows that this behavior is not an artifact of our analysis, but rather it is an inherent property of the CSSP and Nystrom tasks. Finally, using the example of a radial basis function (RBF) kernel, we show that both our improved bounds and the multiple-descent curve can be observed on real datasets simply by varying the RBF parameter.

scholarly journals Multi-Nyström Method Based on Multiple Kernel Learning for Large Scale Imbalanced Classification

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ling Wang ◽  
Hongqiao Wang ◽  
Guangyuan Fu
Keyword(s):  
Kernel Methods ◽  
Large Scale ◽  
Multiple Kernel Learning ◽  
Nonlinear Problems ◽  
Low Rank ◽  
Kernel Learning ◽  
Low Rank Approximation ◽  
Nyström Method ◽  
Nystrom Method ◽  
Multiple Kernel

Extensions of kernel methods for the class imbalance problems have been extensively studied. Although they work well in coping with nonlinear problems, the high computation and memory costs severely limit their application to real-world imbalanced tasks. The Nyström method is an effective technique to scale kernel methods. However, the standard Nyström method needs to sample a sufficiently large number of landmark points to ensure an accurate approximation, which seriously affects its efficiency. In this study, we propose a multi-Nyström method based on mixtures of Nyström approximations to avoid the explosion of subkernel matrix, whereas the optimization to mixture weights is embedded into the model training process by multiple kernel learning (MKL) algorithms to yield more accurate low-rank approximation. Moreover, we select subsets of landmark points according to the imbalance distribution to reduce the model’s sensitivity to skewness. We also provide a kernel stability analysis of our method and show that the model solution error is bounded by weighted approximate errors, which can help us improve the learning process. Extensive experiments on several large scale datasets show that our method can achieve a higher classification accuracy and a dramatical speedup of MKL algorithms.

scholarly journals Low-rank decomposition meets kernel learning: A generalized Nyström method

2017 ◽  
Vol 250 ◽  
pp. 1-15 ◽  
Author(s):  
Liang Lan ◽  
Kai Zhang ◽  
Hancheng Ge ◽  
Wei Cheng ◽  
Jun Liu ◽  
...  
Keyword(s):  
Low Rank ◽  
Kernel Learning ◽  
Nyström Method ◽  
Nystrom Method ◽  
Rank Decomposition ◽  
Low Rank Decomposition

scholarly journals A Gradient System for Low Rank Matrix Completion

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 51 ◽  
Author(s):  
Carmela Scalone ◽  
Nicola Guglielmi
Keyword(s):  
Sparse Matrix ◽  
Matrix Completion ◽  
Low Rank ◽  
Frobenius Norm ◽  
Gradient System ◽  
Large Matrix ◽  
Rank Matrix ◽  
Matrix Differential Equations ◽  
Low Rank Matrix Completion ◽  
Matrix Differential

In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature.

Predictive Nyström method for kernel methods

2017 ◽  
Vol 234 ◽  
pp. 116-125 ◽  
Author(s):  
Jiangang Wu ◽  
Lizhong Ding ◽  
Shizhong Liao
Keyword(s):  
Kernel Methods ◽  
Nyström Method ◽  
Nystrom Method
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