scholarly journals Simultaneous low-rank component and graph estimation for high-dimensional graph signals: Application to brain imaging

2017 ◽  
Author(s):  
Liu Rui ◽  
Hossein Nejati ◽  
Seyed Hamid Safavi ◽  
Ngai-Man Cheung
Keyword(s):  
Brain Imaging ◽  
Low Rank ◽  
High Dimensional ◽  
Graph Estimation ◽  
Dimensional Graph

scholarly journals Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1330
Author(s):  
Raeyong Kim
Keyword(s):  
Group Structure ◽  
Conjugacy Problem ◽  
High Dimensional ◽  
Fundamental Groups ◽  
Graph Manifold ◽  
Solvable Word Problem ◽  
Algorithmic Problems ◽  
Graph Manifolds ◽  
Dimensional Graph ◽  
Solvable Word

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs

2019 ◽  
Vol 19 (1) ◽  
pp. 39-53 ◽  
Author(s):  
Martin Eigel ◽  
Johannes Neumann ◽  
Reinhold Schneider ◽  
Sebastian Wolf
Keyword(s):  
Polynomial Chaos ◽  
Monte Carlo Sampling ◽  
Black Box ◽  
Low Rank ◽  
High Dimensional ◽  
Reconstruction Procedure ◽  
Numerical Examples ◽  
Quasi Monte Carlo ◽  
Generalized Polynomial ◽  
Stochastic Solution

AbstractThis paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions of high-dimensional parametric random PDEs, which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to (Quasi-)Monte Carlo sampling.

Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

2016 ◽  
Vol 6 (2) ◽  
pp. 109-130 ◽  
Author(s):  
Siu-Long Lei ◽  
Xu Chen ◽  
Xinhe Zhang
Keyword(s):  
Krylov Subspace ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Low Rank ◽  
High Dimensional ◽  
Fractional Diffusion Equations ◽  
Implicit Finite Difference Scheme ◽  
Small Norm ◽  
Variable Diffusion ◽  
Variable Diffusion Coefficients

AbstractHigh-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(NlogN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

scholarly journals Learning Markov Models Via Low-Rank Optimization

2021 ◽  
Author(s):  
Ziwei Zhu ◽  
Xudong Li ◽  
Mengdi Wang ◽  
Anru Zhang
Keyword(s):  
Markov Models ◽  
Supply And Demand ◽  
Likelihood Estimation ◽  
Low Rank ◽  
Programming Algorithm ◽  
High Dimensional ◽  
High Dimensions ◽  
Taxi Service ◽  
Low Dimensional ◽  
Nuclear Norm Regularization

Taming high-dimensional Markov models In “Learning Markov models via low-rank optimization”, Z. Zhu, X. Li, M. Wang, and A. Zhang focus on learning a high-dimensional Markov model with low-dimensional latent structure from a single trajectory of states. To overcome the curse of high dimensions, the authors propose to equip the standard MLE (maximum-likelihood estimation) with either nuclear norm regularization or rank constraint. They show that both approaches can estimate the full transition matrix accurately using a trajectory of length that is merely proportional to the number of states. To solve the rank-constrained MLE, which is a nonconvex problem, the authors develop a new DC (difference) programming algorithm. Finally, they apply the proposed methods to analyze taxi trips on the Manhattan island and partition the island based on the destination preference of customers; this partition can help balance supply and demand of taxi service and optimize the allocation of traffic resources.

scholarly journals A Block-Sparse Tensor Train Format for Sample-Efficient High-Dimensional Polynomial Regression

2021 ◽  
Vol 7 ◽  
Author(s):  
Michael Götte ◽  
Reinhold Schneider ◽  
Philipp Trunschke
Keyword(s):  
Polynomial Regression ◽  
Low Rank ◽  
High Dimensional ◽  
Multivariate Polynomials ◽  
Polynomial Space ◽  
Least Squares Regression ◽  
Symmetric Tensors ◽  
Single Block ◽  
Regression Problems ◽  
Block Sizes

Low-rank tensors are an established framework for the parametrization of multivariate polynomials. We propose to extend this framework by including the concept of block-sparsity to efficiently parametrize homogeneous, multivariate polynomials with low-rank tensors. This provides a representation of general multivariate polynomials as a sum of homogeneous, multivariate polynomials, represented by block-sparse, low-rank tensors. We show that this sum can be concisely represented by a single block-sparse, low-rank tensor.We further prove cases, where low-rank tensors are particularly well suited by showing that for banded symmetric tensors of homogeneous polynomials the block sizes in the block-sparse multivariate polynomial space can be bounded independent of the number of variables.We showcase this format by applying it to high-dimensional least squares regression problems where it demonstrates improved computational resource utilization and sample efficiency.

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