Bernstein Concentration Inequalities for Tensors via Einstein Products

A generalization of the Bernstein matrix concentration inequality to random tensors of general order is proposed. This generalization is based on the use of Einstein products between tensors, from which a strong link can be established between matrices and tensors, in turn allowing exploitation of existing results for the former.

[1]  Stefania Bellavia,et al.  Subsampled inexact Newton methods for minimizing large sums of convex functions , 2018, IMA Journal of Numerical Analysis.

[2]  A. Einstein.  The Foundation of the General Theory of Relativity , 1916 .

[3]  Na Li,et al.  Solving Multilinear Systems via Tensor Inversion , 2013, SIAM J. Matrix Anal. Appl..

[4]  B. Sturmfels,et al.  The number of eigenvalues of a tensor , 2010, 1004.4953.

[5]  Aurélien Lucchi,et al.  Sub-sampled Cubic Regularization for Non-convex Optimization , 2017, ICML.

[6]  Tianyi Lin,et al.  Adaptively Accelerating Cubic Regularized Newton's Methods for Convex Optimization via Random Sampling , 2018 .

[7]  Y. Ye,et al.  Linear operators and positive semidefiniteness of symmetric tensor spaces , 2015 .

[8]  L. Qi,et al.  Higher Order Positive Semidefinite Diffusion Tensor Imaging , 2010, SIAM J. Imaging Sci..

[9]  Joel A. Tropp,et al.  An Introduction to Matrix Concentration Inequalities , 2015, Found. Trends Mach. Learn..

[10]  Rudolf Ahlswede,et al.  Strong converse for identification via quantum channels , 2000, IEEE Trans. Inf. Theory.

[11]  STEFANIA BELLAVIA,et al.  Adaptive cubic regularization methods with dynamic inexact Hessian information and applications to finite-sum minimization , 2018, IMA Journal of Numerical Analysis.

[12]  J. Wishart.  THE GENERALISED PRODUCT MOMENT DISTRIBUTION IN SAMPLES FROM A NORMAL MULTIVARIATE POPULATION , 1928 .

[13]  S. Bellavia,et al.  Adaptive Regularization Algorithms with Inexact Evaluations for Nonconvex Optimization , 2018, SIAM J. Optim..

[14]  Peng Xu,et al.  Newton-type methods for non-convex optimization under inexact Hessian information , 2017, Math. Program..

[15]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[16]  Tianyi Lin,et al.  On Adaptive Cubic Regularized Newton's Methods for Convex Optimization via Random Sampling , 2018 .

[17]  Peng Xu,et al.  Second-Order Optimization for Non-Convex Machine Learning: An Empirical Study , 2017, SDM.

[18]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[19]  Alan M. Frieze,et al.  Fast Monte-Carlo algorithms for finding low-rank approximations , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[20]  L. Qi,et al.  Tensor Analysis: Spectral Theory and Special Tensors , 2017 .

[21]  Stefania Bellavia,et al.  Deterministic and stochastic inexact regularization algorithms for nonconvex optimization with optimal complexity , 2018, ArXiv.

[22]  David Rubin,et al.  Introduction to Continuum Mechanics , 2009 .

[23]  Dimitris Achlioptas,et al.  Fast computation of low-rank matrix approximations , 2007, JACM.

[24]  William Thomson,et al.  XXI. Elements of a mathematical theory of elasticity , 1856, Philosophical Transactions of the Royal Society of London.

[25]  P. Forrester.  Log-Gases and Random Matrices , 2010 .