Positive Semidefinite Matrix Supermartingales

• URL: http://arxiv.org/abs/2401.15567v4
• Date: Wed, 29 Jan 2025 00:07:26 GMT
• Title: Positive Semidefinite Matrix Supermartingales
• Authors: Hongjian Wang, Aaditya Ramdas,
• Abstract summary: We explore the convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. Our results lead to new concentration inequalities for either martingale dependent or exchangeable random symmetric matrices under a variety of tail conditions. These inequalities are usually expressed in the boundsner order, are sometimes valid simultaneously for all sample sizes or at an arbitrary data-dependent stopping time, and can often be tightened via an external randomization factor.
• Score: 30.14855064043107
• License:
• Abstract: We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative supermartingales and backward nonnegative submartingales, whose convergence and maximal inequalities are the theoretical foundations for a wide and ever-growing body of results in statistics, econometrics, and theoretical computer science. Our results lead to new concentration inequalities for either martingale dependent or exchangeable random symmetric matrices under a variety of tail conditions, encompassing now-standard Chernoff bounds to self-normalized heavy-tailed settings. Further, these inequalities are usually expressed in the Loewner order, are sometimes valid simultaneously for all sample sizes or at an arbitrary data-dependent stopping time, and can often be tightened via an external randomization factor.

Related papers

• Sharp Matrix Empirical Bernstein Inequalities [30.14855064043107]
  We present two sharp empirical Bernstein inequalities for symmetric random matrices with bounded eigenvalues. By sharp, we mean that both inequalities adapt to the unknown variance in a tight manner.
  arXiv  Detail & Related papers  (2024-11-14T15:27:18Z)
• Efficient conversion from fermionic Gaussian states to matrix product states [48.225436651971805]
  We propose a highly efficient algorithm that converts fermionic Gaussian states to matrix product states. It can be formulated for finite-size systems without translation invariance, but becomes particularly appealing when applied to infinite systems. The potential of our method is demonstrated by numerical calculations in two chiral spin liquids.
  arXiv  Detail & Related papers  (2024-08-02T10:15:26Z)
• Entrywise error bounds for low-rank approximations of kernel matrices [55.524284152242096]
  We derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues. We validate our theory with an empirical study of a collection of synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2024-05-23T12:26:25Z)
• Riemannian statistics meets random matrix theory: towards learning from high-dimensional covariance matrices [2.352645870795664]
  This paper shows that there seems to exist no practical method of computing the normalising factors associated with Riemannian Gaussian distributions on spaces of high-dimensional covariance matrices. It is shown that this missing method comes from an unexpected new connection with random matrix theory. Numerical experiments are conducted which demonstrate how this new approximation can unlock the difficulties which have impeded applications to real-world datasets.
  arXiv  Detail & Related papers  (2022-03-01T03:16:50Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)
• Can Single-Shuffle SGD be Better than Reshuffling SGD and GD? [77.82009268160053]
  We conjecture that the means of matrix products corresponding to with- and without-replacement variants of SGD satisfy a series of spectral norm inequalities. We present theorems that support our conjecture by proving several special cases.
  arXiv  Detail & Related papers  (2021-03-12T04:34:45Z)
• Statistics of the Spectral Form Factor in the Self-Dual Kicked Ising Model [0.0]
  We show that at large enough times the probability distribution agrees exactly with the prediction of Random Matrix Theory. This behaviour is due to a recently identified additional anti-unitary symmetry of the self-dual kicked Ising model.
  arXiv  Detail & Related papers  (2020-09-07T16:02:29Z)
• On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration [115.1954841020189]
  We study the inequality and non-asymptotic properties of approximation procedures with Polyak-Ruppert averaging. We prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity.
  arXiv  Detail & Related papers  (2020-04-09T17:54:18Z)
• Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion [101.83262280224729]
  We develop a relative error bound for nuclear norm regularized matrix completion. We derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix.
  arXiv  Detail & Related papers  (2015-04-26T13:12:16Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.