Flat Minima in Linear Estimation and an Extended Gauss Markov Theorem

• URL: http://arxiv.org/abs/2311.11093v1
• Date: Sat, 18 Nov 2023 14:45:06 GMT
• Title: Flat Minima in Linear Estimation and an Extended Gauss Markov Theorem
• Authors: Simon Segert
• Abstract summary: We derive simple and explicit formulas for the optimal estimator in the cases of Nuclear and Spectral norms. We analytically derive the generalization error in multiple random matrix ensembles, and compare with Ridge regression.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of linear estimation, and establish an extension of the Gauss-Markov theorem, in which the bias operator is allowed to be non-zero but bounded with respect to a matrix norm of Schatten type. We derive simple and explicit formulas for the optimal estimator in the cases of Nuclear and Spectral norms (with the Frobenius case recovering ridge regression). Additionally, we analytically derive the generalization error in multiple random matrix ensembles, and compare with Ridge regression. Finally, we conduct an extensive simulation study, in which we show that the cross-validated Nuclear and Spectral regressors can outperform Ridge in several circumstances.

Related papers

• Low-rank Bayesian matrix completion via geodesic Hamiltonian Monte Carlo on Stiefel manifolds [0.18416014644193066]
  We present a new sampling-based approach for enabling efficient computation of low-rank Bayesian matrix completion. We show that our approach resolves the sampling difficulties encountered by standard Gibbs samplers for the common two-matrix factorization used in matrix completion. Numerical examples demonstrate superior sampling performance, including better mixing and faster convergence to a stationary distribution.
  arXiv  Detail & Related papers  (2024-10-27T03:12:53Z)
• Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation [49.67011673289242]
  This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a smooth manifold. It induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure.
  arXiv  Detail & Related papers  (2023-11-08T15:17:13Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Probabilistic Unrolling: Scalable, Inverse-Free Maximum Likelihood Estimation for Latent Gaussian Models [69.22568644711113]
  We introduce probabilistic unrolling, a method that combines Monte Carlo sampling with iterative linear solvers to circumvent matrix inversions. Our theoretical analyses reveal that unrolling and backpropagation through the iterations of the solver can accelerate gradient estimation for maximum likelihood estimation. In experiments on simulated and real data, we demonstrate that probabilistic unrolling learns latent Gaussian models up to an order of magnitude faster than gradient EM, with minimal losses in model performance.
  arXiv  Detail & Related papers  (2023-06-05T21:08:34Z)
• High-dimensional analysis of double descent for linear regression with random projections [0.0]
  We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem. We first consider the ridge regression estimator and re-interpret earlier results using classical notions from non-parametric statistics. We then compute equivalents of the generalization performance (in terms of bias and variance) of the minimum norm least-squares fit with random projections, providing simple expressions for the double descent phenomenon.
  arXiv  Detail & Related papers  (2023-03-02T15:58:09Z)
• Robust Regularized Low-Rank Matrix Models for Regression and Classification [14.698622796774634]
  We propose a framework for matrix variate regression models based on a rank constraint, vector regularization (e.g., sparsity), and a general loss function. We show that the algorithm is guaranteed to converge, all accumulation points of the algorithm have estimation errors in the order of $O(sqrtn)$ally and substantially attaining the minimax rate.
  arXiv  Detail & Related papers  (2022-05-14T18:03:48Z)
• 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)
• Variational Inference in high-dimensional linear regression [5.065420156125522]
  We study high-dimensional Bayesian linear regression with product priors. We provide sufficient conditions for the leading-order correctness of the naive mean-field approximation to the log-normalizing constant of the posterior distribution.
  arXiv  Detail & Related papers  (2021-04-25T19:09:38Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• Asymptotic Errors for Teacher-Student Convex Generalized Linear Models (or : How to Prove Kabashima's Replica Formula) [23.15629681360836]
  We prove an analytical formula for the reconstruction performance of convex generalized linear models. We show that an analytical continuation may be carried out to extend the result to convex (non-strongly) problems. We illustrate our claim with numerical examples on mainstream learning methods.
  arXiv  Detail & Related papers  (2020-06-11T16:26:35Z)
• 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.