Optimal Estimation in Orthogonally Invariant Generalized Linear Models: Spectral Initialization and Approximate Message Passing

• URL: http://arxiv.org/abs/2602.09240v1
• Date: Mon, 09 Feb 2026 22:08:09 GMT
• Title: Optimal Estimation in Orthogonally Invariant Generalized Linear Models: Spectral Initialization and Approximate Message Passing
• Authors: Yihan Zhang, Hong Chang Ji, Ramji Venkataramanan, Marco Mondelli,
• Abstract summary: This paper proposes optimal spectral estimators and an approximate message passing (AMP) algorithm.<n>Building on the paradigm of spectrally-d iterative optimization, this paper proposes optimal spectral estimators and an approximate message passing (AMP) algorithm.
• Score: 36.35215381372567
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of parameter estimation from a generalized linear model with a random design matrix that is orthogonally invariant in law. Such a model allows the design have an arbitrary distribution of singular values and only assumes that its singular vectors are generic. It is a vast generalization of the i.i.d. Gaussian design typically considered in the theoretical literature, and is motivated by the fact that real data often have a complex correlation structure so that methods relying on i.i.d. assumptions can be highly suboptimal. Building on the paradigm of spectrally-initialized iterative optimization, this paper proposes optimal spectral estimators and combines them with an approximate message passing (AMP) algorithm, establishing rigorous performance guarantees for these two algorithmic steps. Both the spectral initialization and the subsequent AMP meet existing conjectures on the fundamental limits to estimation -- the former on the optimal sample complexity for efficient weak recovery, and the latter on the optimal errors. Numerical experiments suggest the effectiveness of our methods and accuracy of our theory beyond orthogonally invariant data.

Related papers

• Multi-Dimensional Visual Data Recovery: Scale-Aware Tensor Modeling and Accelerated Randomized Computation [51.65236537605077]
  We propose a new type of network compression optimization technique, fully randomized tensor network compression (FCTN)<n>FCTN has significant advantages in correlation characterization and transpositional in algebra, and has notable achievements in multi-dimensional data processing and analysis.<n>We derive efficient algorithms with guarantees to solve the formulated models.
  arXiv  Detail & Related papers  (2026-02-13T14:56:37Z)
• Orthogonal Approximate Message Passing with Optimal Spectral Initializations for Rectangular Spiked Matrix Models [4.705915211123485]
  We propose an approximate message passing (OAMP) algorithm for signal estimation in the rectangular spiked matrix model.<n>We establish a rigorous state evolution that precisely characterizes the algorithm's high-dimensional dynamics.<n>We conjecture that it is statistically optimal within a broad class of iterative estimation methods.
  arXiv  Detail & Related papers  (2025-12-22T12:28:53Z)
• On the Optimal Construction of Unbiased Gradient Estimators for Zeroth-Order Optimization [57.179679246370114]
  A potential limitation of existing methods is the bias inherent in most perturbation estimators unless a stepsize is proposed.<n>We propose a novel family of unbiased gradient scaling estimators that eliminate bias while maintaining favorable construction.
  arXiv  Detail & Related papers  (2025-10-22T18:25:43Z)
• Neural Optimal Transport Meets Multivariate Conformal Prediction [58.43397908730771]
  We propose a framework for conditional vectorile regression (CVQR)<n>CVQR combines neural optimal transport with quantized optimization, and apply it to predictions.
  arXiv  Detail & Related papers  (2025-09-29T19:50:19Z)
• Regularized Projection Matrix Approximation with Applications to Community Detection [1.3761665705201904]
  This paper introduces a regularized projection matrix approximation framework designed to recover cluster information from the affinity matrix. We investigate three distinct penalty functions, each specifically tailored to address bounded, positive, and sparse scenarios. Numerical experiments conducted on both synthetic and real-world datasets reveal that our regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.
  arXiv  Detail & Related papers  (2024-05-26T15:18:22Z)
• Convex Parameter Estimation of Perturbed Multivariate Generalized Gaussian Distributions [18.95928707619676]
  We propose a convex formulation with well-established properties for MGGD parameters. The proposed framework is flexible as it combines a variety of regularizations for the precision matrix, the mean and perturbations. Experiments show a more accurate precision and covariance matrix estimation with similar performance for the mean vector parameter.
  arXiv  Detail & Related papers  (2023-12-12T18:08:04Z)
• Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing [28.91482208876914]
  We consider the problem of parameter estimation in a high-dimensional generalized linear model.<n>Despite their wide use, a rigorous performance characterization, as well as a principled way to preprocess the data, are available only for unstructured designs.
  arXiv  Detail & Related papers  (2023-08-28T11:49:23Z)
• Exploring the Algorithm-Dependent Generalization of AUPRC Optimization with List Stability [107.65337427333064]
  optimization of the Area Under the Precision-Recall Curve (AUPRC) is a crucial problem for machine learning. In this work, we present the first trial in the single-dependent generalization of AUPRC optimization. Experiments on three image retrieval datasets on speak to the effectiveness and soundness of our framework.
  arXiv  Detail & Related papers  (2022-09-27T09:06:37Z)
• Sparse high-dimensional linear regression with a partitioned empirical Bayes ECM algorithm [62.997667081978825]
  We propose a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are used through the use of plug-in empirical Bayes estimates. The proposed approach is implemented in the R package probe.
  arXiv  Detail & Related papers  (2022-09-16T19:15:50Z)
• Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing [21.871513580418604]
  We propose a novel family of approximate message passing (AMP) algorithms for signal estimation. We rigorously characterize their performance in the high-dimensional limit via a state evolution recursion.
  arXiv  Detail & Related papers  (2021-12-08T15:20:04Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Asymptotic Analysis of an Ensemble of Randomly Projected Linear Discriminants [94.46276668068327]
  In [1], an ensemble of randomly projected linear discriminants is used to classify datasets. We develop a consistent estimator of the misclassification probability as an alternative to the computationally-costly cross-validation estimator. We also demonstrate the use of our estimator for tuning the projection dimension on both real and synthetic data.
  arXiv  Detail & Related papers  (2020-04-17T12:47:04Z)

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.