Regularized Variational and Spectral Log-Density-Ratio Estimation in the Gaussian Location Model

• URL: http://arxiv.org/abs/2607.01895v1
• Date: Thu, 02 Jul 2026 08:46:52 GMT
• Title: Regularized Variational and Spectral Log-Density-Ratio Estimation in the Gaussian Location Model
• Abstract summary: We study ridge-regularized log-density-ratio estimation in the Gaussian location model.<n>The variational estimator is the empirical Kullback-Leibler (KL) log-normalized fit with a squared L2-penalty on its nonconstant coefficient.
• Score: 0.554780083433538
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study ridge-regularized log-density-ratio estimation in the Gaussian location model with a common covariance matrix. By affine invariance, the model is written as q $\sim$ N(0, I), p $\sim$ N($Δ$, I), with linear features, where $Δ$ is a mean vector. The variational estimator is the empirical Kullback-Leibler (KL) log-normalized fit with a squared L2-penalty on its nonconstant coefficient, and the spectral estimator recently introduced in [1] replaces a single variational problem by a continuum of ridge-regularized least-squares problems. We derive high-dimensional deterministic asymptotic equivalents when the numbers of observations and dimension tend to infinity with fixed ratios. The regularized variational limit is characterized by a scalar entropy minimization problem derived from the convex-Gaussian-min-max theorem (CGMT), while the regularized spectral limit follows from deterministic equivalents for resolvents of weighted sums of two independent Gaussian sample covariance matrices. We use these formulas to compare population risks, with experiments focused on fixed-signal aspect-ratio sweeps and optimized regularization. Our conclusion is that with many observations, under the criteria and asymptotic regimes analyzed here, the well-specified variational estimator has the smaller risk, while with fewer observations, the spectral estimator is favored because its covariance-based construction has lower variance. We also study how a nuclear penalty can be used and partially analyzed to perform feature learning.

Related papers

• On Higher-Order Geometric Refinements of Classical Covariance Asymptotics: An Approach via Intrinsic and Extrinsic Information Geometry [0.0]
  In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-root parameter spaces, finite-sample behavior can deviate systematically from predictions.<n>We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a ((,g)) with Fisher-constrained-Rao metric.<n>We describe the resolved metric, the role of the real log canonical threshold in learning rates and posterior mean-squared error, and a curvature-based covariance expansion on the resolved space that recovers the regular theory as a special case.
  arXiv  Detail & Related papers  (2026-04-14T13:40:13Z)
• Efficient Covariance Estimation for Sparsified Functional Data [51.69796254617083]
  proposed Random-knots (Random-knots-Spatial) and B-spline (Bspline-Spatial) estimators of the covariance function are computationally efficient.<n>Asymptotic pointwise of the covariance are obtained for sparsified individual trajectories under some regularity conditions.
  arXiv  Detail & Related papers  (2025-11-23T00:50:33Z)
• Asymptotics of Linear Regression with Linearly Dependent Data [28.005935031887038]
  We study the computations of linear regression in settings with non-Gaussian covariates.<n>We show how dependencies influence estimation error and the choice of regularization parameters.
  arXiv  Detail & Related papers  (2024-12-04T20:31:47Z)
• Statistical Inference in Classification of High-dimensional Gaussian Mixture [1.2354076490479515]
  We investigate the behavior of a general class of regularized convex classifiers in the high-dimensional limit. Our focus is on the generalization error and variable selection properties of the estimators.
  arXiv  Detail & Related papers  (2024-10-25T19:58:36Z)
• Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations [51.000851088730684]
  We develop novel modifications of nearest-neighbor and matching estimators which converge at the parametric $sqrt n $-rate.<n>We stress that our estimators do not involve nonparametric function estimators and in particular do not rely on sample-size dependent parameters smoothing.
  arXiv  Detail & Related papers  (2024-07-11T13:28:34Z)
• Optimal Differentially Private PCA and Estimation for Spiked Covariance Matrices [10.377683220196873]
  Estimating a covariance matrix and its associated principal components is a fundamental problem in contemporary statistics. We study optimal differentially private Principal Component Analysis (PCA) and covariance estimation within the spiked covariance model. We propose computationally efficient differentially private estimators and prove their minimax optimality for sub-Gaussian distributions.
  arXiv  Detail & Related papers  (2024-01-08T11:18:14Z)
• Spectrum-Aware Debiasing: A Modern Inference Framework with Applications to Principal Components Regression [1.342834401139078]
  We introduce SpectrumAware Debiasing, a novel method for high-dimensional regression. Our approach applies to problems with structured, heavy tails, and low-rank structures. We demonstrate our method through simulated and real data experiments.
  arXiv  Detail & Related papers  (2023-09-14T15:58:30Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• 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)
• 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)
• Non-asymptotic Optimal Prediction Error for Growing-dimensional Partially Functional Linear Models [0.951828574518325]
  We show the rate-optimal upper and lower bounds of the prediction error. An exact upper bound for the excess prediction risk is shown in a non-asymptotic form. We derive the non-asymptotic minimax lower bound under the regularity assumption of the Kullback-Leibler divergence of the models.
  arXiv  Detail & Related papers  (2020-09-10T08:49:32Z)

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.