Error Propagation in Spectral Functionals of Shrinkage Covariance Estimators: Perturbation Bounds and Calibrated Inference

Ahmad Koman

Home/Archive/arXiv

arXiv · 2607.06373

Error Propagation in Spectral Functionals of Shrinkage Covariance Estimators: Perturbation Bounds and Calibrated Inference

Ahmad Koman

View original source ↗Read source document ↗Export metadata ↓

Abstract

Rolling covariance estimates feed two objects that are routinely treated as market structure. The first is the dominant eigenspace, monitored through the projector movement $\widehat D_{K,t}=\|\widehat P_{K,t}-\widehat P_{K,t-1}\|_F$; the second comprises scalar spectral functionals such as the absorption ratio and the leading-eigenvalue share. Both fluctuate under estimation noise, and shrinkage changes the law of that noise, so reading their movements as structural change requires calibration. For the eigenspace, we derive a first-order null law for $\widehat D_{K,t}$ between overlapping windows that share most of their data and show that it transfers without change to rotation-equivariant shrinkage estimators. A distribution-free Davis-Kahan band gauges whether the eigenspace is identified, an estimator-aware bootstrap provides the calibrated test, and a companion power analysis gives an approximate design rule for the smallest detectable rotation. For the scalar functionals, we show that first-order immunity to elliptical kurtosis holds for scale-invariant functionals and only for them, so that one estimated scalar calibrates the projector null and the absorption-ratio and leading-share intervals across the elliptical family. In high dimensions, where shrinkage cleaning biases the absorption ratio, we give a trace-preserving spike-debiased estimator that removes the bias. The results are verified by simulation under a known population covariance; an equity-panel appendix shows the procedures as diagnostics when the population is unknown.

Explore related subjects

stat.ME math.ST q-fin.ST stat.CO stat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Ahmad Koman. 2026-07-07. Error Propagation in Spectral Functionals of Shrinkage Covariance Estimators: Perturbation Bounds and Calibrated Inference. https://arxiv.org/abs/2607.06373

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Likelihood Based Inference in Fully and Partially Observed Exponential Family Graphical Models with Intractable Normalizing Constants

Probabilistic graphical models that encode an underlying Markov random field are fundamental building blocks of generative modeling to learn latent representations in modern multivariate data sets with complex dependency structures. Among these, the exponential family graphical models are especially popular, given their fairly well-understood statistical properties and computational scalability to high-dimensional data based on pseudo-likelihood methods. These models have been successfully applied in many fields, such as the Ising model in statistical physics and count graphical models in genomics. Another strand of models allows some nodes to be latent, so as to allow the marginal distribution of the observable nodes to depart from exponential family to capture more complex dependence. These approaches form the basis of generative models in artificial intelligence, such as the Boltzmann machines and their restricted versions. A fundamental barrier to likelihood-based (i.e., both maximum likelihood and fully Bayesian) inference in both fully and partially observed cases is the intractability of the likelihood. The usual workaround is via adopting pseudo likelihood-based approaches, following the pioneering work of Besag(1974). The goal of this paper is to demonstrate that full likelihood-based analysis of these models is feasible in a computationally efficient manner under a logarithmically sparse setting. The chief innovation lies in utilizing the tractable independence model underlying an intractable graphical model, to estimate the normalizing constant, as well as its gradient. Extensive numerical results, supporting theory and comparisons with pseudo likelihood-based approaches demonstrate the applicability of the proposed method.

stat.ME↗

arXiv2026

Interpretable Deep Neural Network for Modeling Functional Surrogates

Developing surrogates for computer models has become increasingly important for addressing complex problems in science and engineering. This article introduces an artificial intelligent (AI) surrogate, referred to as the DeepSurrogate, for analyzing functional outputs with vector-valued inputs. The relationship between the functional output and vector-valued input is modeled as an infinite sequence of unknown functions, each representing the relationship at a specific location within the functional domain. These spatially indexed functions are expressed through a combination of basis functions and their corresponding coefficient functions, both of which are modeled using deep neural networks (DNN). The proposed framework accounts for spatial dependencies across locations, while capturing the relationship between the functional output and scalar predictors. It also integrates a Monte Carlo (MC) dropout strategy to quantify prediction uncertainty, enhancing explainability in the deep neural network architecture. The proposed method enables efficient inference on datasets with approximately 50,000 spatial locations and 20 simulations, achieving results in under 10 minutes using standard hardware. The approach is validated on extensive synthetic datasets and a large-scale simulation from the Sea Lake and Overland Surge from Hurricanes (SLOSH) simulator. An open-source Python package implementing the method is made available.

stat.ME↗

arXiv2026

Deep Generative Modeling with Spatial and Network Images: An Explainable AI (XAI) Approach

This article addresses the challenge of modeling the amplitude of spatially indexed low frequency fluctuations (ALFF) in resting state functional MRI as a function of cortical structural features and a multi-task coactivation network in the Adolescent Brain Cognitive Development (ABCD) Study. It proposes a generative model that integrates effects of spatially-varying inputs and a network-valued input using deep neural networks to capture complex non-linear and spatial associations with the output. The method models spatial smoothness, accounts for subject heterogeneity and complex associations between network and spatial images at different scales, enables accurate inference of each images effect on the output image, and allows prediction with uncertainty quantification via Monte Carlo dropout, contributing to one of the first Explainable AI (XAI) frameworks for heterogeneous imaging data. The model is highly scalable to high-resolution data without the heavy pre-processing or summarization often required by Bayesian methods. Empirical results demonstrate its strong performance compared to existing statistical and deep learning methods. We applied the XAI model to the ABCD data which revealed associations between cortical features and ALFF throughout the entire brain. Our model performed comparably to existing methods in predictive accuracy but provided superior uncertainty quantification and faster computation, demonstrating its effectiveness for large-scale neuroimaging analysis. Open-source software in Python for XAI is available.

stat.ME↗