arXiv subjects

Benjamin D. Robinson

Publications and source records attributed to Benjamin D. Robinson.

8 recordsLinked to original sources

arXiv2025

High-Dimensional Sequential Change Detection

We address the problem of detecting a change in the distribution of a high-dimensional multivariate normal time series. Assuming that the post-change parameters are unknown and estimated using a window of historical data, we extend the framework of quickest change detection (QCD) to the highdimensional setting in which the number of variables increases proportionally with the size of the window used to estimate the post-change parameters. Our analysis reveals that an information theoretic quantity, which we call the Normalized High- Dimensional Kullback-Leibler divergence (NHDKL), governs the high-dimensional asymptotic performance of QCD procedures. Specifically, we show that the detection delay is asymptotically inversely proportional to the difference between the NHDKL of the true post-change versus pre-change distributions and the NHDKL of the true versus estimated post-change distributions. In cases of perfect estimation, where the latter NHDKL is zero, the delay is inversely proportional to the NHDKL between the post-change and pre-change distributions alone. Thus, our analysis is a direct generalization of the traditional fixed-dimension, large-sample asymptotic framework, where the standard KL divergence is asymptotically inversely proportional to detection delay. Finally, we identify parameter estimators that asymptotically minimize the NHDKL between the true versus estimated post-change distributions, resulting in a QCD method that is guaranteed to outperform standard approaches based on fixed-dimension asymptotics.

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arXiv2025

Spectrally Robust Covariance Shrinkage for Hotelling's $T^2$ in High Dimensions

We investigate covariance shrinkage for Hotelling's $T^2$ in the regime where the data dimension $p$ and the sample size $n$ grow in a fixed ratio -- without assuming that the population covariance matrix is spiked or well-conditioned. When $p/n\to\phi \in (0,1)$, we propose a practical finite-sample shrinker that, for any maximum-entropy signal prior and any fixed significance level, (a) asymptotically maximizes power under Gaussian data, and (b) asymptotically saturates the Hanson--Wright lower bound on power in the more general sub-Gaussian case. Our approach is to formulate and solve a variational problem characterizing the optimal limiting shrinker, and to show that our finite-sample method consistently approximates this limit by extending recent local random matrix laws. Empirical studies on simulated and real-world data, including the Crawdad UMich/RSS data set, demonstrate up to a $50\%$ gain in power over leading linear and nonlinear competitors at a significance level of $10^{-4}$.

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arXiv2023

Anomaly Search Over Many Sequences With Switching Costs

This paper considers the quickest search problem to identify anomalies among large numbers of data streams. These streams can model, for example, disjoint regions monitored by a mobile robot. A particular challenge is a version of the problem in which the experimenter must suffer a cost each time the data stream being sampled changes, such as the time the robot must spend moving between regions. In this paper, we propose an algorithm which accounts for switching costs by varying a confidence threshold that governs when the algorithm switches to a new data stream. Our main contributions are easily computable approximations for both the optimal value of this threshold and the optimal value of the parameter that determines when a stream must be re-sampled. Further, we empirically show (i) a uniform improvement for switching costs of interest and (ii) roughly equivalent performance for small switching costs when comparing to the closest available algorithm.

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arXiv2023

The Local Ledoit-Peche Law

Ledoit and Peche proved convergence of certain functions of a random covariance matrix's resolvent; we refer to this as the Ledoit-Peche law. One important application of their result is shrinkage covariance estimation with respect to so-called Minimum Variance (MV) loss, discussed in the work of Ledoit and Wolf. We provide an essentially optimal rate of convergence and hypothesize it to be the smallest possible rate of excess MV loss within the shrinkage class.

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arXiv2022

An Improvement on the Hotelling $T^2$ Test Using the Ledoit-Wolf Nonlinear Shrinkage Estimator

Hotelling's $T^2$ test is a classical approach for discriminating the means of two multivariate normal samples that share a population covariance matrix. Hotelling's test is not ideal for high-dimensional samples because the eigenvalues of the estimated sample covariance matrix are inconsistent estimators for their population counterparts. We replace the sample covariance matrix with the nonlinear shrinkage estimator of Ledoit and Wolf 2020. We observe empirically for sub-Gaussian data that the resulting algorithm dominates past methods (Bai and Saranadasa 1996, Chen and Qin 2010, and Li et al. 2020) for a family of population covariance matrices that includes matrices with high or low condition number and many or few nontrivial -- i.e., spiked -- eigenvalues.

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arXiv2021

High-Dimensional Covariance Shrinkage for Signal Detection

In this paper, we consider the problem of determining the presence of a given signal in a high-dimensional observation with unknown covariance matrix by using an adaptive matched filter. Traditionally such filters are formed from the sample covariance matrix of some given training data, but, as is well-known, the performance of such filters is poor when the number of training data $n$ is not much larger than the data dimension $p$. We thus seek a covariance estimator to replace sample covariance. To account for the fact that $n$ and $p$ may be of comparable size, we adopt the "large-dimensional asymptotic model" in which $n$ and $p$ go to infinity in a fixed ratio. Under this assumption, we identify a covariance estimator that is asymptotically detection-theoretic optimal within a general shrinkage class inspired by C. Stein, and we give consistent estimates for conditional false-alarm and detection rate of the corresponding adaptive matched filter.

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arXiv2020

Fisher-Rao distance on the covariance cone

The Fisher-Rao geodesic distance on the statistical manifold consisting of zero-mean p-dimensional multivariate Gaussians appears without proof in several places (such as Steven Smith's "Covariance, Subspace, and Intrinsic Cramer-Rao Bounds"). In this paper, we give a proof using basic Riemannian geometry.

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arXiv2020

Space-Time Adaptive Detection at Low Sample Support

An important problem in space-time adaptive detection is the estimation of the large p-by-p interference covariance matrix from training signals. When the number of training signals n is greater than 2p, existing estimators are generally considered to be adequate, as demonstrated by fixed-dimensional asymptotics. But in the low-sample-support regime (n < 2p or even n < p) fixed-dimensional asymptotics are no longer applicable. The remedy undertaken in this paper is to consider the "large dimensional limit" in which n and p go to infinity together. In this asymptotic regime, a new type of estimator is defined (Definition 2), shown to exist (Theorem 1), and shown to be detection-theoretically ideal (Theorem 2). Further, asymptotic conditional detection and false-alarm rates of filters formed from this type of estimator are characterized (Theorems 3 and 4) and shown to depend only on data that is given, even for non-Gaussian interference statistics. The paper concludes with several Monte Carlo simulations that compare the performance of the estimator in Theorem 1 to the predictions of Theorems 2-4, showing in particular higher detection probability than Steiner and Gerlach's Fast Maximum Likelihood estimator.

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