On Minimax Detection of Gaussian Stochastic Sequences and Gaussian Stationary Signals

On Minimax Robust Detection of Stationary Gaussian Signals in White Gaussian Noise

The problem of detecting a wide-sense stationary Gaussian signal process embedded in white Gaussian noise, where the power spectral density of the signal process exhibits uncertainty, is investigated. The performance of minimax robust detection is characterized by the exponential decay rate of the miss probability under a Neyman-Pearson criterion with a fixed false alarm probability, as the length of the observation interval grows without bound. A dominance condition is identified for the uncertainty set of spectral density functions, and it is established that, under the dominance condition, …

On Minimax Detection of Gaussian Stochastic Sequences with Imprecisely Known Means and Covariance Matrices

We consider the problem of detecting (testing) Gaussian stochastic sequences (signals) with imprecisely known means and covariance matrices. The alternative is independent identically distributed zero-mean Gaussian random variables with unit variances. For a given false alarm (1st-kind error) probability, the quality of minimax detection is given by the best miss probability (2nd-kind error probability) exponent over a growing observation horizon. We explore the maximal set of means and covariance matrices (composite hypothesis) such that its minimax testing can be replaced with testing a sing…

The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising

Let $X_0$ be an unknown $M$ by $N$ matrix. In matrix recovery, one takes $n < MN$ linear measurements $y_1,..., y_n$ of $X_0$, where $y_i = \Tr(a_i^T X_0)$ and each $a_i$ is a $M$ by $N$ matrix. For measurement matrices with Gaussian i.i.d entries, it known that if $X_0$ is of low rank, it is recoverable from just a few measurements. A popular approach for matrix recovery is Nuclear Norm Minimization (NNM). Empirical work reveals a \emph{phase transition} curve, stated in terms of the undersampling fraction $δ(n,M,N) = n/(MN)$, rank fraction $ρ=r/N$ and aspect ratio $β=M/N$. Specifically, a cu…

具有隐私保护的交互式统计决策的最小最大分位数下界

Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy

针对期望准则忽略罕见但严重后果的问题，提出δ-显式最小最大分位数理论，导出高概率交互式Fano和Le Cam方法，并应用于带隐私约束的高斯均值估计和K臂高斯老虎机，得到显式下界。

Computational barriers in minimax submatrix detection

This paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive Gaussian noise. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent to the Gaussian model. Under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size, the following phase transition phenomenon i…

The geometry of hypothesis testing over convex cones: Generalized likelihood tests and minimax radii

We consider a compound testing problem within the Gaussian sequence model in which the null and alternative are specified by a pair of closed, convex cones. Such cone testing problem arise in various applications, including detection of treatment effects, trend detection in econometrics, signal detection in radar processing, and shape-constrained inference in non-parametric statistics. We provide a sharp characterization of the GLRT testing radius up to a universal multiplicative constant in terms of the geometric structure of the underlying convex cones. When applied to concrete examples, thi…