This paper considers a noisy data structure recovery problem. The goal is to investigate the following question: Given a noisy observation of a permuted data set, according to which permutation was the original data sorted? The focus is on scenarios where data is generated according to an isotropic Gaussian distribution, and the noise is additive Gaussian with an arbitrary covariance matrix. This problem is posed within a hypothesis testing framework. The objective is to study the linear regime in which the optimal decoder has a polynomial complexity in the data size, and it declares the permutation by simply computing a permutation-independent linear function of the noisy observations. The main result of the paper is a complete characterization of the linear regime in terms of the noise covariance matrix. Specifically, it is shown that this matrix must have a very flat spectrum with at most three distinct eigenvalues to induce the linear regime. Several practically relevant implications of this result are discussed, and the error probability incurred by the decision criterion in the linear regime is also characterized. A core technical component consists of using linear algebraic and geometric tools, such as Steiner symmetrization.
Volume-Refined Achievability and Converse Approximations for Noisy Permutation Channels
School of Electronics, Peking University（北京大学电子学院）；National Computer Network Emergency Response Technical Team/Coordination Center of China（国家计算机网络应急技术处理协调中心）
Extended Affine (EA) equivalence is the equivalence relation between two vectorial Boolean functions $F$ and $G$ such that there exist two affine permutations $A$ , $B$ , and an affine function $C$ satisfying $G = A \circ F \circ B + C$ . While the problem has a simple formulation, it is very difficult in practice to test whether two functions are EA-equivalent. This problem has two variants: { \em EA-partitioning} deals with partitioning a set of functions into disjoint EA-equivalence classes, and \emph{EA-recovery} is about recovering the tuple $(A,B,C)$ if it exists. In this paper, we prese…
Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting
研究了在高斯测量集成下，恢复一个固定但未知的向量$β^* \in \real^p based on a set of $ n $ noisy observations arises in a variety of settings, including subset selection in regression, graphical model selection, signal denoising, compressive sensing, and constructive approximation. Of interest are conditions on the model dimension $ p $, the sparsity index $ s $ (number of non-zero entries in $ β^* $), and the number of observations $ n $ that are necessary and/or sufficient to ensure asymptotically perfect recovery of the sparsity pattern. This paper focuses on the information-theoretic limits of sparsity recovery: in…
