Recovering Data Permutations From Noisy Observations: The Linear Regime

Abstract

This paper considers the problem of recovering the permutation of an n-dimensional random vector X observed in Gaussian noise. First, a general expression for the probability of error is derived when a linear decoder (i.e., linear estimator followed by a sorting operation) is used. The derived expression holds with minimal assumptions on the distribution of X and when the noise has memory. Second, for the case of isotropic noise (i.e., noise with a diagonal scalar covariance matrix), the rates of convergence of the probability of error are characterized in the high and low noise regimes. In the low noise regime, for every dimension n, the probability of error is shown to behave proportionally to σ, where σ is the noise standard deviation. Moreover, the slope is computed exactly for several distributions and it is shown to behave quadratically in n. In the high noise regime, for every dimension n, the probability of correctness is shown to behave as 1/σ, and the exact expression for the rate of convergence is also provided.

Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms

“…Many statistical seriation problems [6,33,44], that in one way or another aim to find an element in the discrete permutation set optimizing certain objective function, have been studied from various aspects under different settings. These include the well-known consecutive one's problem [29,41,42] that dates back to the 1960s; the feature matching problem [23,38,30] and the noisy ranking problem [11,39,54,20,63,51,55,22]; the matrix seriation problem for various shape-constrained matrices including the monotone or bi-monotone matrices [26,50,47,57], the Robinson matrices [2,27,61,1], and the Monge matrices [36]; and more recently, the seriation problem under the latent space models [31,37]. Many of the existing works have focused on recovering the underlying permutations, estimation of the (disordered) signal structures, or both.…”

Ladder Matrix Recovery from Permutations

“…Besides its theoretical interest, which has attracted several authors (e.g., [6], [7], [8], [9], [10], [11]), unlabeled sensing entertains more than a few potential applications. These include record linkage ( [8], [12], [13]), image and point cloud registration ( [14]), cell sorting ( [15], [16]), metagenomics ( [17]), neuron matching ( [18]), spatial field estimation ( [19]), and target localization ( [20]).…”

Power permutation modulation in multiple‐input multiple‐output systems

“…As a special form of SM, space shift keying is introduced in Reference 27 by encoding the data bits only in the index of transmit antennas where a spatial constellation diagram is used for data modulation. The data permutation concept recently finds applications in some other areas including signal processing, 28,29 biostatistics, 30 and deep learning 31 …”

“…As a special form of SM, space shift keying is introduced in Reference 27 by encoding the data bits only in the index of transmit antennas where a spatial constellation diagram is used for data modulation. The data permutation concept recently finds applications in some other areas including signal processing, 28,29 biostatistics, 30 and deep learning. 31 Non-orthogonal multiple access (NOMA) has been promoted to meet the soaring demands (such as immense spectral/energy efficiency, high connectivity, and low latency) of the current and upcoming communications standards.…”