We establish existence of Stein kernels for probability measures on R d satisfying a Poincaré inequality, and obtain bounds on the Stein discrepancy of such measures. Applications to quantitative central limit theorems are discussed, including a new CLT in Wasserstein distance W 2 with optimal rate and dependence on the dimension. As a byproduct, we obtain a stable version of an estimate of the Poincaré constant of probability measures under a second moment constraint. The results extend more generally to the setting of converse weighted Poincaré inequalities. The proof is based on simple arguments of calculus of variations.Further, we establish two general properties enjoyed by the Stein discrepancy, holding whenever a Stein kernel exists: Stein discrepancy is strictly decreasing along the CLT, and it controls the skewness of a random vector.

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Consider a noisy linear observation model with an unknown permutation, based on observing y = Π * Ax * + w, where x * ∈ R d is an unknown vector, Π * is an unknown n × n permutation matrix, and w ∈ R n is additive Gaussian noise. We analyze the problem of permutation recovery in a random design setting in which the entries of the matrix A are drawn i.i.d. from a standard Gaussian distribution, and establish sharp conditions on the SNR, sample size n, and dimension d under which Π * is exactly and approximately recoverable. On the computational front, we show that the maximum likelihood estimate of Π * is NP-hard to compute, while also providing a polynomial time algorithm when d = 1.

The multivariate linear regression model with shuffled data and additive Gaussian noise arises in various correspondence estimation and matching problems. Focusing on the denoising aspect of this problem, we provide a characterization the minimax error rate that is sharp up to logarithmic factors. We also analyze the performance of two versions of a computationally efficient estimator, and establish their consistency for a large range of input parameters. Finally, we provide an exact algorithm for the noiseless problem and demonstrate its performance on an image point-cloud matching task. Our analysis also extends to datasets with outliers.1 We refer to the setting W = 0 a.s. as the noiseless case.

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