Optimal Rates of Statistical Seriation
This work addresses a problem in shape-constrained estimation and permutation learning, with applications in areas like graph denoising and ranking, but it is incremental as it builds on existing statistical frameworks.
The paper tackles the statistical seriation problem of permuting rows of a noisy matrix to achieve columns with specific shapes like monotonicity, showing that the least squares estimator is optimal up to logarithmic factors for unimodal or monotone columns and adapts to natural structures. It also proposes a computationally efficient estimator for the monotonic case, with theoretical and experimental validation.
Given a matrix the seriation problem consists in permuting its rows in such way that all its columns have the same shape, for example, they are monotone increasing. We propose a statistical approach to this problem where the matrix of interest is observed with noise and study the corresponding minimax rate of estimation of the matrices. Specifically, when the columns are either unimodal or monotone, we show that the least squares estimator is optimal up to logarithmic factors and adapts to matrices with a certain natural structure. Finally, we propose a computationally efficient estimator in the monotonic case and study its performance both theoretically and experimentally. Our work is at the intersection of shape constrained estimation and recent work that involves permutation learning, such as graph denoising and ranking.