# Optimal Rates of Statistical Seriation

arXiv: https://arxiv.org/abs/1607.02435 · HTML: https://mlsanity.com/paper/1607.02435/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2016.

## Built on by (most cited first, 10 of 10)

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- [Permuted and Unlinked Monotone Regression in ℝd: an approach based on mixture modeling and optimal transport](https://mlsanity.com/paper/2201.03528/) (2022)
- [Statistical and Computational Efficiency for Smooth Tensor Estimation with Unknown Permutations](https://mlsanity.com/paper/2111.04681/) (2021)
- [Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms](https://mlsanity.com/paper/2201.06438/) (2022)
- [Regularization for Shuffled Data Problems via Exponential Family Priors on the Permutation Group](https://mlsanity.com/paper/2111.01767/) (2021)
- [Matching Map Recovery with an Unknown Number of Outliers](https://mlsanity.com/paper/2210.13354/) (2022)
- [Covariance alignment: from maximum likelihood estimation to Gromov-Wasserstein](https://mlsanity.com/paper/2311.13595/) (2023)
- [Modeling and Correcting Bias in Sequential Evaluation](https://mlsanity.com/paper/2205.01607/) (2022)
- [Statistical and computational challenges in ranking](https://mlsanity.com/paper/2512.21111/) (2025)

## Abstract (arXiv)

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.