  Motivated by applications in single-cell biology and metagenomics, we
investigate the problem of matrix reordering based on a noisy disordered
monotone Toeplitz matrix model. We establish the fundamental statistical limit
for this problem in a decision-theoretic framework and demonstrate that a
constrained least squares estimator achieves the optimal rate. However, due to
its computational complexity, we analyze a popular polynomial-time algorithm,
spectral seriation, and show that it is suboptimal. To address this, we propose
a novel polynomial-time adaptive sorting algorithm with guaranteed performance
improvement. Simulations and analyses of two real single-cell RNA sequencing
datasets demonstrate the superiority of our algorithm over existing methods.
