# Spectral thresholding for the estimation of Markov chain transition operators

arXiv: https://arxiv.org/abs/1808.08153 · HTML: https://mlsanity.com/paper/1808.08153/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Aug 2018.

## Built on by (most cited first, 2 of 2)

- [Learning Good State and Action Representations via Tensor Decomposition](https://mlsanity.com/paper/2105.01136/) (2021)
- [Minimax estimation of the pseudo-spectral gap of continuous Markov chains, with applications to empirical concentration bounds](https://mlsanity.com/paper/2610.20007/) (2026)

## Abstract (arXiv)

We consider nonparametric estimation of the transition operator P of a Markov chain and its transition density p where the singular values of P are assumed to decay exponentially fast. This is for instance the case for periodised, reversible multi-dimensional diffusion processes observed in low frequency. We investigate the performance of a spectral hard thresholded Galerkin-type estimator for P and p, discarding most of the estimated singular triplets. The construction is based on smooth basis functions such as wavelets or B-splines. We show its statistical optimality by establishing matching minimax upper and lower bounds in L2-loss. Particularly, the effect of the dimensionality d of the state space on the nonparametric rate improves from 2d to d compared to the case without singular value decay.