From Continuous to Discrete: a No-U-Turn Sampler for Permutations

TL;DR

This work introduces a discrete No-Underrun Sampler (NURS) as a permutation-space analogue of No-U-Turn sampling, enabling local adaptivity on the symmetric group $S_n$ for Mallows distributions with $P_\beta(\sigma) \propto e^{-\beta d(\sigma,\sigma_0)}$. By replacing Hamiltonian trajectories with measure-preserving group orbits and employing a randomized dyadic doubling and no-underrun stopping rule, NURS achieves a rejection-free, reversible kernel via a Gibbs Self-Tuning (GIST) framework. The authors prove reversibility, develop a shift-coupling argument, and establish a path-coupling based mixing-time bound of $O(n^2\log n)$ in a high-temperature, Lipschitz-energy regime, with explicit edge-wise contraction factors. Numerical experiments across Kendall, $L^1$, $L^2$, Hamming, Cayley, and Ulam distances corroborate the theory, demonstrating effective exploration in warm regimes and revealing the impact of direction laws (uniform, block-shuffle, local) on mixing. The framework paves the way for orbit-based reversible sampling beyond permutations, linking discrete analogues of Hamiltonian ideas to structured discrete spaces.

Abstract

We introduce a discrete-space analogue of the No-U-Turn sampler on the symmetric group $S_n$, yielding a locally adaptive and reversible Markov chain Monte Carlo method for $\mathrm{Mallows}(d,σ_0)$. Here $d:S_n\times S_n\to[0,\infty)$ is any fixed distance on $S_n$, $σ_0\in S_n$ is a fixed reference permutation, and the target distribution on $S_n$ has mass function $π(σ)\propto e^{-βd(σ,σ_0)}$ where $β>0$ is the inverse temperature. The construction replaces Hamiltonian trajectories with measure-preserving group-orbit exploration. A randomized dyadic expansion is used to explore a one-dimensional orbit until a probabilistic \emph{no-underrun} criterion is met, after which the next state is sampled from the explored orbit with probability proportional to the target weights. On the theory side, embedding this transition within the Gibbs self-tuning (GIST) framework provides a concise proof of reversibility. Moreover, we construct a \emph{shift coupling} for orbit segments and prove an explicit edge-wise contraction in the Cayley distance under a mild Lipschitz condition on the energy $E(σ)=d(σ,σ_0)$. A path-coupling argument then yields an $O(n^2\log n)$ total-variation mixing-time bound.

Figures (14)

• Figure 1: One realization of orbit construction in NURS by random doubling. Starting from $\sigma$, the orbit expands forward, then backward, then forward again. The construction halts once either a Stop or SubStop condition is met, or when the maximum allowed trajectory length $2^M$ is reached, yielding the final orbit $\mathcal{O}$.
• Figure 2: Binary-tree representation of the dyadic orbit construction. Starting from $\sigma$, each bit $B_j\in\{0,1\}$ specifies a forward ($1$) or backward ($0$) doubling at level $j$. The concatenation $B=(B_1,\dots,B_m)$ determines the rightmost index $b=\sum_{j=1}^m B_j\,2^{\,j-1}$ and hence the orbit interval $[a,b]$ with $a=b-2^m+1$. The highlighted path illustrates the case $B=(1,0,1)$, yielding rightmost index $b=5$ and orbit interval $[a,b]=[-2,5]$.
• Figure 3: Aligned case $(I,J)=(i,j)$ with $\mathop{\mathrm{ord}}\nolimits(\eta)=2m=6$, so $m=3$. The two rows show the orbits $\{\sigma \eta^{t}\}_{t=0}^{5}$ and $\{\sigma' \eta^{t}\}_{t=0}^{5}$. Since $\eta^{3} = \tau_{ij}$, we have $\sigma' = \sigma \tau_{ij} = \sigma \eta^{3}$, and thus for all $t$$\sigma' \eta^{t} = \sigma \eta^{t+3}$. In particular, the lower row is just a cyclic shift of the upper row by three positions, so the two orbits induce the same categorical law on indices.
• Figure 4: If the orbit lengths are of fixed length $2^M$, and the indices are sampled uniformly along the constructed orbit (equivalently, all orbit weights are equal), then the signed index $k$ has the symmetric triangular pmf $p(k)=\frac{2^M+1-|k|}{(2^M+1)^2}$ for $|k|\le 2^M$. In our experiments $M=7$, hence $2^M=128$.
• Figure 5: Signed index $k$ for NURS with Uniform directions with scaling of $\beta=1/E_{\max}$.

Theorems & Definitions (44)

• Definition 2.1: Direction law
• Definition 2.2: Orbit
• Definition 2.3: Orbit Concatenation
• Definition 2.4: Stop (no--underrun condition)
• Definition 2.5: Dyadic sub-orbits
• Definition 2.6: SubStop condition
• Example 2.7: Dyadic decomposition of an orbit
• Theorem 3.1: Reversibility of GIST transitions BouRabeeCarpenterMarsden2024
• proof : Sketch of proof
• Theorem 3.2