The Geometry of Efficient Nonconvex Sampling

Abstract: We present an efficient algorithm for uniformly sampling from an arbitrary compact body $\mathcal{X} \subset \mathbb{R}^n$ from a warm start under isoperimetry and a natural volume growth condition. Our result provides a substantial common generalization of known results for convex bodies and star-shaped bodies. The complexity of the algorithm is polynomial in the dimension, the Poincaré constant of the uniform distribution on $\mathcal{X}$ and the volume growth constant of the set $\mathcal{X}$.

• The paper demonstrates that efficient sampling from nonconvex bodies is achievable by enforcing a Poincaré inequality and controlled volume growth.
• It introduces the In-and-Out sampler, a reversible alternating Gibbs sampler that manages rejection bias in complex nonconvex domains.
• The work establishes polynomial-time mixing guarantees, broadening efficient sampling beyond convex and star-shaped bodies.

The Geometry of Efficient Nonconvex Sampling: An Essay

Overview and Motivation

Sampling uniformly from high-dimensional bodies is central in computational geometry, convex analysis, and high-dimensional probability. Prior algorithmic theory is essentially restricted to convex bodies and star-shaped sets, which places severe constraints on the geometric complexity of the domains amenable to efficient sampling. "The Geometry of Efficient Nonconvex Sampling" (2603.25622) systematically expands this frontier by analyzing the sampling problem for general nonconvex compact bodies, solely under two assumptions: a Poincaré isoperimetric inequality for the uniform distribution on the set, and a controlled volume growth condition. The work thereby abstracts away from specific geometric structure, instead focusing on functional-analytic properties sufficient for efficient mixing and implementation of Markov Chain Monte Carlo (MCMC) approaches.

Generalizing Sampling Theory Beyond Convexity

In the classical setting, efficient algorithms—such as Hit-and-Run or Ball Walk—operate assuming either convexity or logconcavity, which respectively guarantee fast geometric mixing and measure concentration. While nonconvex sampling is NP-hard in the worst case, this hardness is pathological: many natural bodies are nonconvex but not adversarial, and often possess favorable isoperimetric behavior and manageable boundary growth. Prior to this work, the only family of nonconvex sets with tractable polynomial-time sampling algorithms were star-shaped bodies, where the mixing time depends on the volume fraction of a convex "core." However, many reasonable sets, such as unions of convex bodies or bodies with holes, are not star-shaped yet appear amenable to polynomial time sampling. Figure 1 in the paper illustrates this gap between what is currently covered and geometries that have resisted prior analysis.

Figure 1: Examples of bodies $X \subset \mathbb{R}^n$—(a) convex and (b) star-shaped, for which efficient algorithms are known, and (c,d) nonconvex non-star-shaped sets that fall outside existing theory.

The present work definitively closes this gap by isolating two analytic properties: the existence of a Poincaré (or Cheeger) inequality for the uniform measure, and a mild volume growth control, that together suffice for polynomial-time sampling without regard to convexity or even star-shapedness.

Algorithmic Framework: In-and-Out Sampler

The core algorithm is the "In-and-Out" sampler, first proposed for convex bodies by Kook, Vempala, and Zhang, and here extended well beyond that regime. It is a reversible alternating Gibbs sampler, whose steps are:

1. Forward step: Sample $y_{i} \sim N(x_{i}, h I_n)$.
2. Backward step: Repeat until acceptance or $N$ attempts: sample $x_{i+1} \sim N(y_i, h I_n)$, and accept if $x_{i+1} \in X$.

Notably, the backward step is implemented with bounded rejection sampling (truncated after $N$ trials per iteration) to keep expected cost and bias manageable even in the presence of complex boundaries.

Convergence and Output Guarantees

The distribution after $T$ successful iterations approximates the uniform measure on $X$ in Rényi divergence (with index $q \geq 2$), which via information-theoretic inequalities controls total variation and KL-divergence—thus ensuring strong output quality. The step size, number of iterations, and rejection threshold $N$ are all parametrized in terms of the Poincaré constant, the volume growth parameters, the dimension $n$, and the warmness of the initial point.

Geometric and Analytic Conditions

Poincaré Isoperimetry

The Poincaré inequality is fundamental to the analysis of diffusion processes and mixing times. For the uniform distribution on $X$, it stipulates that $$\operatorname{Var}_\pi(\phi) \leq C \cdot \mathbb{E}_\pi[\|\nabla \phi\|^2]$$ for all sufficiently regular $\phi$. This property excludes bottleneck geometries (e.g., dumbbells), which defeat mixing, but is stable under moderate geometric perturbations and holds for all convex and star-shaped bodies (with explicit constants).

Figure 2: A comparison of (a) a dumbbell-shaped body with poor isoperimetry (large Poincaré constant) and (b) a cylinder with good isoperimetry but potentially large volume growth.

Volume Growth Condition

A critical insight is that isoperimetry alone is insufficient to guarantee tractable sampling for all $X$: certain sets with thin tendrils or narrow tunnels may satisfy Poincaré but frustrate efficient random walk implementations due to local escape probabilities. The required regularity is formalized as an $(\alpha, \beta)$-volume growth condition: $$\operatorname{Vol}(X \oplus B_t) \leq \alpha (1 + t \beta)^n \operatorname{Vol}(X)$$ for all $t > 0$. This controls the relative growth of the Minkowski thickening and excludes sets with vanishingly thin fingers or spines. The authors demonstrate that this property is preserved under set unions and exclusions, so their results apply to very broad nonconvex constructions.

Main Results and Technical Contributions

The main theorem establishes that, starting from an $M$-warm initial point, "In-and-Out" produces an $\varepsilon$-approximate uniform sample using a number of membership oracle calls and arithmetic operations bounded in expectation by

$$\tilde{O}\left(q\, \alpha\, \beta^2\, M\, n^3\, \log^4 (1/\varepsilon) \right)$$

where $\tilde{O}$ hides logarithmic dependencies on the parameters. The dependence on $n$ arises due to the high-dimensional geometry of nonconvex boundaries. For convex bodies, this result recovers prior polynomial mixing bounds, albeit with higher complexity ($n^3$ vs. $n^2$ for the isotropic convex case). For star-shaped bodies, the algorithm not only improves the dependence on $\varepsilon$ from polynomial to logarithmic, but also lifts the error metric to stronger divergences.

The analysis proceeds in two principal steps:

• Outer mixing analysis: Extending convergence proofs under Poincaré for the Proximal Sampler to the truncated, practical In-and-Out sampler, controlling bias and ensuring output quality.
• Escape probability and per-iteration analysis: The technical crux is bounding the stationary probability that a sampler step exits $X$ or requires excessive rejection—crucially, volume growth replaces the convexity-based halfspace separation used in prior proofs, see Figure 3.

Figure 3: The difference of containment guarantees in the (a) convex case, where halfspace separation suffices, and (b) nonconvex case, relying on ball separation—a key analytical innovation for general domains.

Implications, Practical and Theoretical

This work provides a functional-analytic foundation for sampling algorithms robust to set-theoretic and geometric complexity; in effect, it shows that convexity is not the essential barrier—poor isoperimetry and uncontrolled boundary growth are. The class of efficiently sampleable bodies now includes an expansive family constructed via unions, set exclusions, and other natural operations, provided one can verify or bound the relevant analytic constants.

From a practical standpoint, the result yields a blueprint for implementing efficient uniform samplers with only a membership oracle, even in complex nonconvex settings arising in statistical physics, Bayesian inference, and computational geometry.

On the theoretical side, this work analogizes modern results in nonconvex optimization, where certain regularity (PL inequalities) can substitute for convexity, but notably, sampling is tractable for a much broader class than global optimization. The theoretical characterization of sampling through Poincaré and volume growth parameters opens directions for further work on comparing MCMC methods, warm-start generation, and extensions to non-uniform log-concave or multimodal distributions.

Conclusion

"The Geometry of Efficient Nonconvex Sampling" (2603.25622) delivers a unifying theoretical framework for efficient uniform sampling from a vast class of high-dimensional nonconvex sets, under analytic rather than geometric restrictions. By focusing on isoperimetry and Minkowski volume growth, the work generalizes and strictly subsumes prior results on convex and star-shaped bodies. The techniques are likely to inspire further development of Markov chain methods robust to complex domains and new analysis of MCMC convergence beyond the confines of classical convexity.