arXiv subjects

Zhihan Huang

Publications and source records attributed to Zhihan Huang.

8 recordsLinked to original sources

arXiv2026

Saddlepoint approximations for plug-in resampling

Resampling-based procedures can improve on normal approximations in sparse, large-scale testing problems, but their computational cost can be prohibitive. We recognize that several existing procedures belong to a faster plug-in resampling subclass, fixing fitted nuisance parameters during resampling. When the resampled statistic is a sum of conditionally independent terms, the saddlepoint approximation (SPA) for the resampling $p$-value offers further acceleration, replacing resampling with an analytical tail approximation. However, standard Edgeworth-based approximation-error bounds impose regularity conditions that are hard to verify for plug-in resampling laws. We use an alternative approach to establish a finite-sample relative-error bound for the Lugannani-Rice approximation under more tractable conditions, which we apply in two contexts. In statistical genetics, we identify response resampling procedures as the targets of existing SPAs and establish guarantees in a representative setting. In conditional independence testing, we introduce spaCRT, an SPA for the distilled conditional randomization test (dCRT), which has been applied successfully in biology. Our rates quantify the effects of sparsity and signal strength, with matching lower bounds in special cases. We additionally establish asymptotic Type-I error control of the corresponding plug-in resampling procedures under growing sparsity. In simulations and single-cell CRISPR data analysis, spaCRT closely approximates dCRT $p$-values and preserves its statistical performance while accelerating computation by up to 250-fold.

stat.ME↗

arXiv2026

Provably adaptive sampling with uniform and remasking discrete diffusion models

Discrete diffusion models offer a promising alternative to autoregressive generation by enabling parallel updates, but their sampling efficiency can depend strongly on the choice of the forward process and the sampler. For the uniform forward process, existing lower bounds for the standard $τ$-leaping sampler scale linearly with the ambient dimension $d$, raising the question of whether this dependence is intrinsic to the forward process. We answer this question in the negative. We consider a first-order sampler based on the leave-one-out denoiser for uniform and remasking processes whose coordinate updates can be performed in parallel. In both cases, the sampler can correct denoising mistakes during the sampling process, which becomes necessary when many coordinates are updated together. Our main result establishes an adaptive sampling guarantee: up to logarithmic factors, $N = O(\mathrm{DTC}(X_0) / \varepsilon)$ discretization steps suffice to achieve sampling error $O(\varepsilon_{\mathrm{score}}+\varepsilon)$, where $\varepsilon_{\mathrm{score}}$ is the error in score estimation. Thus, the sampling complexity is governed by the intrinsic dependence structure of the target distribution, as measured by its dual total correlation $\mathrm{DTC}(X_0)$, rather than directly by the ambient dimension $d$. Our analysis proceeds through a Bayes-optimal auxiliary sampler that separates discretization error from score-estimation error. We also derive an exact information-theoretic representation of the discretization error in terms of the mutual information between different coordinates of the forward process at different times. This representation applies to general forward processes and, in the uniform and remasking cases, can be controlled by $\mathrm{DTC}(X_0)$. Numerical experiments on structured synthetic distributions illustrate the predicted dimension-adaptive behavior.

cs.LG↗

arXiv2026

Low-dimensional adaptation of diffusion models: Convergence in total variation

This paper investigates how diffusion generative models leverage (unknown) low-dimensional structure to accelerate sampling. Focusing on two mainstream samplers -- the denoising diffusion implicit model (DDIM) and the denoising diffusion probabilistic model (DDPM), we prove that their iteration complexities under exact score functions are at most the order of $k/\varepsilon$ (up to log factor), where $\varepsilon$ is the precision in total variation distance and $k$ is some intrinsic dimension of the target distribution. We further extend these convergence guarantees to the setting in which the score functions are learned from data rather than known exactly, showing that the convergence performance degrades gracefully under suitable score estimation assumptions. We then show that these assumptions are attainable via kernel-based score estimators with finite-sample guarantees that also adapt to the low-dimensional structure. Our results apply to a broad family of target distributions without requiring smoothness or log-concavity. Our findings provide the first rigorous evidence for the adaptivity of the DDIM-type samplers to unknown low-dimensional structure, and improve over the state-of-the-art DDPM theory regarding total variation convergence.

stat.ML↗

arXiv2026

Efficient Sampling with Discrete Diffusion Models: Sharp and Adaptive Guarantees

Diffusion models over discrete spaces have recently shown striking empirical success, yet their theoretical foundations remain incomplete. In this paper, we study the sampling efficiency of score-based discrete diffusion models under a continuous-time Markov chain (CTMC) formulation, with a focus on $τ$-leaping-based samplers. We establish sharp convergence guarantees for attaining $\varepsilon$ accuracy in Kullback-Leibler (KL) divergence for both uniform and masking noising processes. For uniform discrete diffusion, we show that the $τ$-leaping algorithm achieves an iteration complexity of order $\tilde O(d/\varepsilon)$, with $d$ the ambient dimension of the target distribution, eliminating linear dependence on the vocabulary size $S$ and improving existing bounds by a factor of $d$; moreover, we establish a matching algorithmic lower bound showing that linear dependence on the ambient dimension is unavoidable in general. For masking discrete diffusion, we introduce a modified $τ$-leaping sampler whose convergence rate is governed by an intrinsic information-theoretic quantity, termed the effective total correlation, which is bounded by $d \log S$ but can be sublinear or even constant for structured data. As a consequence, the sampler provably adapts to low-dimensional structure without prior knowledge or algorithmic modification, yielding sublinear convergence rates for various practical examples (such as hidden Markov models, image data, and random graphs). Our analysis requires no boundedness or smoothness assumptions on the score estimator beyond control of the score entropy loss.

cs.LG↗

arXiv2026

Semiparametric KSD test: unifying score and distance-based approaches for goodness-of-fit testing

Goodness-of-fit (GoF) tests are fundamental for assessing model adequacy. Score-based tests are appealing because they require fitting the model only once under the null. However, extending them to powerful nonparametric alternatives is difficult due to the lack of suitable score functions. Through a class of exponentially tilted models, we show that the resulting score-based GoF tests are equivalent to the tests based on integral probability metrics (IPMs) indexed by a function class. When the class is rich, the test is universally consistent. This simple yet insightful perspective enables reinterpretation of classical distance-based testing procedures-including those based on Kolmogorov-Smirnov distance, Wasserstein-1 distance, and maximum mean discrepancy-as arising from score-based constructions. Building on this insight, we propose a new nonparametric score-based GoF test through a special class of IPM induced by kernelized Stein's function class, called semiparametric kernelized Stein discrepancy (SKSD) test. Compared with other nonparametric score-based tests, the SKSD test is computationally efficient and accommodates general nuisance-parameter estimators, supported by a generic parametric bootstrap procedure. The SKSD test is universally consistent and attains Pitman efficiency. Moreover, SKSD test provides simple GoF tests for models with intractable likelihoods but tractable scores with the help of Stein's identity and we use two popular models, kernel exponential family and conditional Gaussian models, to illustrate the power of our method. Our method achieves power comparable to task-specific normality tests such as Anderson-Darling and Lilliefors, despite being designed for general nonparametric alternatives.

stat.ML↗

arXiv2026

Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality

The denoising diffusion probabilistic model (DDPM) has emerged as a mainstream generative model in generative AI. While sharp convergence guarantees have been established for the DDPM, the iteration complexity is, in general, proportional to the ambient data dimension, resulting in overly conservative theory that fails to explain its practical efficiency. This has motivated the recent work Li and Yan (2024a) to investigate how the DDPM can achieve sampling speed-ups through automatic exploitation of intrinsic low dimensionality of data. We strengthen this line of work by demonstrating, in some sense, optimal adaptivity to unknown low dimensionality. For a broad class of data distributions with intrinsic dimension $k$, we prove that the iteration complexity of the DDPM scales nearly linearly with $k$, which is optimal when using KL divergence to measure distributional discrepancy. Notably, our work is closely aligned with the independent concurrent work Potaptchik et al. (2024) -- posted two weeks prior to ours -- in establishing nearly linear-$k$ convergence guarantees for the DDPM.

cs.LG↗

arXiv2024

Towards a mathematical theory for consistency training in diffusion models

Consistency models, which were proposed to mitigate the high computational overhead during the sampling phase of diffusion models, facilitate single-step sampling while attaining state-of-the-art empirical performance. When integrated into the training phase, consistency models attempt to train a sequence of consistency functions capable of mapping any point at any time step of the diffusion process to its starting point. Despite the empirical success, a comprehensive theoretical understanding of consistency training remains elusive. This paper takes a first step towards establishing theoretical underpinnings for consistency models. We demonstrate that, in order to generate samples within $\varepsilon$ proximity to the target in distribution (measured by some Wasserstein metric), it suffices for the number of steps in consistency learning to exceed the order of $d^{5/2}/\varepsilon$, with $d$ the data dimension. Our theory offers rigorous insights into the validity and efficacy of consistency models, illuminating their utility in downstream inference tasks.

stat.ML↗

arXiv2022

Azadkia-Chatterjee's correlation coefficient adapts to manifold data

In their seminal work, Azadkia and Chatterjee (2021) initiated graph-based methods for measuring variable dependence strength. By appealing to nearest neighbor graphs, they gave an elegant solution to a problem of Rényi (Rényi, 1959). Their idea was later developed in Deb et al. (2020) and the authors there proved that, quite interestingly, Azadkia and Chatterjee's correlation coefficient can automatically adapt to the manifold structure of the data. This paper furthers their study in terms of calculating the statistic's limiting variance under independence and showing that it only depends on the manifold dimension.

math.ST↗