Hidden Markov models (HMMs) offer an important capability to interpret fitted latent states and transitions. When emission distributions nearly coincide, a small observation-density error does not ensure accurate estimation of the latent quantities needed for interpretation. This motivates us to characterize convergence rates for latent recovery that account for emission separation. Our main contribution lies in deriving deterministic inverse inequalities that convert Hellinger error between fitted and true densities into bounds on latent recovery errors. We develop these bounds for multivariate Gaussian HMMs with a common, known emission covariance matrix and combine them with a Hellinger estimation guarantee for the global maximum likelihood estimator to obtain latent recovery rates. Although the joint observation distribution is a Gaussian mixture with exponentially many path components, they share the same emission centers across time. Leveraging this shared structure, we show that the separation exponents are governed by the maximum number of simultaneously approaching centers rather than the number of path components. Under exact specification, the global bounds on state-sequence probability discrepancies and the 1-Wasserstein loss for latent location distributions exhibit sharp separation exponents that do not increase with the number of selected times. Moreover, sufficiently small observation-density error guarantees unique matching between fitted and true emission centers, which yields error bounds for individual emission centers and transition rows. Under over-specification, individual split-state transition rows are not necessarily identifiable, so we quantify aggregate latent recovery errors and show that state-sequence probability bounds incur larger separation exponents compared to exact specification. We conduct synthetic experiments to validate our theoretical results.
We study finite sample guarantees for transport based mutual information estimation. Mutual information can be estimated by first learning the conditional and marginal densities and then evaluating their log-density ratio by Monte Carlo. Given training samples, we use triangular transport maps to learn these densities and then compute the mutual information estimate on an independent batch of evaluation samples. For conditional density estimation using triangular transport maps, we establish non-asymptotic guarantees over realizable H\"older classes by viewing the conditional density as an ordinary density with respect to the product of the law of the conditioning variable and Lebesgue measure. This allows standard density estimation techniques to be applied directly to the conditional density estimation. These results yield finite sample guarantees for the resulting mutual information estimator and characterize how a fixed sample budget should be allocated between training and evaluation in order to minimize the mean squared error. Numerical experiments support the predicted training and evaluation error rates and illustrate the resulting sample allocation tradeoff.
Large generative models are commonly trained on data collected from the open web, which may contain data produced by previous generations of the same model. Previous work has shown that recursively training on model-generated data can progressively degrade model performance, a phenomenon known as model collapse. Yet, generative models continue to improve dramatically over time. Existing theory suggests that this is enabled by the availability of new real data at each iteration, kept separate from generated data. However, exact separation is not realistic, and uncurated data is still commonly used. We bridge this gap by considering a more realistic setup in which real and generated data are not handled differently during training. Specifically, we present a comprehensive theoretical study of iterative estimation of the mean and covariance of a multivariate Gaussian distribution. At each iteration, samples are drawn from a mixture of the real data distribution and the estimated distribution from the previous iteration, but are used indistinguishably. We provide a single equivalent condition for consistency of the mean and covariance estimators in Euclidean norms, and convergence to zero of the expected squared 2-Wasserstein distance between the estimated and true distributions. Using this result, we show that consistency holds, i.e., model collapse is avoided, for any fixed real data fraction above zero, and even for a fraction that vanishes across iterations, provided it does not vanish too quickly. However, more real data (or data curation) remains important at practical finite times: the convergence rate of the mean estimator deteriorates as the real data fraction approaches zero. We support our results with an empirical investigation of the theoretical setup, and with experiments on iterative training of a diffusion model on CIFAR-10 that correlate with our theory.
Conditional generative models parameterized by triangular transport maps are widely used for conditional sampling, yet theoretical guarantees on the quality of the conditional measures they produce remain scarce. We address this question for minimum divergence estimators of triangular maps. We use MMD-based objectives as a tractable setting in which the gap between joint and conditional errors can be analyzed precisely, and we measure performance through the average conditional estimation error, defined as the squared maximum mean discrepancy (MMD) between the estimated and the true conditionals, averaged over the conditioning variable. We first show that the common practice of minimizing a joint MMD may fail to recover the conditionals: writing the joint and the conditional criteria as quadratic energies of the same conditional mean embedding residual, we prove that, for parametric classes, joint minimizers are generically inconsistent in the misspecified setting. We then analyze an estimator that directly minimizes the conditional criterion through a surrogate loss computable from joint target samples and reference samples. We prove a non-asymptotic error bound that decomposes the average conditional estimation error into approximation, estimation, and reference sampling errors, with fast rates independent of the ambient dimension in the parametric setting. Finally, for joint MMDs built from concentrating Gaussian kernels on the conditioning variable, we show that the joint criterion approximates a density-reweighted version of the conditional one, and we derive an error bound for the joint minimizer whose rate suffers from a curse of dimensionality in the conditioning variable. Our results suggest that training criteria for conditional generative models should target the conditional rather than the joint error, a principle that extends to other divergences and to dynamic transport.
Transformers have become a central architecture for in-context learning (ICL), particularly through their state-of-the-art performance in large language models. This success motivates understanding how transformers exploit intrinsic geometry of real-world data. However, existing nonparametric ICL theory has largely focused on simplified settings, such as Euclidean domains or single manifold models, failing to fully explain its phenomenal empirical success in complex tasks. To address this gap, we study the prediction problem under unknown heterogeneous geometry, modeled by sample size-dependent mixtures of manifolds with heterogeneous dimensions, smoothness, and sampling masses. Under the growing geometric complexity model, we establish a minimax lower bound capturing the aggregate difficulty of the components and construct an oracle tangent local polynomial estimator with a matching upper bound. This estimator is connected to a structure-informed, two-stage softmax transformer with a geometric preconditioner and local polynomial solvers. The transformer achieves negligible approximation error relative to the minimax rate with logarithmic depth and polynomial size. Finally, we derive an in-context generalization bound for near empirical risk minimizers (ERM) over this class. Together, these results identify conditions under which the resulting predictor exploits local geometry and attains minimax rate.
Diffusion-guided methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data. Understanding their accuracy requires quantifying both (i) finite-scale approximation error relative to the true local intrinsic dimension (LID) and (ii) estimation error from finite samples. We address these two questions for the FLIPD field (Kamkari et al., 2024; arXiv:2406.03537), defined at the population level through the logarithmic scale derivative of a Gaussian-smoothed density and admitting a Fokker--Planck representation. For $\alpha$-smooth densities on a regular $d$-dimensional manifold, we establish a uniform $O(\sigma^2)$ approximation bound relative to the manifold dimension $d$. Under expected integrated squared loss, we prove a minimax lower bound of order $(n\sigma^d)^{-1}$ for estimating the FLIPD field from $n$ observations when $n^{-1/(2\alpha+d)}\le\sigma\le\sigma_0$. A plug-in FLIPD estimator based on the empirical denoising score matching (DSM) solution attains a matching upper bound. Thus the minimax rate is of order $(n\sigma^d)^{-1}$ and it becomes $n^{-2\alpha/(2\alpha+d)}$ at its smallest scale.
We study ordinary kernel ridge regression over full H\"older-Zygmund balls, measuring errors in Sobolev norms and in pointwise and uniform losses for mixed derivatives. For targets of smoothness $s$, every RKHS norm equivalent to $H^{s+d/2}$ gives the optimal $H^r$ risk for $0\le r<s$; integer choices of $r$ include integrated squared derivative losses. Pointwise and uniform guarantees require additional kernel structure. A localized wavelet kernel attains the sharp derivative rates with the same regularization orders as for function estimation, whereas equivalent changes of the RKHS norm can destroy the population bias rates. For the ordinary Gaussian kernel with a shrinking bandwidth, we prove a uniform derivative risk bound with the optimal polynomial rate and an explicit, unoptimized logarithmic factor. The results allow bounded measurable design densities bounded away from zero and conditionally sub-Gaussian noise.
We develop a quantitative statistical theory of transformers in the large-context regime by adopting the abstraction of contextual flow maps (CFMs): dynamical systems that evolve a distinguished token in the presence of a contextual measure across a stack of attention blocks. Within this framework, the finite-context model approximates an idealized infinite-context system in which the contextual measure is replaced by its underlying population, so that the context length $n$ becomes a statistical resource. Exploiting the McKean--Vlasov structure of the dynamics and the classical machinery of propagation of chaos, we establish a forward bound controlling the deviation between the finite- and infinite-context CFMs uniformly along depth, and a backward bound controlling the deviation between the corresponding training trajectories uniformly across iterations of online gradient descent. Both bounds achieve the optimal Wasserstein rate $n^{-1/d}$ for general CFMs and parametric rate $n^{-1/2}$ for a restricted class of CFMs that includes transformers as a special case. The analysis rests on a new Eulerian adjoint formulation of the loss gradient and stability estimates for the resulting forward--adjoint system, both of which may be of independent interest.
In recent years, Rectified flow (RF) has gained considerable popularity largely due to its generation efficiency and state-of-the-art performance. In this paper, we investigate how well RF automatically adapts to the intrinsic low dimensionality of the support of the target distribution to accelerate sampling. We show that, using a carefully designed \blue{end-point adapted} time-discretization scheme and with sufficiently accurate drift estimates, the RF sampler enjoys an iteration complexity of order $O(k/\varepsilon)$ (up to log factors), where $\varepsilon$ is the precision in total variation distance and $k$ is the intrinsic dimension of the target distribution. In addition, we show that the denoising diffusion probabilistic model (DDPM) procedure is equivalent to a stochastic version of RF by establishing a novel connection between these processes and stochastic localization. Building on this connection, we further design a stochastic RF sampler that also adapts to the low-dimensionality of the target distribution under mild requirements on the accuracy of the drift estimates, and also with a specific time schedule. We illustrate the efficacy of newly designed time-discretization schedules with simulations on the synthetic data and text-to-image (T2I) data experiments.
Inference-time alignment for diffusion models aims to adapt a pre-trained reference diffusion model toward a target distribution without retraining the reference score network, thereby preserving the generative capacity of the reference model while enforcing desired properties at inference time. A central mechanism for achieving such alignment is guidance, which modifies the sampling dynamics through an additional drift term. In this work, we introduce variationally stable Doob's matching, a novel framework for provable guidance estimation grounded in Doob's $h$-transform. Our approach formulates guidance as the gradient of the logarithm of an underlying Doob's $h$-function and employs gradient-regularized regression to simultaneously estimate both the $h$-function and its gradient, resulting in a consistent estimator of the guidance. Theoretically, we establish non-asymptotic convergence rates for the estimated guidance. Moreover, we analyze the resulting controllable diffusion processes and prove non-asymptotic convergence guarantees for the generated distributions in the Wasserstein-2 distance. Finally, we show that variationally stable guidance estimators are adaptive to unknown low dimensionality, effectively mitigating the curse of dimensionality under low-dimensional linear subspace assumptions. Numerical experiments demonstrate the effectiveness of our proposed method and show the advantages compared to the baselines.
We study the sequential root-finding problem with repeated noisy measurements. We propose sequential-test bisection STB, which uses a nonparametric sequential sign test to determine when to move to a new design point. Compared with variants of stochastic approximation, STB offers broader adaptivity and faster convergence. When the regression function satisfies $|f(x)|\geq \textsf{c}_0|x-\theta|^\gamma$ in a neighborhood of $\theta$ for constants $\textsf{c}_0,\gamma>0$ where $\theta$ is the root, i.e. $f(\theta) = 0$, we establish non-asymptotic guarantees showing that STB, without using the knowledge of $\gamma$, achieves the minimax adaptive optimal convergence rate $\widetilde{O}(n^{-1/(2\gamma)})$, outperforming Robbins--Monro and related stochastic-approximation methods in convergence rate. Under sub-Gaussian noise, STB provides finite-sample confidence sequences for the root $\theta$ without estimating a derivative. In the positive-slope model, we augment STB with {randomized weighted-section and a line fitted to all observations}, leading to asymptotic normality and consistent variance estimation. We formalize the efficiency by a convolution theorem of H\'ajek-Le Cam type. Furthermore, when the regression function is discontinuous at the root, we show that Robbins--Monro converges at rate $1/n$, whereas STB converges super-polynomially. We extend this construction to multivariate nonparametric level-set estimation by sampling along rays. As a technical byproduct, we establish a new central limit theorem under random stopping, with proof techniques that may be of independent interest.
Variational inference, as an alternative to Markov chain Monte Carlo sampling, has played a transformative role in enabling scalable computation for complex Bayesian models. Nevertheless, existing approaches often depend on either rigid model-specific formulations or stochastic black-box optimization routines. Tangent approximation is a principled class of structured variational methods that exploit the geometry of the underlying probability model. However, its utility has largely been confined to logistic regression and related modeling regimes. In this article, we propose a novel variational framework based on tangent transformation for a broad class of probability models characterized by strongly super-Gaussian likelihoods. Our method leverages convex duality to construct tangent minorants of the log-likelihood, thereby inducing conjugacy with Gaussian priors over model parameters in an otherwise intractable setup. Under mild assumptions on the data-generating mechanism, we establish algorithmic convergence guarantees, a contribution that stands in contrast to the limited theoretical assurances typically available for black-box variational methods. Additionally, we derive near-parametric variational risk bounds. Superior performance of our proposed methodology is illustrated on simulated and real-data scenarios that challenge state-of-the-art variational algorithms in terms of scalability and their ability to consistently capture complex underlying data structure.
Objective functions based on Hellinger distance yield robust and efficient estimators of model parameters. Motivated by privacy and regulatory requirements encountered in contemporary applications, we derive in this paper \emph{private minimum Hellinger distance estimators}. The estimators satisfy a new privacy constraint, namely, Hellinger differential privacy, while retaining the robustness and efficiency properties. We demonstrate that Hellinger differential privacy shares several features of standard differential privacy while allowing for sharper inference. Additionally, for computational purposes, we also develop Hellinger differentially private gradient descent and Newton-Raphson algorithms. We illustrate the behavior of our estimators in finite samples using numerical experiments and verify that they retain robustness properties under gross-error contamination.
We study the subspace-constrained Tyler's estimator (STE) for recovering a low-dimensional linear subspace from data contaminated by outliers. Under deterministic conditions on inlier spread and outlier alignment, we prove linear convergence to the true subspace from a sufficiently accurate initialization. The guarantee also applies when the dimension-scaled signal-to-noise ratio is below one, where general robust subspace recovery is SSE-hard. When initializing STE from a candidate subspace, we provide a sufficient recovery condition based on the initial angular error and the size of the identity shift applied to the subspace projector. For noisy inliers, we prove a subspace error proportional to the noise level within an explicit iteration bound. Finally, under an asymptotic, anisotropic Gaussian inlier--outlier model, we show that Tyler's M-estimator (TME) provides a successful STE initialization even when TME's own leading eigenspace differs from the target subspace.
We consider nonparametric maximum likelihood estimation of the intensity function of a Poisson process, allowing the intensity to depend on observed covariates. Such models describe how event rates vary with external conditions, including conditions under which the intensity becomes arbitrarily small or vanishes. However, risk analysis based on the Kullback-Leibler (KL) divergence typically requires the true intensity to be bounded away from zero, a restrictive assumption in applications. We develop a unified approach that avoids this assumption and derive an oracle inequality for the covariate-averaged Hellinger distance between the true intensity and the nonparametric maximum likelihood estimator (NPMLE). As an application of the oracle inequality, we establish convergence rates for NPMLEs based on deep neural networks (DNNs). Our results show that, as in nonparametric regression, the ability of DNNs to capture underlying structure leads to improved convergence rates. Specifically, DNN-based NPMLEs mitigate the curse of dimensionality when the true intensity has a compositional structure and automatically adapt to a low-dimensional Riemannian manifold structure of the event space. For intensity functions with a compositional structure, DNN-based NPMLEs achieve nearly minimax-optimal convergence rates without explicitly incorporating this structure into the network architecture. Simulations demonstrate that DNN-based estimators outperform kernel estimators in most settings, while manifold kernels remain competitive in some designs. Finally, we analyze stimulus-evoked neural spiking using data from the Allen Brain Observatory Visual Coding Neuropixels project. DNN-based NPMLEs provide better out-of-sample performance than a conditional kernel estimator, as evaluated by held-out log-likelihood.
Instance-dependent limits for bandit regret, sequential tests, best-arm identification and Markov decision processes are proved by one chain. It runs from a divergence decomposition through data processing to a statistic, a testing step that turns a class guarantee into a requirement on the history, and optimization over alternatives and allocations. Only the first step is an identity. This work writes the chain as an exact ledger. By the chain rule, data processing loses exactly a conditional divergence, which vanishes for arms in natural exponential families at the per-arm counts and reward sums. A one-parameter family of martingale identities extends the decomposition to every tilt between two instances. The classical limits follow by relaxations with named defects. All defects but the asymptotic step's can be computed from an algorithm's own runs, the uniform quantification's only when the class guarantee is a number. On seven published sources, from movie ratings to engine replacement, the ledger locates the difference between standard algorithms and the classical limits. For identification among trained models and for Wald's test the largest term is usually the uniform quantification's.
Learning a shared subspace in multitask learning is an important statistical problem, yet exact rates remain poorly understood in many of the formal settings considered in the literature. To address this gap, which arises from limitations of existing upper-bound techniques, we propose a general multitask lemma. The lemma provides sufficient conditions for fast subspace estimation in generic multitask settings, in the form of geometric conditions on the task regression functions and their estimators. We subsequently show that the multitask lemma yields sharp minimax rates---previously unknown---for linear regression and classification in the regimes considered, and rates matching up to logarithmic factors for additive-noise regression.
We extend the recently introduced e-closure principle for multiple testing to general state and action spaces and to arbitrary, including continuously indexed, families of risks. A collection of e-values indexed by states and risks certifies a menu of actions, and every data-dependent selection from this menu controls each expected loss at its prescribed level. Conversely, every valid procedure is certified by its own normalized losses, which often leave part of the risk budget unused. We introduce a method called recertification, which spends this unused budget. Using known distributional structure, it enlarges the certifying e-values while preserving validity and selects a less conservative action from the enlarged menu. One particularly compelling application is to the popular framework of conformal risk control (CRC) with a known loss family. Here, recertification redistributes unused risk budget among the held-out points of each hypothetical augmented sample. The resulting controller is valid under every finite exchangeable law and less (usually strictly) conservative than CRC. For convex losses on an interval of actions, repeated recertification converges geometrically to an admissible controller. As a second example involving false discovery rate control with e-values, recertifying randomized e-BH yields a closed-form procedure that rejects a superset of randomized e-BH for every realization of the randomization and maximizes the expected number of rejections under the same constraints. Recertifying with the larger e-collection of closed e-BH gives a second closed-form procedure that rejects a superset of the first. Experiments on CIFAR-100 hierarchical classification and simulations demonstrate improved budget utilization and gains in informativeness and power over the corresponding baselines.
Accurate and efficient representation of non-Gaussian state distributions is critical for estimation, control, and decision-making, especially when the system is highly nonlinear. This paper develops a mathematical framework for estimating non-Gaussian uncertainty via semi-nonparametric (SNP), or Gallant-Nychka, densities. SNP densities use squared probabilists' Hermite polynomials to guarantee nonnegative probability densities while representing non-Gaussian behavior. For uncertainty propagation, once fitted to propagated state samples, SNP densities enable direct analytical density and probability evaluations without needing to tune additional hyperparameters besides the polynomial order. To enable their application to nonlinear dynamical systems, we derive analytic marginal and cumulative distribution functions and develop a lossless convex formulation of the coefficient estimation problem along a given sign branch. The approach is demonstrated on the chaotic Lorenz system and compared against Gaussian mixture models and Edgeworth approximations. Results demonstrate that SNP densities can accurately represent non-Gaussian state distributions while enabling direct probability and quantile evaluation, making them a promising tool for density estimation and uncertainty quantification in nonlinear estimation and control.
We show that outlier eigenvectors of the Bethe Hessian achieve weak recovery in the stochastic block model down to the Kesten-Stigum threshold, for every fixed common expected degree $d>1$. This extends previous recovery guarantees, which required $d$ to be sufficiently large. This threshold is information-theoretically optimal for the symmetric two-community model and is conjectured to mark the limit of efficient recovery in broader symmetric models. Our result matches the recovery guarantees of non-backtracking spectral methods while using Hermitian $n\times n$ matrices. To do this, we identify recursive formulas for the negative outlier eigenvalues and show that their eigenspaces have nonvanishing overlap with the corresponding planted eigenspaces. The proof constructs approximate eigenvectors from the planted labels on the limiting Poisson tree, with weights given by its Green's function, and transfers these approximations to the finite graph.