arXiv subjects

Alessandro Rinaldo

Publications and source records attributed to Alessandro Rinaldo.

At least 19 recordsLinked to original sources

arXiv2026

Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective

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.

stat.ML↗

arXiv2026

Setting Things Straight: On the Straightness and Non-Straightness of Two-Step Rectified Flow

Rectified flow (RF) is an ODE-based generative modeling framework that has achieved remarkable success in practice over the past few years. By design, RF seeks to produce flows that have a high degree of straightness, in the sense that the resulting ODE trajectories can be obtained with relatively few (ideally, one) sampling steps. However, a rigorous analysis of straightness or near-straightness of RF flows remains missing in the literature. In this paper, we study settings in which the coupling generated by two-step RF is provably straight at the population level. We provide a complete characterization in two popular classes of data distributions: (i) smooth, strongly log-concave densities and (ii) Gaussian mixture models. Our main results yield sharp conditions for the coupling generated by two-step RF to be straight. Non-straight couplings are constructed when these conditions fail, which demonstrate the sharpness of these conditions. Using the theory developed on the straightness of two-step RF, we propose a notion of approximate straightness that measures the straightness of the learned couplings and can be estimated from the data. We introduce a procedure for constructing asymptotic confidence intervals for our proposed notion of approximate straightness and propose a data-driven method for selecting the number of reflow rounds in multiple iterations of RF.

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arXiv2026

When Does Sparse Routing Help? The Geometry of Expert Separation in Mixture of Experts

Sparse mixtures of experts combine multiple prediction models while evaluating only a small subset for each input. Although this architecture reduces computation, how expert separation and routing sparsity jointly govern expert recovery remains an open problem. In this paper, we address this question for location Gaussian mixtures of experts with a fixed number of experts by establishing how expert separation and spatial arrangement determine recovery rates. For dense models, our results reveal that when experts form well-separated clusters, the separation geometry is governed by the size of the largest cluster; without a prescribed cluster structure, this geometry is instead driven by the total number of experts. For sparse models, the separation geometry on retained routing regions is determined by the number of active experts, while the contribution of omitted regions is controlled by their total probability. Consequently, activating fewer experts reduces the separation complexity in these bounds. Furthermore, Top-1 routing yields a particularly striking result: experts can be recovered at a rate independent of the separation quantity. Our technical approach combines polynomial interpolation and Gaussian reproducing-kernel Hilbert space (RKHS) test functions with quantitative restriction inequalities that extend pointwise comparisons to expert functions.

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arXiv2026

The Statistical Geometry of Latent Recovery in Hidden Markov Models

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.

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arXiv2026

On Statistical-Computational Gaps in Graph Combinatorial Testing Problems

We study high-dimensional graph combinatorial hypothesis testing problems in which a signal is supported on the edge set of an unknown subgraph drawn from a class invariant under vertex relabeling. For Gaussian observations, we characterize sharp statistical and computational thresholds (at polynomial order) for the signal magnitude required for strong detection across various combinatorial graph alternatives. Our analysis is based on bounding the second moments of the full and low-degree likelihood ratios in terms of the overlap of two independent latent structures and comparing the resulting lower bounds with scan, average, and coordinatewise maximum tests. Under an appropriate low-degree conjecture for computational hardness, we obtain planted-clique thresholds matching those known for Gaussian submatrix detection and identify new statistical-computational gaps for multiple planted cliques and hypercliques, thick cycles and thick paths. We also establish a novel transfer principle across natural exponential family observation models: matching the one-coordinate chi-square divergence preserves full likelihood-ratio second moments exactly and makes low-degree second moments equivalent up to constant factors under a degree condition. Under local regularity, we obtain corresponding testing upper bounds in terms of an effective signal strength. When these upper and lower bounds match, they yield universal threshold scales in the local alternative regime. As a consequence, statistical-computational gaps for planted-clique detection in Erdős--Rényi settings can be deduced from the Gaussian model.

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arXiv2026

Weak Limits of Parameter Estimation in Strongly Identifiable Mixture Models

We study the asymptotic behavior of global maximum likelihood estimation in finite mixtures with an unknown number of components bounded by $k$. Although convergence rates are well understood under strong identifiability, they do not determine the limiting distribution of mixing-measure loss. Under second-order strong identifiability, we establish two geometric results that connect density asymptotics to uncertainty about the mixing measure. First, we characterize the exact attainable density tangent set, accounting for the component budget shared by nonlocal vanishing components and local splitting directions. Second, we derive a sequential first-order expansion of squared Wasserstein loss, showing that its leading transportation cost is uniquely determined by the relative-density direction. Together with a Gaussian likelihood argmax theorem, these results yield joint weak convergence of the relative-density maximum likelihood estimator and $\sqrt n W_2^2(\widehat G_n,G_*)$. The density limit maximizes a Gaussian likelihood process over the attainable tangent set, and the induced limit of $n^{1/4}W_2(\widehat G_n,G_*)$ is generally non-Gaussian. We establish consistency of an $m$-out-of-$n$ parametric bootstrap when $m\to\infty$ and $m/n\to0$. Numerical studies assess the approximation and distinguish bootstrap and optimization errors; a flow-cytometry application illustrates uncertainty summaries invariant to component labeling under a working mixture model.

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arXiv2026

Streaming PCA: averaging from a geometric perspective

We study principal component analysis (PCA) under memory constraints, a setting that is increasingly important in large-scale data analysis. Our focus is on Oja's algorithm, which is a one-pass, memory-efficient algorithm requiring only $O(p)$ storage in the rank-one case and $O(pk)$ storage for $k$- PCA. The main goal is to develop a procedure based on Oja's algorithm that is optimal without prior knowledge of the eigengap, which is otherwise needed to tune the learning rates effectively. We do this by first introducing a geometric perspective on the convergence of $k$-PCA: by embedding the Oja iterates into the exterior space, we establish an exact equivalence between $k$-PCA in the ambient space and $1$-PCA in the exterior space. This geometric viewpoint enables us to establish several convergence guarantees, including one based only on a single learning-rate schedule. Building on this geometric perspective, we develop an averaging theory for $k$-PCA and show that the resulting averaged estimator is adaptive: it achieves the nearly optimal convergence rate without prior knowledge of the eigengap. As an application, we construct memory-efficient estimators for elliptical component analysis (ECA) \cite{han2014scale,han2018eca}. Simulation studies and real data analysis are conducted to demonstrate the benefits of our proposed algorithms.

stat.ME↗

arXiv2026

Sparsity Saves Compute, Not Samples: Oracle Inequalities for Dense and Sparse Mixture of Experts

Mixture-of-Experts (MoE) models combine specialized experts through an input-dependent routing mechanism, and their sparse variants underpin many modern large-scale systems. Yet the statistical cost of adaptive routing remains only partially understood. We study the estimation error of least-squares MoE estimators when both the experts and the gating network are Lipschitz in their parameters. We represent each MoE as a gate--expert pair in a Gated Hilbert Space, whose norm $\|\cdot\|_Δ$ separates the contributions of routing and expert discrepancies. Using its empirical representation, we derive an oracle inequality with a fast rate of order $\tilde{\mathcal O}\big((Kd_{\mathrm{pr},e}+d_{\mathrm{pr},g})/n\big)$ for dense MoEs with $K$ experts. For sparse Top-$k$ and Top-$p$ routing, the selection map is discontinuous in the gating parameters. We handle this by partitioning the gating class according to the selection patterns it induces on the sample. The resulting bounds match the dense rate up to an additive routing cost governed by the Natarajan dimension of the selector. We show that such a routing-complexity condition cannot be dropped: Lipschitz continuity in the parameters alone does not suffice. For linear softmax gating, the routing cost is of the same order as the gating term. Finally, for linear experts, we establish a matching minimax lower bound of order $Kd_{\mathrm{pr},e}/n$, showing that the rates are optimal up to logarithmic factors.

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arXiv2026

Characterizing Heterogeneous Rates in Finite Mixture Estimation via Partial Optimal Transport

Parameter estimation in finite mixture models can exhibit highly heterogeneous convergence behavior: locally isolated components may be estimated substantially faster than groups of competing components. Existing analyses based on Wasserstein distances typically characterize only the worst-case rate and therefore do not fully capture this local heterogeneity. In this paper, we introduce a Voronoi-based partial optimal transport (VPOT) framework for obtaining refined local and global convergence guarantees for the maximum likelihood estimator of the mixing measure. The key geometric idea is to localize the comparison of two mixing measures to extended Voronoi neighborhoods and use partial optimal transport to accommodate the unequal masses of their local restrictions. Within each neighborhood, the first-order POT discrepancy is raised to a power determined by the number of locally competing atoms, allowing the resulting loss to adapt to the local degree of singularity. Under suitable regularity and strong identifiability conditions, we establish uniform local and global upper bounds for a maximum likelihood estimator under the VPOT loss. These bounds reveal a configuration-dependent form of parameter estimation: less singular local configurations admit faster convergence, whereas the most singular configuration recovers the classical worst-case behavior characterized by Wasserstein-based analyses. We further establish a minimax lower bound showing that the convergence rate for estimating the mixing measure under the VPOT loss is optimal. Our results hold in arbitrary fixed dimension without requiring mixing proportions to be uniformly bounded away from zero or prior knowledge of the true number of mixture components. Overall, VPOT provides a configuration-adaptive framework for capturing heterogeneous parameter-estimation behavior in finite mixture models.

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arXiv2026

Berry-Esseen bounds for multivariate martingale difference sequences in the Kolmogorov distance

We derive new Gaussian approximations for finite martingale difference sequences in $\mathbb{R}^d$ with respect to the Kolmogorov distance. Under appropriate conditions, our bounds exhibit a dependence of order $n^{-1/4}$ on the length of the sequence and of order $\mathrm{polylog}(d)$ on the dimension. As an application, we derive a high-dimensional Berry-Esseen bound over hyper-rectangles for martingale sequences generated from Markov chains.

math.PR↗

arXiv2026

Partial Differential Equation Barriers to Identifiability in Infinite Mixture Models

We study identifiability of mixing measures in infinite mixture models. We show that, in many common cases, lack of identifiability can be characterized in terms of certain differential structures of the kernel family with respect to its parameters. In our main results, we prove that when the kernel is annihilated by a non-trivial differential or difference-differential operator over the parameter space, there exist infinitely many distinct mixing measures yielding the same mixture density. We give verifiable conditions for such operators to exist, covering many common cases, including the location-scale Gaussian, location-scale Student-t, Gamma, Beta, Dirichlet, negative binomial and non-central Chi-squared families. Furthermore, our conditions apply to any exponential family whose parameter dimension exceeds the dimension of its sufficient statistic and, more generally, to kernels with polynomially-growing score functions. We complement our results with a minimax lower bound on the estimation error for the mixing measure in the Wasserstein distance under non-identifiability. On the flips side, we describe three classes of kernels for which identifiability is preserved and nonparametric statistical inference remains possible.

math.ST↗

arXiv2026

Prediction-Only Distillation in Linear and Logistic Regression

Self-distillation (SD) is typically studied when the student is retrained on the teacher's original training inputs. In many practical deployments, however, the labeled training data are no longer available, and one has access only to the trained predictor and fresh unlabeled covariates. We study SD in this prediction-only regime through a fresh-X prediction-mixed scheme: a pure-distilled student is trained on fresh covariates pseudo-labeled by the teacher, and the final predictor is an affine combination of the teacher and student predictions. For ridge regression under proportional asymptotics, we derive deterministic equivalents for the optimally mixed prediction risk under general anisotropic covariance and deterministic signal. We show that this risk is strictly smaller than the teacher risk for almost every pair of teacher and student regularization levels, including when the fresh covariates are out-of-distribution and even when their covariance is isotropic. We further show that the optimal mixing weight cannot be identified from unlabeled data alone, but can be consistently estimated in a single post-training step using a small independent labeled calibration set, without additional model fitting. Finally, for binary logistic regression, we show that prediction mixing can outperform both the teacher and the pure-distilled classifier.

math.ST↗

arXiv2026

Convergence Rates for Latent Mixing Measures in Infinite Homoscedastic Location-Scale Mixture Models

We study posterior contraction rates for mixing measures in homoscedastic location-scale mixture models with infinitely many components. While posterior convergence at the level of densities is well understood, ensuring convergence of the latent mixing measure is more challenging and has remained an open problem in settings where both location and scale parameters are unknown. We address this by deriving novel lower-bounds that connect the $L^1$ distance between mixture densities to discrepancies, based on the Wasserstein distances and the operator norm, between the underlying mixing measures and scale matrices. Our approach combines the dual formulation of the $W_1$ distance with functional-analytic approximation techniques. This leads to general inequalities, whose strength is determined (i) by the smoothness of the mixture kernel via the rate of decay of its characteristic function, and (ii) by a key lower-bound on the $L^1$ metric involving the operator norm discrepancy between scale parameters. Moreover, a novel PDE inversion condition yields a sharper inequality for important ordinary-smooth cases. We specialize these bounds to popular mixtures based on multivariate Gaussian, Cauchy, and Laplace kernels. As a consequence, we obtain first-of-their-kind contraction rates in the context of Dirichlet process mixtures with an unknown scale parameter shared across components. As a byproduct of our inequalities, we can distinguish the convergence behavior of the location mixing measure from that of the scale parameter across a range of kernel choices, leading to nuanced insights into their respective rates.

math.ST↗

arXiv2026

Statistical Inference for Policy Evaluation with Temporal Difference Learning

We investigate the statistical properties of Temporal Difference (TD) learning with Polyak-Ruppert averaging, arguably one of the most widely used algorithms in reinforcement learning, for the task of estimating the parameters of the optimal linear approximation to the value function. Assuming independent samples, we make three theoretical contributions that improve upon the current state-of-the-art results: (i) we establish refined high-dimensional Berry-Esseen bounds over the class of convex sets, achieving faster rates than the best known results, and (ii) we propose and analyze a novel, computationally efficient online plug-in estimator of the asymptotic covariance matrix; (iii) we derive sharper high probability convergence guarantees that depend explicitly on the asymptotic variance and hold under weaker conditions than those adopted in the literature. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function approximation, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.

stat.ML↗

arXiv2026

On the Geometry of Separation in Finite Gaussian Mixtures

We study an open problem of understanding the effects of the minimum component separation on the convergence rates of parameter estimation in finite Gaussian mixtures. We address this by developing a unified geometric framework based on novel Hellinger lower bounds that directly relate discrepancies between mixture densities directly to Wasserstein distances between their underlying mixing measures, with explicit dependence on both the minimum separation and the minimum weight. Our approach combines carefully designed interpolation polynomials with confluent divided difference techniques to construct specialized moment-extraction test functions. When the number of components is known, these bounds uncover a localization phenomenon: the separation complexity is driven strictly by the spatial configuration of mixture components, namely, whether they are concentrated in a single cluster, partitioned into multiple clusters separated by a macroscopic gap, or arranged without any structural constraints. On the other hand, when the number of components becomes unknown and is over-specified, the separation complexity is slightly reduced, while the minimum mixture weight disappears entirely from the convergence rates due to a transition from first-order to second-order Wasserstein geometry. As a consequence, we obtain separation-dependent convergence rates that continuously interpolate between point-wise and uniform estimation regimes, thereby settling the fundamental limits of parameter recovery in finite Gaussian mixtures.

math.ST↗

arXiv2026

On the Convergence and Straightness of Rectified Flow

Flow Matching has become a cornerstone of modern generative models like Stable Diffusion 3, largely due to the efficiency of its Rectified Flow (RF) variant. The success of RF hinges on iteratively learning straight trajectories, pushing generation towards fewer sampling steps. However, the theoretical link between path geometry and sampling efficiency has been underexplored. This paper fills this gap by introducing a novel \textit{Piecewise Straightness} parameter, $γ_{2,T}$. We establish the first Wasserstein convergence bound that explicitly links the discretization error of \textit{any} general flow-model to $γ_{2,T}$, proving that minimizing curvature is the key to achieving high-fidelity, one-step sampling. Building on this theory, we establish the first theoretical framework to analyze the straightness of RF. We begin by offering intuitive geometric arguments for simple cases before identifying sufficient conditions under which a single rectification step (1-RF) yields a perfectly straight or even a Monge optimal coupling. While whether these sufficient conditions are met depends on the problem geometry, they enable the first concrete proofs in this area. Critically, fulfilling these conditions makes the subsequent flow (2-RF) perfectly straight ($γ_{2,T}=0$). This eliminates the discretization error in our bound and makes flawless, single-step sampling possible.

cs.LG↗

arXiv2026

Uncertainty quantification for Markov chain induced martingales with application to temporal difference learning

We establish novel and general high-dimensional concentration inequalities and Berry-Esseen bounds for vector-valued martingales induced by Markov chains. We apply these results to analyze the performance of the Temporal Difference (TD) learning algorithm with linear function approximations, a widely used method for policy evaluation in Reinforcement Learning (RL), obtaining a sharp high-probability consistency guarantee that matches the asymptotic variance up to logarithmic factors. Furthermore, we establish an $O(T^{-\frac{1}{4}}\log T)$ distributional convergence rate for the Gaussian approximation of the TD estimator, measured in convex distance. Our martingale bounds are of broad applicability, and our analysis of TD learning provides new insights into statistical inference for RL algorithms, bridging gaps between classical stochastic approximation theory and modern RL applications.

stat.ML↗

arXiv2026

On Bayesian Softmax-Gated Mixture-of-Experts Models

Mixture-of-experts models provide a flexible framework for learning complex probabilistic input-output relationships by combining multiple expert models through an input-dependent gating mechanism. These models have become increasingly prominent in modern machine learning, yet their theoretical properties in the Bayesian framework remain largely unexplored. In this paper, we study Bayesian mixture-of-experts models, focusing on the ubiquitous softmax-based gating mechanism. Specifically, we investigate the asymptotic behavior of the posterior distribution for three fundamental statistical tasks: density estimation, parameter estimation, and model selection. First, we establish posterior contraction rates for density estimation, both in the regimes with a fixed, known number of experts and with a random learnable number of experts. We then analyze parameter estimation and derive convergence guarantees based on tailored Voronoi-type losses, which account for the complex identifiability structure of mixture-of-experts models. Finally, we propose and analyze two complementary strategies for selecting the number of experts. Taken together, these results provide one of the first systematic theoretical analyses of Bayesian mixture-of-experts models with softmax gating, and yield several theory-grounded insights for practical model design.

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