This paper studies P-trainable neural networks, where a proportion P of hidden neurons are trainable and the rest stay fixed as random features, bridging random feature methods and fully trainable networks. The authors establish two-layer approximation bounds showing that a small trainable fraction can capture most of the gain when initialization mismatch is large, yielding a phase transition in rates of the form $\mathcal{O}(((P/\kappa_1^*+(1-P)/\kappa_0^*)m/\log m)^{-\alpha})$. They verify these assumptions explicitly for spherical biased-ReLU features and present numerical experiments showing analogous phase transitions in training losses across architectures with up to 15 layers.

Classical analyses of the Unadjusted Langevin Algorithm (ULA) in Wasserstein-2 distance predict a dimension-dependent mixing time and bias, yet empirically ULA often estimates low-dimensional statistical functionals accurately without scaling the step size with dimension. This paper proposes using the subspace robust Wasserstein (SRW) distance to analyze ULA on ridge-separable Bayesian posteriors, proving that iteration complexity can be dimension-free in both the overparameterized regime ($N \lesssim d$) and the large-sample regime ($N \gtrsim d^2$), while showing such guarantees are impossible near the proportional regime $N \asymp d$. The results explain a bell-shaped bias profile and provide a rotationally invariant alternative to prior $\ell^\infty$-based dimension-free analyses that fail under isotropy.

The paper studies minimax estimation of the pseudo-spectral gap $\gamma_{ps}$ of a continuous-state Markov chain on $[0,1]^d$ from a single trajectory. The core idea is to discretize the state space into a regular grid, approximate the transition density by a histogram, and adapt finite-state estimators to this compressed chain. The result matters because it yields fully empirical concentration inequalities for non-iid data, enabling UCB bandit algorithms and PAC-Bayes bounds without prior knowledge of mixing coefficients.

This paper studies the infinite-generation limit of autoregressive transformers. Rather than fixing sequence length, the authors let generation time tend to infinity and ask whether the model's empirical state converges. Their core result is that, for a broad class of architectures, the stochastic generation process tracks a deterministic ODE on a slow time scale, and every fixed transformer has a finite temperature threshold above which it converges to a unique limit. However, no uniform temperature works for all models; for any finite temperature, some transformer can fail to converge, even with just one layer and three tokens.

Bayesian posterior sampling offers a potentially 'free' source of differential privacy, but existing guarantees are fragmented and often require strong boundedness assumptions. This paper develops a unified Rényi-divergence framework that characterizes the inherent privacy loss of One-Posterior-Sampling (OPS) through exponential moments of the single-record log-likelihood ratio. The results yield exact, tighter bounds for a broader class of models and clarify how prior strength, sample size, and neighboring relations jointly shape privacy.

Many applications require sampling from high-dimensional categorical distributions with local dependence, but discrete diffusion models lack end-to-end statistical guarantees. This paper introduces a pinning decomposition of the discrete score under uniform noising, showing that time dependence separates multiplicatively from target dependence for Markov random fields of order $d$. Building on this, the authors design a weight-sharing neural score learner and prove a sample-complexity rate of $\widetilde O_{\mathbb P}(S^d/n)$, replacing the ambient dimension $D$ with the interaction order $d$.

This paper studies model collapse in diffusion models under a fixed-budget data regime, where synthetic data accumulates but each generation trains on a constant-size subsample from the growing pool. Unlike prior work on total replacement (rapid collapse) or unbounded accumulation (collapse avoided), the authors argue that fixed budgets yield partial, feature-selective collapse: "fragile" features with large eigenvalues in a linearized parameter dynamics model decay quickly, while "robust" features with small eigenvalues persist for exponentially many generations. They support this with theory (stochastic recursion giving per-feature halving times) and experiments on 2D spirals, MNIST, Fashion-MNIST, and CIFAR-10.

Counterfactual generation aims to sample the distribution of outcomes that would occur if a treatment were applied to an entire population, but observational data suffer from selection bias because treated individuals differ systematically from the target population. This paper develops a flow-matching method that learns to transport observed outcomes to counterfactual targets using a learned source-target coupling, a doubly robust training objective, and a score-corrected stochastic sampler with Gaussian smoothing. The core theoretical result is a finite-step KL bound governed by the coupling displacement moment $D_5$ rather than worst-case regularity, yielding near-linear dimension scaling and an end-to-end nonparametric sample-complexity guarantee; experiments on synthetic and semi-synthetic image data support the predicted coupling dependence and show that stochastic sampling can outperform deterministic counterparts at limited discretization budgets.

The nonparametric maximum likelihood estimator (NPMLE) of a Gaussian mixture is a convex program over an infinite-dimensional space, yet its maximizers can be nonunique and are only known to have at most $n$ atoms. This paper shows that a vanishingly small random reweighting of the likelihood---using independent Gamma weights that concentrate around one---selects a unique maximizer with polylogarithmically many atoms in any fixed dimension $d$. The resulting estimator retains a Hellinger rate that is parametric up to logarithmic factors, solving the open problem of exact sparsity for multivariate Gaussian NPMLEs without any support penalty.

Feature incremental clustering addresses dynamic scenarios where data arrives in expanding feature spaces—such as activity recognition systems that acquire new sensors over time. This paper proposes four k-means-based algorithms (FIC-FT, FIC-DR, FIC-DA, FIC-MR) tailored to different data-access constraints, from full historical access to model-only reuse. The core theoretical contribution establishes generalization error bounds for all four settings, revealing that model reuse (FIC-MR) can achieve a fast $\tilde{\mathcal{O}}(1/n_2)$ convergence rate when the pre-trained model aligns well with the current distribution.

This paper studies nonparametric regression for learning degree-$k_0$ spherical polynomials on the unit sphere $\mathbb{S}^{d-1}$ using over-parameterized two-layer neural networks. The authors propose a novel Gradient Descent with Projection (GDP) algorithm that constrains learning to the top $r_0 = \Theta(d^{k_0})$ eigenspaces of the Neural Tangent Kernel (NTK). The main result establishes a nearly minimax optimal risk bound of order $\log(4/\delta) \cdot \Theta(d^{k_0}/n)$, improving the sample complexity from previous polynomial-in-$1/\varepsilon$ rates to linear $1/\varepsilon$ scaling.