We study ordinary kernel ridge regression over full Hölder-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.
Tensor factor models provide low-dimensional representations of tensor-valued observations while preserving their multiway structure. When these observations are held by different clients, privacy constraints may prevent them from being pooled for centralized analysis. We study estimation of common loading spaces in Tucker tensor factor models under federated differential privacy, allowing heterogeneous sample sizes, privacy budgets, and factor strengths. The proposed method combines private stability testing, Huber aggregation, and Gaussian perturbation at each client with weighted aggregation at the server. We establish non-asymptotic upper bounds governed by a scaled harmonic mean of the squared local error rates, and prove minimax optimality up to logarithmic factors under strong factors, fixed factor ranks, and fixed noise variance. We further develop a private calibration procedure that selects the local tuning parameters and aggregation weights without knowledge of the noise level or client-specific factor strengths. The resulting adaptive estimator preserves the privacy guarantee and attains the error rate obtained with oracle tuning. We illustrate the finite-sample performance of the proposed estimators through simulation experiments and an application to a Meditation Traits EEG dataset.
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.
Using the Feature Space Decomposition framework, we establish non-asymptotic matching upper and lower bounds for the expected Kullback-Leibler estimation error of the Add-$λ$ estimator of a discrete probability distribution, for any signal, any dimension, any sample size, and any $λ>0$.
Bayesian inverse problems require both informative prior models and computationally feasible posterior inference. Score-based diffusion models provide expressive learned priors, but their use for inversion introduces a conditioning problem: the score of a noised posterior is generally unavailable from a prior score alone. This selective survey reviews the principal approaches to that problem, including likelihood guidance, spectral and projection methods, sequential Monte Carlo corrections, and amortized conditional training. We organize methods by how they split cost between offline preparation and online inference, and compare their assumptions, advantages and limitations, with emphasis on computational speedups. Some ongoing directions and open questions are gathered at the end.
Gaussian differential privacy (GDP) and zero-concentrated differential privacy (zCDP) are two competing privacy accounting frameworks, but they share similar minimax lower bounds in various estimation problems. Moreover, it is known that a mechanism satisfying $μ$-GDP satisfies $(μ^2/2)$-zCDP, whereas the converse fails to hold at the same calibration. In this paper, we establish the optimal reverse conversion, showing that every $ρ$-zCDP mechanism satisfies $μ_{\mathrm{zCDP}}(ρ)$-GDP, where $μ_{\mathrm{zCDP}}(ρ)=2Φ^{-1}((1+e^{-t_ρ})^{-1})$ and $t_ρ$ uniquely solves the equation $ρ=t_ρ\tanh(t_ρ/2)$. We also obtain simplified bounds $\sqrt{πρ}\leqμ_{\mathrm{zCDP}}(ρ)\leq\sqrt{8ρ}$, with sharp constants, which imply equivalence of the two frameworks up to constants. This explains why their minimax risks agree up to a constant rescaling of the privacy budget. We show that the squared conversion is strictly convex, which implies that converting zCDP to GDP before applying composition or group privacy yields a tighter GDP guarantee than converting afterward. We also derive the sharp conversion from bounded-range and logistic DP mechanisms (e.g., exponential mechanism) to GDP, where converting before composition but after group privacy gives the tighter guarantee.
Operators choosing among decision rules, such as when a battery charges or where an idle taxi goes, rank them by estimated long-run value. A standard estimator, linear temporal-difference (TD) learning, fits values from one stream of correlated observations. Large fits often use an adaptive optimizer, such as adaptive moment estimation (Adam), with fixed averaging rates. We show that these rates can change what is estimated, not just how fast: the same observations drive the updates and the optimizer's step-size scaling, moving the fit away from the intended TD value. The shift differs across rules, so it need not cancel in a comparison, and more data only makes a wrong ranking look more certain. Our remedy refits the same data at a second averaging rate that halves the effective gain, the number to which the leading shift is proportional. Twice the new fit minus the original removes that shift. In New York taxi and battery models calibrated to real data, common optimizer settings leave the selected rule unchanged but order close rules wrongly and misstate its value. Our remedy corrects nearly every wrong ordering, with one extra fit when the shift is moderate and two when it is larger.
Self-regularization and structural adaptation of NPMLE for Gaussian mixture models in all dimensions
Yanjun Han, Yandi Shen, Yihong Wu
AI 速览：作者证明：均值位于超立方体内的 d 维高斯混合模型，其 NPMLE 能以高概率自动拟合出 O((log n/log log n)^d) 个分量，并在有限混合数据下使拟合分量数依概率有界。
We establish structural properties of the nonparametric maximum likelihood estimator (NPMLE) for $d$-dimensional Gaussian mixtures whose means lie in a hypercube. We show that, with high probability, the NPMLE based on $n$ independent observations self-regularizes by fitting a mixture model with $O((\frac{\log n}{\log\log n})^d)$ components, achieving the optimal model complexity without explicit model selection. Furthermore, the NPMLE exhibits a structural adaptation property: For data drawn from a finite mixture, the number of fitted components is bounded in probability as $n\to\infty$. Our proof applies a multivariate Jensen-type bound to control the typical number of critical points of the KKT function associated with the first-order optimality conditions. The key step is to lower bound its gradient norm uniformly on complex spheres. We establish this lower bound by decomposing the KKT function into an empirical fluctuation term and a relative density estimation error term, combining small-ball probability estimates for the former with polynomial approximation and Hellinger error bounds for the latter.
Albert Gong, Annabelle Michael Carrell, Raaz Dwivedi, Lester Mackey
AI 速览：提出 Express 工具，将非因果注意力近似转化为因果近似，并结合 Thinformer 实现更优的因果注意力保证与高效实现。
We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.
Optimal Neural Estimation for High-Dimensional Nonparametric Regression under Multisource Covariate Shift
Chenghao Xu, Haotian Xu, Shivam Kumar, Daren Wang
AI 速览：研究多源协变量偏移下高维非参数回归，提出oracle加权神经估计并给出minimax最优收敛率，同时给出无需已知噪声方差与偏移参数的自适应方法。
We study high-dimensional nonparametric regression under multisource covariate shift, where $\mathrm J$ sources of training data share a common regression function but differ in their covariate distributions, sample sizes, and noise variances. Our goal is to combine these sources to estimate the regression function accurately under a target covariate distribution. We propose an oracle weighted neural estimator and establish its minimax optimality up to logarithmic factors through matching upper and lower bounds. The resulting rate is $\mathsf n_{\rm eff}^{-2p/(2p+K)}$, where $p$ and $K$ denote the effective smoothness and dimension of the regression function, and the effective sample size is \begin{align} \mathsf n_{\rm eff} =\sum_{j=1}^{\mathrm J} \frac{n_j}{(1+σ_{ρ,j})σ_{ε,j}^{2}}. \nonumber \end{align} Here, $n_j$ is the sample size of source $j$, $σ_{ε,j}^{2}$ is its noise variance, and $σ_{ρ,j}$ controls the deviation between the covariate distribution of source $j$ and the target covariate distribution. We also develop an adaptive procedure that requires no knowledge of source noise variances or covariate-shift parameters. Under suitable conditions, this procedure attains the oracle minimax rate up to logarithmic factors.
Synthetic data are widely used for training modern AI systems. However, their statistical understanding remains limited. It is well recognized in statistics that repeated self-training on synthetic data can amplify errors inherited from pre-trained models, leading to model collapse. Verification plays a crucial role in selecting high-quality synthetic data to help prevent model collapse and improve model performance. We study verified self-training in a linear regression framework, where a pre-trained estimator generates synthetic data with injected generation noise and a verifier selects samples according to a verification threshold. Post-training on the accepted samples updates either all coordinates (full fine-tuning) or only an active subset (partial fine-tuning). Our analysis reveals three potentially counterintuitive effects. We establish that tighter verification does not always lead to lower prediction risk. We also find that a suitable level of injected generation noise can reduce prediction risk, while excessive noise can increase it. Moreover, full and partial fine-tuning can respond differently to the same verification rule, since pre-training error in the frozen coordinates affects both sample selection and post-training risk. We analyze these effects by deriving finite sample risk bounds for four separate regimes determined by the relative scales of the verification threshold, generation noise, and pre-training error. The bounds precisely characterize how verification affects prediction risk across these regimes. In particular, we quantify how the generation noise, the effective accepted sample size, and the covariance of the accepted covariates affect prediction risk. The analysis extends to iterative verified full fine-tuning, where sufficiently tight verification reduces the initial error contribution at a geometric rate.
A robot searching for the source of an odor must eventually answer a notably hard question: have I arrived? Near the source, concentration gradients homogenize, turbulence corrupts the signal, and a small robot often lacks the positional accuracy to map the plume. Here, we describe a lightweight remedy inspired by a World War II statistics conundrum: the German tank problem. To approximate German armament production, Allied analysts developed an estimator based on serial numbers from captured tanks. Analogously, we use the same procedure over sampled chemical sensor readings from a robot to create estimates for the peak concentration of an odor plume, and therefore the odor's source. Treating each sensor reading as an observation, our unbiased estimator needs only a running maximum and a sample counter. A confidence bound turns both numbers into a principled rule that the robot or UAV can use to determine whether it should keep searching for the odor source. We derive the estimator and its bound, demonstrate its sample efficiency, and examine its robustness even when its assumptions fail. Finally, we summarize the estimator's role as the termination criterion for olfactory navigation.
Flow Matching has recently gained attention in generative modeling as a simple and flexible alternative to diffusion models. While existing statistical guarantees adapt tools from the analysis of diffusion models, we take a different perspective by connecting Flow Matching to kernel density estimation. We first verify that the kernel density estimator matches the optimal rate of convergence in Wasserstein distance up to logarithmic factors, improving existing bounds for the Gaussian kernel. Based on this result, we prove that for sufficiently large networks, Flow Matching achieves the optimal rate up to logarithmic factors. If the target distribution lies on a lower-dimensional manifold, we show that the kernel density estimator profits from the smaller intrinsic dimension on a small tube around the manifold. The faster rate also applies to Flow Matching, providing a theoretical foundation for its empirical success in high-dimensional settings.
The Sinkhorn divergence, the debiased entropic optimal transport cost, determines a metric tensor on probability measures defined along curves, due to Lavenant et al. We construct a real-analytic Hilbert manifold of measures, the smooth stratum, on which this tensor is a weak Riemannian metric, and we continue Eguchi's construction of a geometry from a divergence to the third order. The stratum consists of the measures with a positive Sobolev density on a closed Riemannian manifold, and the strata form a scale whose intersection is the Fréchet manifold of smooth positive densities of Ay. Following it, we characterize the metric through the gradients of mean values: its cotangent form is a spectral reweighting of the Dirichlet form of a Markov chain, so the metric is of the same kind as the Fisher-Rao and the Otto metric, and on the stratum it is bounded by a multiple of the Fisher-Rao metric (but not conversely). At third order, the Sinkhorn divergence is self-dual: both connection forms of Eguchi's construction coincide with the Levi-Civita form of the metric, and the cubic tensor vanishes, as a consequence of the transposition symmetry of the transport problem. On every stratum, the Levi-Civita form is represented, so the stratum has a real-analytic Levi-Civita connection, which on the Fréchet manifold is the connection of Ay and Schwachhöfer.
Szabolcs Szentpéteri, Péter Kovács, Balázs Csanád Csáji
This paper studies robust uncertainty quantification for separable nonlinear least squares (SNLLS) problems. We propose two identification methods, (i) Residual-Perturbed Gradient (RPG) and (ii) Global-Fit-Gap (GFG), to construct joint distribution-free confidence regions for both the linear and nonlinear parameters. By exploiting assumed group invariance of the noise (such as symmetry or exchangeability), we apply random perturbations (such as sign-changes or permutations) to build the regions based on resampling and ranking. We prove that both methods guarantee non-asymptotically exact coverage (type I error). Finally, we illustrate the constructions through two numerical experiments: (i) identifying generalized finite impulse response (GFIR) systems using Laguerre polynomials with an adjustable decay rate and (ii) fitting Gaussian radial basis function (RBF) models with variable knots and widths.
Large language model (LLM) distillation can propagate teacher bias under covariate shift when target reward feedback is unavailable. We propose Iterative Calibration and Learning (IterCaL), which alternates source-reward teacher calibration through token-level branching with student learning on target prompts. We prove polynomial convergence in expected average KL to the oracle student, the reference-regularized target-reward maximizer within a potentially misspecified student class; a quantified source-to-target calibration discrepancy gives an explicit additive true-return term. Our separation shows that regularized direct teacher matching retains positive oracle-KL error at its population optimum for every shared reference penalty, even when the teacher is stronger than every student. On the same instance, IterCaL converges to the oracle using source rewards alone.
Survival analysis provides statistical methods to model the time until an event occurs. Reporting delays arise when event times are not observed at their occurrence but are only revealed upon reporting. This issue is particularly critical for timely risk evaluation when the observation window is short due to administrative censoring. In this study, we incorporate right-censored reporting delays by jointly modeling parametric hazards for the event and reporting processes. We then construct a consistent estimator for the model parameters and develop a Monte Carlo expectation-maximization algorithm to compute it. To address the challenges posed by administrative censoring, we leverage these findings and propose a transfer-learning procedure. Experimental results demonstrate that our method improves the accuracy of timely risk evaluation under administrative censoring.
Learning independence is fundamental to statistical inference and machine learning, yet learning independence between entire time series by a test remains challenging when only a single observation record is available. Existing guarantees often rely on structural or mixing assumptions, while stationarity and ergodicity alone do not permit consistent testing. In this work, we identify a class-level boundary governed by shared nontrivial Koopman eigenvalues, which represent common persistent oscillatory frequencies of the two processes. When no such eigenvalues are shared, we extend shift-based rank calibration with finite-window dependence witnesses and a data-adaptive threshold to construct a deterministic, weakly consistent test with finite-sample false-positive control, without quantitative mixing assumptions. Conversely, we show that no possibly randomized test can achieve vanishing false-positive probabilities under every independent jointly totally ergodic law while consistently detecting every dependent jointly totally ergodic law with a shared nontrivial eigenvalue. This impossibility already holds for binary observations and persists within any prescribed nonempty open eigenvalue window, identifying a spectral obstruction to universal learning of full-process independence.
Estimating conditional parameters, such as heterogeneous treatment effects, often requires rich covariate adjustment, even when the parameters are reported along only a few dimensions. Fitting nuisance functions on the full sample makes all observations available for adjustment, but creates dependence between fitted scores and the observations at which they are evaluated, which is typically avoided by cross-fitting. We develop pointwise inference with full-sample nuisance predictions for exactly identified conditional moment equations, recovering conditional average treatment effects as a special case. We localize the estimating equations around the reporting point using distributional nearest neighbors (DNN), a U-statistic-based averaging procedure. Under sufficient conditions on the local design, orthogonality, prediction accuracy and quantitative deletion stability, the estimator is asymptotically normal, and a frozen-score jackknife consistently estimates its limiting covariance with nuisance predictions held fixed. In simulations, full-sample and cross-fitted inference perform similarly, with coverage shortfalls from nuisance approximation error, spatial bias and heavy studentized tails. An application to 401(k) eligibility using Survey of Income and Program Participation data finds similar estimates across nuisance prediction methods, with wide normal-reference intervals.
Neural networks are powerful memorizers which are often able to interpolate their training data even when the labels have been randomized. While there exists an extensive literature on the memorization capacity of fully connected networks, in practice more specific architectures with problem-dependent properties are preferred, such as convolutional networks for image classification. In this paper we consider convolutional networks which are invariant with respect to a given arbitrary finite group of orthogonal transformations, which may include translations, reflections, and/or rotations of images and invariances of higher order tensors. We prove upper bounds on the size of an invariant convolutional neural network with threshold activations that is required to interpolate a given dataset with two classes. Our bounds are formulated in terms of a complexity measure that takes the invariant structure into account and is instance-specific, i.e., it is related to the mutual arrangement of the classes in the given dataset and their geometric properties rather than the number of samples. Our proof is constructive: for given input data, instances of an interpolating network can be computed via a randomized algorithm with high probability. In numerical experiments we investigate the practical effectiveness of this algorithm.
Principal-agent hypothesis testing studies statistical contracts in which a principal rewards an agent according to data the agent generates. Existing work focuses on the single-shot setting, where the agent produces a single dataset. We extend the framework to sequential settings, where evidence accumulates over time, rewards may be deferred to the end or disbursed incrementally, and the agent may stop the contract at a strategically chosen stopping time. We call a contract null-deterring if no ''null agent'' (those whose data-generating distribution lies in the null hypothesis) can profit in expectation under any stopping rule, and our main conceptual result shows that such contracts are characterized by e-processes. We then turn to the agent's best response and identify the optimal stopping rule through the Snell envelope of the reward process. We develop simpler, more interpretable approximations that often require the agent to have only partial knowledge of its distribution.
We study in-context learning problems where a Transformer is pretrained on tasks drawn from a mixture distribution $π=\sum_{α\in\mathcal{A}} λ_α π_α$, called the pretraining prior, in which each mixture component $π_α$ is a distribution on tasks of a specific difficulty level indexed by $α$. Our goal is to understand the performance of the pretrained Transformer when evaluated on a different test distribution $μ$, consisting of tasks of fixed difficulty $β\in\mathcal{A}$, and with potential distribution shift relative to $π_β$, subject to the chi-squared divergence $χ^2(μ,π_β)$ being at most $κ$. In particular, we consider nonparametric regression problems with random smoothness, and multi-index models with both random smoothness and random effective dimension. We prove that a large Transformer pretrained on sufficient data achieves the optimal rate of convergence corresponding to the difficulty level $β$, uniformly over test distributions $μ$ in the chi-squared divergence ball. Thus, the pretrained Transformer is able to achieve faster rates of convergence on easier tasks and is robust to distribution shift at test time. Finally, we prove that even if an estimator had access to the test distribution $μ$, the convergence rate of its expected risk over $μ$ could not be faster than that of our pretrained Transformers, thereby providing a more appropriate optimality guarantee than minimax lower bounds.
Let $P$ be an irreducible reversible Markov kernel on an $m$-state space $Ω$, and denote its right spectral gap by $\Gap(P) =1-λ_2(P)$. For a stationary trajectory, let $K_t$ be the occupation-count vector of the length-$n$ window beginning at time $t$. The stationary pair $(K_0,K_1)$ defines a reversible projected count kernel $\widetilde P_n$. For each fixed $m\ge2$, define \[ c_m^\star= \inf_{\substack{|Ω|=m,\ P\ \mathrm{irreducible\ and\ reversible} n\ge2}} \frac{n\Gap(\widetilde P_n)}{\Gap(P)}. \] We prove \[ \frac1{1080m}\le c_m^\star\le q_{m-2}, \qquad q_0=\frac14,\quad q_{r+1}=q_r(1-q_r). \] We also show $q_{m-2}=(m+\log m+O(1))^{-1}$, which implies that $c_m^\star=Θ(m^{-1})$. Thus, the reciprocal optimal comparison coefficient $C_m^{\mathrm{opt}}=(c_m^\star)^{-1}$ is of the order $m$. In the two-state case, we determine the exact values $c_2^\star=1/4$ and $C_2^{\mathrm{opt}}=4$. %the exact constants for $m\ge3$ remain open. The lower bound also holds for $n=1$ and is uniform in $P$, including sparse and periodic kernels. The proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson--Hardy inequality. A nested rare-state construction produces finite $m$-state witnesses whose normalized Rayleigh quotients approach $q_{m-2}$ through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, $\Gap(\widetilde P_n)=Θ_P(n^{-1})$.
We study the problem of finding the minimum $\mathbb{L}\_2$-norm element in a transportation polytope. It corresponds to minimizing the index of coincidence or maximizing R{é}nyi entropy over joint probability distributions with prescribed marginals. The solution is straightforward under a feasibility condition on the marginals. We make three contributions. First, we decompose the basic solution into a mixture of three interpretable couplings and derive a closed-form sampling procedure, revealing its role as the $\mathbb{L}\_2$-projection of the uniform distribution onto the transportation polytope. Second, we show that the feasibility condition becomes exponentially rare as the number of modalities grows, which highlights the practical necessity of a more general construction. Third, we provide a KKT-based iterative construction of the optimal coupling for arbitrary marginals with explicit Lagrange multipliers.
Binary Iterative Hard Thresholding (BIHT) is a simple, yet effective, greedy method for recovering a sparse vector from one-bit sign measurements. In its original form, BIHT performs a ``gradient-descent'' step, followed by hard thresholding. A convergence analysis of this algorithm was left open in the introductory work of Jacques, Laska, Boufounos, and Baraniuk [IEEE Trans. Inform. Theory (2013)] and has remained unresolved for over a decade, with subsequent sharp analyses studying a normalized variant instead, that additionally projects every iterate onto the unit sphere. This paper resolves that gap and characterizes when per-iterate normalization is algorithmically necessary. In the noiseless setting, we prove a universal, sample-optimal convergence theorem for the original BIHT algorithm. Specifically, with $\widetilde O(s/ε)$ measurements, a deterministic, computable finite-time iterate has directional error at most $ε$, simultaneously for every $s$-sparse unit vector. This matches the optimal sample dependence achieved by normalized BIHT in prior work. Under sign corruptions, we prove a sharp separation. If at most a $τ$ fraction of signs are flipped adversarially, then BIHT, without per-iterate normalization, still reaches the robust error floor at an early iterate with a matching $\widetilde O(s/ε)$ sample complexity rate as its normalized variant. This recovery, however, is not stable. We prove a scalar lower bound showing that any nontrivial corruption pattern, even one that involves only one flipped sign together with one clean sign, forces the iterates to oscillate indefinitely. Consequently, no general last-iterate convergence theorem can hold for BIHT under sign corruptions, while its normalized surrogate provably escapes this instance.
For each time $t$, the regression function $m_t(x)=E[Y_t\mid X_t=x]$ of an outcome on contemporaneous covariates can be regarded as a field on covariate space. When the covariates of units move as a diffusion and each unit's outcome has its own dynamics, we show that $m_t$ satisfies a second-order evolution equation whose diffusion term is $\frac12\mathrm{tr}(Σ\nabla^2 m_t)$, the Hessian of the regression function weighted by the covariance $Σ$ of covariate increments, whose transport term is the drift of the time-reversed covariate process, and whose source term is the conditional mean rate of change of the outcome; this yields a Feynman-Kac representation along reversed covariate paths. We relate this Hessian to quadratic regression (under Gaussian design the best quadratic approximation has Hessian $E[\nabla^2 m(X)]$) and to kernel smoothing bias. For bistable replicator dynamics on covariate space, travelling fronts are exactly logistic in $(x,t)$, so one logistic regression identifies the diffusion-reaction ratio and the invading strategy. We then ask whether spread in covariate space reflects fluctuating unit characteristics or imitation among similar units. Imitation acts through exactly the same operator as covariate fluctuation, so the two are observationally equivalent in the regression field; panel data separate them because they identify $Σ$ directly. The two cannot be separated on a travelling front, and we propose a split-sample instrumental-variables test that works on seeded, non-front profiles, with size and power evaluated in agent-based experiments.
Kernel methods face two recurrent bottlenecks: the cubic cost of inverting an $n\times n$ kernel matrix, as in kernel ridge regression, and the evaluation of kernel integrals, as required in kernel quadrature. Existing low-rank kernel and Gaussian process approximations primarily address the former bottleneck, but do not control the downstream quadrature error when both the kernel matrix and kernel mean are approximated, particularly when the integrand lies outside the kernel reproducing kernel Hilbert space. We address both challenges through an operator-adapted finite-feature approximation. Specifically, we construct a finite-dimensional working kernel from the eigenvalues and eigenfunctions of a second-order differential operator. We establish convergence guarantees for the resulting finite-feature estimators in kernel quadrature, with extension to other related kernel methods. Our results show that, with $m \lesssim n$ basis functions, the truncated estimator attains the corresponding benchmark statistical rates. The theory covers fixed/random designs, noisy/noiseless observations, oversmoothing/undersmoothing regimes, and Matérn/Gaussian-type weights. We provide explicit error bounds that quantify the roles of the spectral cutoff, regularization parameter, lengthscale, sampling geometry, and operator boundary condition. We verify the convergence results in numerical experiments.
We consider the tensor principal components analysis (tensor PCA) problem on the sphere, in which we are asked to recover an unknown vector in $\mathcal{S}_N=\{x\in\mathbb{R}^N:\lVert x\rVert^2=N\}$ from a noisy observation of its $k$-th tensor power, for a fixed $k\ge3$. Tensor PCA exhibits a well-studied information-computation gap: global algorithms such as spectral methods and sum-of-squares work above an algorithmic threshold, below which we know matching lower bounds (against these and some other restricted models of computation). In contrast, local algorithms such as Markov chains were only known to succeed from a random initialization at a polynomially larger signal-to-noise ratio, and were believed to be inferior to global algorithms for tensor PCA. Continuing the program initiated in Part I [arXiv:2610.16705], we show that spherical Langevin dynamics, when supplemented with annealing and ergodic averaging, achieves polynomial-time strong recovery above the algorithmic threshold from a uniformly random initialization. Together with Part I, our results demonstrate that local Markov chains can succeed all the way down to the algorithmic threshold from an uninformative cold start.
We study conformal prediction under covariate shift with a known covariate density ratio, seeking prediction sets with minimum expected length under the target distribution. For every calibration sample size, we prove that the rule combining the inverse conditional-density score with randomized weighted conformal calibration minimizes this expected length among all rules valid over the full weighted-exchangeability class. We approximate the oracle using quantile and kernel histograms, including forest-weighted estimators, and select estimators and tuning parameters by resampling to estimate calibrated target length. On a bounded response domain, we bound excess length linearly in $L^1$ density error at fixed calibration size, without requiring a uniform positive lower bound on the true density. We also bound selection-pool and resampling errors. Under explicit growth conditions, the selected procedure approaches the same oracle while the calibration size remains fixed. Experiments on 36 known-density configurations and four public regression datasets compare coverage, prediction-set length, and runtime with existing methods. In targeted low-coverage and strong-shift comparisons, inverse-density selection yields shorter mean sets in most cases and lower mean runtime than the evaluated weighted baselines, with some differences in achieved coverage.
Efron et al. (2004) introduced least angle regression (LAR) as an algorithm for linear predictions, intended as an alternative to forward selection with connections to penalized regression. However, LAR has remained somewhat of a "black box," where some basic behavioral properties of LAR output are not well understood, including an appropriate termination point for the algorithm. We provide a novel framework for inference with LAR, which also allows LAR to be understood from new perspectives with several newly developed mathematical properties. The LAR algorithm at a data level can viewed as estimating a population counterpart "path" that organizes a response mean along regressor variables which are ordered according to a decreasing series of population "correlation" parameters; such parameters are shown to have meaningful interpretations for explaining variable contributions whereby zero correlations denote unimportant variables. In the output of LAR, estimates of all non-zero population correlations turn out to have independent normal distributions for use in inference, while estimates of zero-valued population correlations have a certain non-normal joint distribution. These properties help to provide a formal rule for stopping the LAR algorithm. While the standard bootstrap for regression can fail for LAR, a modified bootstrap provides a practical and formally justified tool for interpreting the entrance of variables and quantifying uncertainty in estimation. The LAR inference method is studied through simulation and illustrated with data examples.