Papers

We present a simple Gaussian approximation to the finite-sample distribution of the classical ridge regression estimator. Our approximation captures the fact that, in finite samples, the ridge regression estimator trades off bias and variance to reduce estimation and prediction error. Our approximation is based on nonstandard asymptotics where $i)$ we let the estimator's regularization parameter grow proportionally to the sample size; and $ii)$ we treat the population regression coefficients as local to the reference vector that defines the estimator's direction of shrinkage. In contrast to other asymptotic approximations in the literature, we allow for general forms of heteroskedasticity and autocorrelation in the data generating process (at the cost of considering a low-dimensional model where the number of covariates is not allowed to grow with the sample size). We use our simple Gaussian approximation to propose two new strategies to select the regularization parameter for the ridge regression estimator. The suggested strategies select the regularization parameter to minimize either average or worst-case excess prediction risk, where risk is computed using our suggested Gaussian approximation.

This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on $\mathbb{R}$ with mean in $[-\lambda,\lambda]$ and absolute $k$-th central moment at most $\sigma^k$, where $k>1$ is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy $\epsilon$ and confidence $1-\delta$, its sample complexity scales as \[ \log\frac{\lambda}{\sigma} + \begin{cases} (\sigma/\epsilon)^2\log(1/\delta), & k>2,\\ (\sigma/\epsilon)^2\log(\sigma/\epsilon)\log(1/\delta), & k=2,\\ (\sigma/\epsilon)^{k/(k-1)}\log(1/\delta), & 1<k<2, \end{cases} \] up to constants depending only on $k$. In the range covered by the known lower bound, this rate is minimax optimal even among fully adaptive protocols. This gives a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation with general queries \citep[Open Problem~1]{lau2026open}.

We characterize the finite sample behavior of the log-likelihood ratio statistic in binary logistic regression, uniformly over both the design and the target parameter. For $n\geq d\geq 3$, we determine, up to universal constants, its worst case $(1-\delta)$ quantile over all fixed collections of design vectors and all target parameters: \[ d\log\left(\frac{e n}{d}\right)+\log\left(\frac{1}{\delta}\right). \] This is a nonasymptotic analogue of the Wilks $\chi^2_d$ phenomenon and requires no regularity assumptions on the design. The low dimensional cases exhibit unusual behavior. The worst case quantile in dimension $d=2$ is sharply of order \[ \log\log\log n+\log\left(\frac{1}{\delta}\right). \] The worst case quantile in dimension $d=1$ is of order $\log(1/\delta)$, with no dependence on $n$. Finally, i.i.d. Gaussian design vectors recover the classical Wilks scale. In the regime $n\gtrsim d+\log(1/\delta)$, we prove the sharp bound \[ d+\log\left(\frac{1}{\delta}\right). \] Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on $n$, $d$, and $\delta$.

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying c-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. In contrast, under suitable compactness and uniform-integrability assumptions, iterative c-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative c-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).

Very often, accuracy of analysis and forecasting (multiple coefficient of regression and residual means) obtained for a sample used to formulate a regression model is not equal to the accuracy achieved for another homogeneous sample. Indeed, accuracy of analysis and forecasting based on another sample is much worse. This is explained by the discrepancy between a postulated and a real model. This discrepancy is caused by the redundancy of the number of terms of an approximating series. It describes a phenomenon under study with the sample noise. To filter this noise it is necessary to correctly choose the terms of an approximating series and to determine their number, which in turn depends on sample size. It is possible to include in this approximating series not only independent variables but also various functions of the value of the independent variables (e.g., squares, cubes, algorithms, etc.). This paper gives the procedure for selection of an approximating series.

Distinguishing long memory behaviour from nonstationarity can be very difficult as in both cases the sample autocovariance function decays very slowly. Available stationarity tests either do not include long memory or fare poorly in terms of empirical size, especially near the boundary between long memory and nonstationarity. We propose a testing procedure based on evaluating periodograms at different epochs. Limiting distributions established here are easily tractable as sum of weighted independent $\chi^2$ random variables. Moreover, numerical studies are provided to show that the proposed approach seems to outperform existing methods.

A Gaussian mixture density can have more modes than components. It has been conjectured that the maximum number of modes of a $d$-variate $k$-component Gaussian mixture density is $\binom{d+k-1}{d}$, which equals six for $(d,k)=(2,3)$. We construct an explicit family of equally weighted heteroscedastic three-component bivariate Gaussian mixture densities with at least seven distinct nondegenerate modes, showing that this conjectured upper bound fails for $(d,k)=(2,3)$. To the best of our knowledge, this provides the first counterexample to the conjecture across all pairs $(d,k)$.

We analyze approximate message passing (AMP) with an independent Gaussian initialization for noiseless phase retrieval in the proportional asymptotic regime. A random initialization has overlap of order $d^{-1/2}$ with the signal, and AMP requires a growing number of iterations to attain non-vanishing overlap. Thus, its precise behavior cannot be characterized by classical fixed-time state evolution. We prove a Gaussian decomposition of the AMP trajectory and control its error over the horizons required for recovery. The resulting analysis shows that random initialization attains the weak-recovery threshold $\delta_{\rm weak}=1/2$. For $\delta\in(\delta_{\rm weak},\delta_{\rm str})$, where $\delta_{\rm str}\approx1.13$, the signal strength follows state evolution and approaches its stable finite fixed point uniformly for \(n^{1/3}/\operatorname{polylog}(n)\) iterations. For $\delta>\delta_{\rm str}$, AMP reaches any prescribed fixed recovery accuracy within $O_{\delta,\varepsilon}(\log n)$ iterations. The majority of our analysis applies more generally to generalized AMP for single-index models.

Datasets in many settings naturally partition into clusters arising from sub-populations, batch effects, or aggregation across multiple sources. A common response to such heterogeneity is to ensemble learners trained on each cluster rather than fit a single model to the pooled data. Prior work motivating such approaches has typically considered settings in which both the covariate distribution and the conditional outcome model differ across clusters; the role of cluster-aware partitioning and ensembling based solely on the covariate distribution remains to be explored. We address this case for ridge-regularized least-squares regression under a linear outcome model and consider all ridge penalty values $\lambda \geq 0$, including the special case of the ridgeless predictor at $\lambda = 0$. By considering both fixed-effects and random-effects models, we argue that under random effects, an optimally tuned pooled ridge predictor always outperforms ensembles of individually optimally tuned predictors. For fixed effects, we derive a general formula for the pooled and ensembled predictors to characterize the role of both regression coefficients as well as the predictor distribution shifts. Together, these results generalize prior risk analyses of bagging and random-partition estimation using ridge and ridgeless regression predictors from the i.i.d. setting to encompass covariate shift and heterogeneity-aware partition structure.

We propose method-of-moments estimators for the eigenvalues of variance component covariance matrices in multivariate mixed effects models. Assuming a parametric form for the eigenvalue distribution, we focus on the high-dimensional regime where the number of predictors is large and comparable to the number of realizations of each random effect. In this setting, we show that the empirical moments of sum-of-squares matrices (e.g., MANOVA estimators of the covariance matrices) can be closely approximated by deterministic functions of the underlying parameters. This relationship enables the construction of consistent and asymptotically normal estimators via moment matching. Our approach is motivated by applications in quantitative genetics, where estimating genetic covariance components across multiple phenotypic traits is of central interest. We implement our method in a new python package mlmm-mom, and demonstrate how our method adapts to several common experimental designs in this domain.