We develop a near-optimal testing procedure under the framework of Gaussian differential privacy for simple as well as one- and two-sided tests under monotone likelihood ratio conditions. Our mechanism is based on a private mean estimator with data-driven clamping bounds, whose population risk matches the private minimax rate up to logarithmic factors. Using this estimator, we construct private test statistics that achieve the same asymptotic relative efficiency as the non-private, most powerful tests while maintaining conservative type I error control. In addition to our theoretical results, our numerical experiments show that our private tests outperform competing DP methods and offer comparable power to the non-private most powerful tests, even at moderately small sample sizes and privacy loss budgets.

We consider the problem of graph matching for a sequence of graphs generated under a time-dependent Markov chain noise model. Our edgelighter error model, a variant of the classical lamplighter random walk, iteratively corrupts the graph $G_0$ with edge-dependent noise, creating a sequence of noisy graph copies $(G_t)$. Much of the graph matching literature is focused on anonymization thresholds in edge-independent noise settings, and we establish novel anonymization thresholds in this edge-dependent noise setting when matching $G_0$ and $G_t$. Moreover, we also compare this anonymization threshold with the mixing properties of the Markov chain noise model. We show that when $G_0$ is drawn from an Erdős-Rényi model, the graph matching anonymization threshold and the mixing time of the edgelighter walk are both of order $Θ(n^2\log n)$. We further demonstrate that for more structured model for $G_0$ (e.g., the Stochastic Block Model), graph matching anonymization can occur in $O(n^α\log n)$ time for some $α<2$, indicating that anonymization can occur before the Markov chain noise model globally mixes. Through extensive simulations, we verify our theoretical bounds in the settings of Erdős-Rényi random graphs and stochastic block model random graphs, and explore our findings on real-world datasets derived from a Facebook friendship network and a European research institution email communication network.

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 the degree to which 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 choice of the 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 milder requirements on the accuracy of the drift estimates, and also with a specific time schedule. We illustrate with simulations on the synthetic data and text-to-image data experiments the improved performance of the proposed samplers implementing the newly designed time-discretization schedules.

Identifying low-dimensional sufficient structures in nonlinear sufficient dimension reduction (SDR) has long been a fundamental yet challenging problem. Most existing methods lack theoretical guarantees of exhaustiveness in identifying lower dimensional structures, either at the population level or at the sample level. We tackle this issue by proposing a new method, generative sufficient dimension reduction (GenSDR), which leverages modern generative models. We show that GenSDR is able to fully recover the information contained in the central $σ$-field at both the population and sample levels. In particular, at the sample level, we establish a consistency property for the GenSDR estimator from the perspective of conditional distributions, capitalizing on the distributional learning capabilities of deep generative models. Moreover, by incorporating an ensemble technique, we extend GenSDR to accommodate scenarios with non-Euclidean responses, thereby substantially broadening its applicability. Extensive numerical results demonstrate the outstanding empirical performance of GenSDR and highlight its strong potential for addressing a wide range of complex, real-world tasks.

High-dimensional covariance estimation is notoriously sensitive to outliers. While statistically optimal estimators exist for general heavy-tailed distributions, they often rely on computationally expensive techniques like semidefinite programming or iterative M-estimation ($O(d^3)$). In this work, we target the specific regime of \textbf{Sub-Weibull distributions} (characterized by stretched exponential tails $\exp(-t^α)$). We investigate a computationally efficient alternative: the \textbf{Cross-Fitted Norm-Truncated Estimator}. Unlike element-wise truncation, our approach preserves the spectral geometry while requiring $O(Nd^2)$ operations, which represents the theoretical lower bound for constructing a full covariance matrix. Although spherical truncation is geometrically suboptimal for anisotropic data, we prove that within the Sub-Weibull class, the exponential tail decay compensates for this mismatch. Leveraging weighted Hanson-Wright inequalities, we derive non-asymptotic error bounds showing that our estimator recovers the optimal sub-Gaussian rate $\tilde{O}(\sqrt{r(Σ)/N})$ with high probability. This provides a scalable solution for high-dimensional data that exhibits tails heavier than Gaussian but lighter than polynomial decay.

We revisit the $(f,g)$-clustering problem that we introduced in a recent work [SODA'25], and which subsumes fundamental clustering problems such as $k$-Center, $k$-Median, Min-Sum of Radii, and Min-Load $k$-Clustering. This problem assigns each of the $k$ clusters a cost determined by the monotone, symmetric norm $f$ applied to the vector distances in the cluster, and aims at minimizing the norm $g$ applied to the vector of cluster costs. Previously, we focused on certain special cases for which we designed constant-factor approximation algorithms. Our bounds for more general settings left, however, large gaps to the known bounds for the basic problems they capture. In this work, we provide a clearer picture of the approximability of these more general settings. First, we design an $O(\log^2 n)$-approximation algorithm for $(f, L_{1})$-clustering for any $f$. This improves upon our previous $\widetilde{O}(\sqrt{n})$-approximation. Second, we provide an $O(k)$-approximation for the general $(f,g)$-clustering problem, which improves upon our previous $\widetilde{O}(\sqrt{kn})$-approximation algorithm and matches the best-known upper bound for Min-Load $k$-Clustering. We then design an approximation algorithm for $(f,g)$-clustering that interpolates, up to polylog factors, between the best known bounds for $k$-Center, $k$-Median, Min-Sum of Radii, Min-Load $k$-Clustering, (Top, $L_{1}$)-clustering, and $(L_{\infty},g)$-clustering based on a newly defined parameter of $f$ and $g$.

Motivated by recent work involving the analysis of leveraging spatial correlations in sparsified mean estimation, we present a novel procedure for constructing covariance estimator. The proposed Random-knots (Random-knots-Spatial) and B-spline (Bspline-Spatial) estimators of the covariance function are computationally efficient. Asymptotic pointwise of the covariance are obtained for sparsified individual trajectories under some regularity conditions. Our proposed nonparametric method well perform the functional principal components analysis for the case of sparsified data, where the number of repeated measurements available per subject is small. In contrast, classical functional data analysis requires a large number of regularly spaced measurements per subject. Model selection techniques, such as the Akaike information criterion, are used to choose the model dimension corresponding to the number of eigenfunctions in the model. Theoretical results are illustrated with Monte Carlo simulation experiments. Finally, we cluster multi-domain data by replacing the covariance function with our proposed covariance estimator during PCA.

Protein structure is central to biological function, and enabling multimodal protein models requires joint reasoning over sequence, structure, and function. A key barrier is the lack of principled protein structure tokenizers (PSTs): existing approaches fix token size or rely on continuous vector codebooks, limiting interpretability, multi-scale control, and transfer across architectures. We introduce GeoBPE, a geometry-grounded PST that transforms continuous, noisy, multi-scale backbone conformations into discrete ``sentences'' of geometry while enforcing global constraints. Analogous to byte-pair encoding, GeoBPE generates a hierarchical vocabulary of geometric primitives by iteratively (i) clustering Geo-Pair occurrences with k-medoids to yield a resolution-controllable vocabulary; (ii) quantizing each Geo-Pair to its closest medoid prototype; and (iii) reducing drift through differentiable inverse kinematics that optimizes boundary glue angles under an $\mathrm{SE}(3)$ end-frame loss. GeoBPE offers compression ($>$10x reduction in bits-per-residue at similar distortion rate), data efficiency ($>$10x less training data), and generalization (maintains test/train distortion ratio of $1.0-1.1$). It is architecture-agnostic: (a) its hierarchical vocabulary provides a strong inductive bias for coarsening residue-level embeddings from large PLMs into motif- and protein-level representations, consistently outperforming leading PSTs across $12$ tasks and $24$ test splits; (b) paired with a transformer, GeoBPE supports unconditional backbone generation via language modeling; and (c) tokens align with CATH functional families and support expert-interpretable case studies, offering functional meaning absent in prior PSTs. Code is available at https://github.com/shiningsunnyday/PT-BPE/.

While data scaling laws of large language models (LLMs) have been widely examined in the one-pass regime with massive corpora, their form under limited data and repeated epochs remains largely unexplored. This paper presents a theoretical analysis of how a common workaround, training for multiple epochs on the same dataset, reshapes the data scaling laws in linear regression. Concretely, we ask: to match the performance of training on a dataset of size $N$ for $K$ epochs, how much larger must a dataset be if the model is trained for only one pass? We quantify this using the \textit{effective reuse rate} of the data, $E(K, N)$, which we define as the multiplicative factor by which the dataset must grow under one-pass training to achieve the same test loss as $K$-epoch training. Our analysis precisely characterizes the scaling behavior of $E(K, N)$ for SGD in linear regression under either strong convexity or Zipf-distributed data: (1) When $K$ is small, we prove that $E(K, N) \approx K$, indicating that every new epoch yields a linear gain; (2) As $K$ increases, $E(K, N)$ plateaus at a problem-dependent value that grows with $N$ ($Θ(\log N)$ for the strongly-convex case), implying that larger datasets can be repeated more times before the marginal benefit vanishes. These theoretical findings point out a neglected factor in a recent empirical study (Muennighoff et al. (2023)), which claimed that training LLMs for up to $4$ epochs results in negligible loss differences compared to using fresh data at each step, \textit{i.e.}, $E(K, N) \approx K$ for $K \le 4$ in our notation. Supported by further empirical validation with LLMs, our results reveal that the maximum $K$ value for which $E(K, N) \approx K$ in fact depends on the data size and distribution, and underscore the need to explicitly model both factors in future studies of scaling laws with data reuse.

We develop a formal statistical framework for classical multidimensional scaling (CMDS) applied to noisy dissimilarity data. We establish distributional convergence results for the embeddings produced by CMDS for various noise models, which enable the construction of \emph{bona~fide} uniform confidence sets for the latent configuration, up to rigid transformations. We further propose bootstrap procedures for constructing these confidence sets and provide theoretical guarantees for their validity. We find that the multiplier bootstrap adapts automatically to heteroscedastic noise such as multiplicative noise, while the empirical bootstrap seems to require homoscedasticity. Either form of bootstrap, when valid, is shown to substantially improve finite-sample accuracy. The empirical performance of the proposed methods is demonstrated through numerical experiments.