Shuheng Zhou

Shuheng Zhou

In this paper, we consider the problem of partitioning a small data sample of size $n$ drawn from a mixture of $2$ sub-gaussian distributions. Our work is motivated by the application of clustering individuals according to their population of origin using markers, when the divergence between the two populations is small. We are interested in the case that individual features are of low average quality $γ$, and we want to use as few of them as possible to correctly partition the sample. We consider semidefinite relaxation of an integer quadratic program which is formulated essentially as finding the maximum cut on a graph where edge weights in the cut represent dissimilarity scores between two nodes based on their features. A small simulation result in Blum, Coja-Oghlan, Frieze and Zhou (2007, 2009) shows that even when the sample size $n$ is small, by increasing $p$ so that $np= Ω(1/γ^2)$, one can classify a mixture of two product populations using the spectral method therein with success rate reaching an ``oracle'' curve. There the ``oracle'' was computed assuming that distributions were known, where success rate means the ratio between correctly classified individuals and the sample size $n$. In this work, we show the theoretical underpinning of this observed concentration of measure phenomenon in high dimensions, simultaneously for the semidefinite optimization goal and the spectral method, where the input is based on the gram matrix computed from centered data. We allow a full range of tradeoffs between the sample size and the number of features such that the product of these two is lower bounded by $1/{γ^2}$ so long as the number of features $p$ is lower bounded by $1/γ$.

Yan Hong, Kedong Xiu, Wei Li et al.

We study controlled post-training refusal suppression in routed MoE and hybrid-MoE foundation models, aiming to increase non-refusal target-response behavior while preserving general capability under a compact intervention footprint. Existing broad direction-based edits can perturb general-purpose computation, whereas support-only expert edits often lack sufficient capacity to correct heterogeneous refusal representations. To address this limitation, we introduce Localized Multidirectional Correction (LoMC), a support-gated intervention framework that follows a support-then-correction execution order: it first identifies a compact edit support, then aggregates prototype correction directions into layer-wise correction directions, and finally applies rank-one layer-wise correction only within the selected support. By using the edit support as a structural gating constraint, LoMC increases correction capacity without expanding the intervention scope. Experiments on text-only and multimodal safety benchmarks across four routed backbones show that LoMC substantially improves non-refusal target-response behavior while maintaining general capability under a compact intervention footprint.

Yan Hong, Wei Li, Kedong Xiu et al.

Knowledge injection updates pretrained MLLMs with new factual or domain-specific knowledge, but fitting full authoritative answers can cause drift in non-updated behavior. Online distillation mitigates this drift by training on model-generated rollouts, yet uniform reference-conditioned distillation provides coarse supervision: it can under-emphasize reference-supported rollout tokens and supervise omitted facts only indirectly. We introduce RoCo-ACE, a rollout-conditioned online distillation objective for knowledge injection. RoCo uses same-rollout reference-free/reference-conditioned likelihood contrast to reallocate additional distillation weight to reference-supported rollout tokens, while ACE adds sparse reference-side anchored correction for authoritative anchors omitted from the rollout without full-answer imitation. Across three knowledge-injection settings, six retention benchmarks, multiple baselines, and multiple base models, RoCo-ACE achieves the best injected-knowledge accuracy among compared methods while keeping evaluated retention close to the base model.

Shuheng Zhou

In this paper, we consider the problem of partitioning a small data sample of size $n$ drawn from a mixture of 2 sub-gaussian distributions in $\R^p$. We consider semidefinite programming relaxations of an integer quadratic program that is formulated as finding the maximum cut on a graph, where edge weights in the cut represent dissimilarity scores between two nodes based on their $p$ features. We are interested in the case that individual features are of low average quality $γ$, and we want to use as few of them as possible to correctly partition the sample. Denote by $Δ^2:=p γ$ the $\ell_2^2$ distance between two centers (mean vectors) in $\R^p$. The goal is to allow a full range of tradeoffs between $n, p, γ$ in the sense that partial recovery (success rate $< 100%$) is feasible once the signal to noise ratio $s^2 := \min{np γ^2, Δ^2}$ is lower bounded by a constant. For both balanced and unbalanced cases, we allow each population to have distinct covariance structures with diagonal matrices as special cases. In the present work, (a) we provide a unified framework for analyzing three computationally efficient algorithms, namely, SDP1, BalancedSDP, and Spectral clustering; and (b) we prove that the misclassification error decays exponentially with respect to the SNR $s^2$ for SDP1. Moreover, for balanced partitions, we design an estimator $\bf {BalancedSDP}$ with a superb debiasing property. Indeed, with this new estimator, we remove an assumption (A2) on bounding the trace difference between the two population covariance matrices while proving the exponential error bound as stated above. These estimators and their statistical analyses are novel to the best of our knowledge. We provide simulation evidence illuminating the theoretical predictions.

Roger Fan, Byoungwook Jang, Yuekai Sun et al.

Estimating conditional dependence graphs and precision matrices are some of the most common problems in modern statistics and machine learning. When data are fully observed, penalized maximum likelihood-type estimators have become standard tools for estimating graphical models under sparsity conditions. Extensions of these methods to more complex settings where data are contaminated with additive or multiplicative noise have been developed in recent years. In these settings, however, the relative performance of different methods is not well understood and algorithmic gaps still exist. In particular, in high-dimensional settings these methods require using non-positive semidefinite matrices as inputs, presenting novel optimization challenges. We develop an alternating direction method of multipliers (ADMM) algorithm for these problems, providing a feasible algorithm to estimate precision matrices with indefinite input and potentially nonconvex penalties. We compare this method with existing alternative solutions and empirically characterize the tradeoffs between them. Finally, we use this method to explore the networks among US senators estimated from voting records data.

Mark Rudelson, Shuheng Zhou

Suppose that we observe $y \in \mathbb{R}^n$ and $X \in \mathbb{R}^{n \times m}$ in the following errors-in-variables model: \begin{eqnarray*} y & = & X_0 β^* +ε\\ X & = & X_0 + W, \end{eqnarray*} where $X_0$ is an $n \times m$ design matrix with independent subgaussian row vectors, $ε\in \mathbb{R}^n$ is a noise vector and $W$ is a mean zero $n \times m$ random noise matrix with independent subgaussian column vectors, independent of $X_0$ and $ε$. This model is significantly different from those analyzed in the literature in the sense that we allow the measurement error for each covariate to be a dependent vector across its $n$ observations. Such error structures appear in the science literature when modeling the trial-to-trial fluctuations in response strength shared across a set of neurons. Under sparsity and restrictive eigenvalue type of conditions, we show that one is able to recover a sparse vector $β^* \in \mathbb{R}^m$ from the model given a single observation matrix $X$ and the response vector $y$. We establish consistency in estimating $β^*$ and obtain the rates of convergence in the $\ell_q$ norm, where $q = 1, 2$. We show error bounds which approach that of the regular Lasso and the Dantzig selector in case the errors in $W$ are tending to 0. We analyze the convergence rates of the gradient descent methods for solving the nonconvex programs and show that the composite gradient descent algorithm is guaranteed to converge at a geometric rate to a neighborhood of the global minimizers: the size of the neighborhood is bounded by the statistical error in the $\ell_2$ norm. Our analysis reveals interesting connections between computational and statistical efficiency and the concentration of measure phenomenon in random matrix theory. We provide simulation evidence illuminating the theoretical predictions.

Michael Hornstein, Roger Fan, Kerby Shedden et al.

It has been proposed that complex populations, such as those that arise in genomics studies, may exhibit dependencies among observations as well as among variables. This gives rise to the challenging problem of analyzing unreplicated high-dimensional data with unknown mean and dependence structures. Matrix-variate approaches that impose various forms of (inverse) covariance sparsity allow flexible dependence structures to be estimated, but cannot directly be applied when the mean and covariance matrices are estimated jointly. We present a practical method utilizing generalized least squares and penalized (inverse) covariance estimation to address this challenge. We establish consistency and obtain rates of convergence for estimating the mean parameters and covariance matrices. The advantages of our approaches are: (i) dependence graphs and covariance structures can be estimated in the presence of unknown mean structure, (ii) the mean structure becomes more efficiently estimated when accounting for the dependence structure among observations; and (iii) inferences about the mean parameters become correctly calibrated. We use simulation studies and analysis of genomic data from a twin study of ulcerative colitis to illustrate the statistical convergence and the performance of our methods in practical settings. Several lines of evidence show that the test statistics for differential gene expression produced by our methods are correctly calibrated and improve power over conventional methods.

Mark Rudelson, Shuheng Zhou

Suppose that we observe $y \in \mathbb{R}^f$ and $X \in \mathbb{R}^{f \times m}$ in the following errors-in-variables model: \begin{eqnarray*} y & = & X_0 β^* + ε\\ X & = & X_0 + W \end{eqnarray*} where $X_0$ is a $f \times m$ design matrix with independent subgaussian row vectors, $ε\in \mathbb{R}^f$ is a noise vector and $W$ is a mean zero $f \times m$ random noise matrix with independent subgaussian column vectors, independent of $X_0$ and $ε$. This model is significantly different from those analyzed in the literature in the sense that we allow the measurement error for each covariate to be a dependent vector across its $f$ observations. Such error structures appear in the science literature when modeling the trial-to-trial fluctuations in response strength shared across a set of neurons. Under sparsity and restrictive eigenvalue type of conditions, we show that one is able to recover a sparse vector $β^* \in \mathbb{R}^m$ from the model given a single observation matrix $X$ and the response vector $y$. We establish consistency in estimating $β^*$ and obtain the rates of convergence in the $\ell_q$ norm, where $q = 1, 2$ for the Lasso-type estimator, and for $q \in [1, 2]$ for a Dantzig-type conic programming estimator. We show error bounds which approach that of the regular Lasso and the Dantzig selector in case the errors in $W$ are tending to 0.

Theodoros Tsiligkaridis, Alfred O. Hero, Shuheng Zhou

This paper studies iteration convergence of Kronecker graphical lasso (KGLasso) algorithms for estimating the covariance of an i.i.d. Gaussian random sample under a sparse Kronecker-product covariance model and MSE convergence rates. The KGlasso model, originally called the transposable regularized covariance model by Allen ["Transposable regularized covariance models with an application to missing data imputation," Ann. Appl. Statist., vol. 4, no. 2, pp. 764-790, 2010], implements a pair of $\ell_1$ penalties on each Kronecker factor to enforce sparsity in the covariance estimator. The KGlasso algorithm generalizes Glasso, introduced by Yuan and Lin ["Model selection and estimation in the Gaussian graphical model," Biometrika, vol. 94, pp. 19-35, 2007] and Banerjee ["Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data," J. Mach. Learn. Res., vol. 9, pp. 485-516, Mar. 2008], to estimate covariances having Kronecker product form. It also generalizes the unpenalized ML flip-flop (FF) algorithm of Dutilleul ["The MLE algorithm for the matrix normal distribution," J. Statist. Comput. Simul., vol. 64, pp. 105-123, 1999] and Werner ["On estimation of covariance matrices with Kronecker product structure," IEEE Trans. Signal Process., vol. 56, no. 2, pp. 478-491, Feb. 2008] to estimation of sparse Kronecker factors. We establish that the KGlasso iterates converge pointwise to a local maximum of the penalized likelihood function. We derive high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. Our results establish that KGlasso has significantly faster asymptotic convergence than Glasso and FF. Simulations are presented that validate the results of our analysis.