Ming Yuan

Runshi Tang, Ming Yuan, Anru R. Zhang

This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two steps: Average Subspace Capture (ASC) and Alternating Projection (AP). These steps are specifically designed to capture the row-wise and column-wise dimension-reduced subspaces which contain the most informative features of the data. ASC utilizes a novel average projection operator as initialization and achieves exact recovery in the noiseless setting. We analyze the convergence and non-asymptotic error bounds of MOP-UP, introducing a blockwise matrix eigenvalue perturbation bound that proves the desired bound, where classic perturbation bounds fail. The effectiveness and practical merits of the proposed framework are demonstrated through experiments on both simulated and real datasets. Lastly, we discuss generalizations of our approach to higher-order data.

Sichao Wang, Ming Yuan, Chuang Zhang et al.

In V2X collaborative perception, the domain gaps between heterogeneous nodes pose a significant challenge for effective information fusion. Pose errors arising from latency and GPS localization noise further exacerbate the issue by leading to feature misalignment. To overcome these challenges, we propose V2X-DGPE, a high-accuracy and robust V2X feature-level collaborative perception framework. V2X-DGPE employs a Knowledge Distillation Framework and a Feature Compensation Module to learn domain-invariant representations from multi-source data, effectively reducing the feature distribution gap between vehicles and roadside infrastructure. Historical information is utilized to provide the model with a more comprehensive understanding of the current scene. Furthermore, a Collaborative Fusion Module leverages a heterogeneous self-attention mechanism to extract and integrate heterogeneous representations from vehicles and infrastructure. To address pose errors, V2X-DGPE introduces a deformable attention mechanism, enabling the model to adaptively focus on critical parts of the input features by dynamically offsetting sampling points. Extensive experiments on the real-world DAIR-V2X dataset demonstrate that the proposed method outperforms existing approaches, achieving state-of-the-art detection performance. The code is available at https://github.com/wangsch10/V2X-DGPE.

Weijing Tang, Ming Yuan, Zongqi Xia et al.

Latent space models are widely used for analyzing high-dimensional discrete data matrices, such as patient-feature matrices in electronic health records (EHRs), by capturing complex dependence structures through low-dimensional embeddings. However, estimation becomes challenging in the imbalanced regime, where one matrix dimension is much larger than the other. In EHR applications, cohort sizes are often limited by disease prevalence or data availability, whereas the feature space remains extremely large due to the breadth of medical coding system. Motivated by the increasing availability of external semantic embeddings, such as pre-trained embeddings of clinical concepts in EHRs, we propose a knowledge-embedded latent projection model that leverages semantic side information to regularize representation learning. Specifically, we model column embeddings as smooth functions of semantic embeddings via a mapping in a reproducing kernel Hilbert space. We develop a computationally efficient two-step estimation procedure that combines semantically guided subspace construction via kernel principal component analysis with scalable projected gradient descent. We establish estimation error bounds that characterize the trade-off between statistical error and approximation error induced by the kernel projection. Furthermore, we provide local convergence guarantees for our non-convex optimization procedure. Extensive simulation studies and a real-world EHR application demonstrate the effectiveness of the proposed method.

Arnab Auddy, Ming Yuan

We study spectral learning for orthogonally decomposable (odeco) tensors, emphasizing the interplay between statistical limits, optimization geometry, and initialization. Unlike matrices, recovery for odeco tensors does not hinge on eigengaps, yielding improved robustness under noise. While iterative methods such as tensor power iterations can be statistically efficient, initialization emerges as the main computational bottleneck. We investigate perturbation bounds, non-convex optimization analysis, and initialization strategies, clarifying when efficient algorithms attain statistical limits and when fundamental barriers remain.

Shuo Shuo Liu, Shikun Wang, Yuxuan Chen et al.

Background: Spatial transcriptomics have emerged as a powerful tool in biomedical research because of its ability to capture both the spatial contexts and abundance of the complete RNA transcript profile in organs of interest. However, limitations of the technology such as the relatively low resolution and comparatively insufficient sequencing depth make it difficult to reliably extract real biological signals from these data. To alleviate this challenge, we propose a novel transfer learning framework, referred to as TransST, to adaptively leverage the cell-labeled information from external sources in inferring cell-level heterogeneity of a target spatial transcriptomics data. Results: Applications in several real studies as well as a number of simulation settings show that our approach significantly improves existing techniques. For example, in the breast cancer study, TransST successfully identifies five biologically meaningful cell clusters, including the two subgroups of cancer in situ and invasive cancer; in addition, only TransST is able to separate the adipose tissues from the connective issues among all the studied methods. Conclusions: In summary, the proposed method TransST is both effective and robust in identifying cell subclusters and detecting corresponding driving biomarkers in spatial transcriptomics data.

Ming Yuan, Sichao Wang, Chuang Zhang et al.

The depth completion task is a critical problem in autonomous driving, involving the generation of dense depth maps from sparse depth maps and RGB images. Most existing methods employ a spatial propagation network to iteratively refine the depth map after obtaining an initial dense depth. In this paper, we propose DenseFormer, a novel method that integrates the diffusion model into the depth completion task. By incorporating the denoising mechanism of the diffusion model, DenseFormer generates the dense depth map by progressively refining an initial random depth distribution through multiple iterations. We propose a feature extraction module that leverages a feature pyramid structure, along with multi-layer deformable attention, to effectively extract and integrate features from sparse depth maps and RGB images, which serve as the guiding condition for the diffusion process. Additionally, this paper presents a depth refinement module that applies multi-step iterative refinement across various ranges to the dense depth results generated by the diffusion process. The module utilizes image features enriched with multi-scale information and sparse depth input to further enhance the accuracy of the predicted depth map. Extensive experiments on the KITTI outdoor scene dataset demonstrate that DenseFormer outperforms classical depth completion methods.

Hao Li, Chenming Wu, Ming Yuan et al.

Comprehensive testing of autonomous systems through simulation is essential to ensure the safety of autonomous driving vehicles. This requires the generation of safety-critical scenarios that extend beyond the limitations of real-world data collection, as many of these scenarios are rare or rarely encountered on public roads. However, evaluating most existing novel view synthesis (NVS) methods relies on sporadic sampling of image frames from the training data, comparing the rendered images with ground-truth images. Unfortunately, this evaluation protocol falls short of meeting the actual requirements in closed-loop simulations. Specifically, the true application demands the capability to render novel views that extend beyond the original trajectory (such as cross-lane views), which are challenging to capture in the real world. To address this, this paper presents a synthetic dataset for novel driving view synthesis evaluation, which is specifically designed for autonomous driving simulations. This unique dataset includes testing images captured by deviating from the training trajectory by $1-4$ meters. It comprises six sequences that cover various times and weather conditions. Each sequence contains $450$ training images, $120$ testing images, and their corresponding camera poses and intrinsic parameters. Leveraging this novel dataset, we establish the first realistic benchmark for evaluating existing NVS approaches under front-only and multicamera settings. The experimental findings underscore the significant gap in current approaches, revealing their inadequate ability to fulfill the demanding prerequisites of cross-lane or closed-loop simulation.

Shun Xu, Ming Yuan

In this note, we investigate how well we can reconstruct the best rank-$r$ approximation of a large matrix from a small number of its entries. We show that even if a data matrix is of full rank and cannot be approximated well by a low-rank matrix, its best low-rank approximations may still be reliably computed or estimated from a small number of its entries. This is especially relevant from a statistical viewpoint: the best low-rank approximations to a data matrix are often of more interest than itself because they capture the more stable and oftentimes more reproducible properties of an otherwise complicated data-generating model. In particular, we investigate two agnostic approaches: the first is based on spectral truncation; and the second is a projected gradient descent based optimization procedure. We argue that, while the first approach is intuitive and reasonably effective, the latter has far superior performance in general. We show that the error depends on how close the matrix is to being of low rank. Both theoretical and numerical evidence is presented to demonstrate the effectiveness of the proposed approaches.

Arnab Auddy, Ming Yuan

In this paper, we study the estimation of a rank-one spiked tensor in the presence of heavy tailed noise. Our results highlight some of the fundamental similarities and differences in the tradeoff between statistical and computational efficiencies under heavy tailed and Gaussian noise. In particular, we show that, for $p$ th order tensors, the tradeoff manifests in an identical fashion as the Gaussian case when the noise has finite $4(p-1)$ th moment. The difference in signal strength requirements, with or without computational constraints, for us to estimate the singular vectors at the optimal rate, interestingly, narrows for noise with heavier tails and vanishes when the noise only has finite fourth moment. Moreover, if the noise has less than fourth moment, tensor SVD, perhaps the most natural approach, is suboptimal even though it is computationally intractable. Our analysis exploits a close connection between estimating the rank-one spikes and the spectral norm of a random tensor with iid entries. In particular, we show that the order of the spectral norm of a random tensor can be precisely characterized by the moment of its entries, generalizing classical results for random matrices. In addition to the theoretical guarantees, we propose estimation procedures for the heavy tailed regime, which are easy to implement and efficient to run. Numerical experiments are presented to demonstrate their practical merits.

Ming Yuan, Vikas Kumar, Muhammad Aurangzeb Ahmad et al.

Fairness in AI and machine learning systems has become a fundamental problem in the accountability of AI systems. While the need for accountability of AI models is near ubiquitous, healthcare in particular is a challenging field where accountability of such systems takes upon additional importance, as decisions in healthcare can have life altering consequences. In this paper we present preliminary results on fairness in the context of classification parity in healthcare. We also present some exploratory methods to improve fairness and choosing appropriate classification algorithms in the context of healthcare.

Yuetian Luo, Garvesh Raskutti, Ming Yuan et al.

In this paper, we develop novel perturbation bounds for the high-order orthogonal iteration (HOOI) [DLDMV00b]. Under mild regularity conditions, we establish blockwise tensor perturbation bounds for HOOI with guarantees for both tensor reconstruction in Hilbert-Schmidt norm $\|\widehat{\bcT} - \bcT \|_{\tHS}$ and mode-$k$ singular subspace estimation in Schatten-$q$ norm $\| \sin Θ(\widehat{\U}_k, \U_k) \|_q$ for any $q \geq 1$. We show the upper bounds of mode-$k$ singular subspace estimation are unilateral and converge linearly to a quantity characterized by blockwise errors of the perturbation and signal strength. For the tensor reconstruction error bound, we express the bound through a simple quantity $ξ$, which depends only on perturbation and the multilinear rank of the underlying signal. Rate matching deterministic lower bound for tensor reconstruction, which demonstrates the optimality of HOOI, is also provided. Furthermore, we prove that one-step HOOI (i.e., HOOI with only a single iteration) is also optimal in terms of tensor reconstruction and can be used to lower the computational cost. The perturbation results are also extended to the case that only partial modes of $\bcT$ have low-rank structure. We support our theoretical results by extensive numerical studies. Finally, we apply the novel perturbation bounds of HOOI on two applications, tensor denoising and tensor co-clustering, from machine learning and statistics, which demonstrates the superiority of the new perturbation results.

Arnab Auddy, Ming Yuan

We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl, Davis, Kahan and Wedin. Our bounds demonstrate intriguing differences between matrices and higher-order tensors. Most notably, they indicate that for higher-order tensors perturbation affects each essential singular value/vector in isolation, and its effect on an essential singular vector does not depend on the multiplicity of its corresponding singular value or its distance from other singular values. Our results can be readily applied and provide a unified treatment to many different problems in statistics and machine learning involving spectral learning of higher-order orthogonally decomposable tensors. In particular, we illustrate the implications of our bounds in the context of high dimensional tensor SVD problem, and how it can be used to derive optimal rates of convergence for spectral learning.

Anru Zhang, Yuetian Luo, Garvesh Raskutti et al.

In this paper, we develop a novel procedure for low-rank tensor regression, namely \emph{\underline{I}mportance \underline{S}ketching \underline{L}ow-rank \underline{E}stimation for \underline{T}ensors} (ISLET). The central idea behind ISLET is \emph{importance sketching}, i.e., carefully designed sketches based on both the responses and low-dimensional structure of the parameter of interest. We show that the proposed method is sharply minimax optimal in terms of the mean-squared error under low-rank Tucker assumptions and under randomized Gaussian ensemble design. In addition, if a tensor is low-rank with group sparsity, our procedure also achieves minimax optimality. Further, we show through numerical study that ISLET achieves comparable or better mean-squared error performance to existing state-of-the-art methods while having substantial storage and run-time advantages including capabilities for parallel and distributed computing. In particular, our procedure performs reliable estimation with tensors of dimension $p = O(10^8)$ and is $1$ or $2$ orders of magnitude faster than baseline methods.

Tong Li, Ming Yuan

Nonparametric tests via kernel embedding of distributions have witnessed a great deal of practical successes in recent years. However, statistical properties of these tests are largely unknown beyond consistency against a fixed alternative. To fill in this void, we study here the asymptotic properties of goodness-of-fit, homogeneity and independence tests using Gaussian kernels, arguably the most popular and successful among such tests. Our results provide theoretical justifications for this common practice by showing that tests using Gaussian kernel with an appropriately chosen scaling parameter are minimax optimal against smooth alternatives in all three settings. In addition, our analysis also pinpoints the importance of choosing a diverging scaling parameter when using Gaussian kernels and suggests a data-driven choice of the scaling parameter that yields tests optimal, up to an iterated logarithmic factor, over a wide range of smooth alternatives. Numerical experiments are also presented to further demonstrate the practical merits of the methodology.

Dong Xia, Ming Yuan

We introduce a flexible framework for making inferences about general linear forms of a large matrix based on noisy observations of a subset of its entries. In particular, under mild regularity conditions, we develop a universal procedure to construct asymptotically normal estimators of its linear forms through double-sample debiasing and low-rank projection whenever an entry-wise consistent estimator of the matrix is available. These estimators allow us to subsequently construct confidence intervals for and test hypotheses about the linear forms. Our proposal was motivated by a careful perturbation analysis of the empirical singular spaces under the noisy matrix completion model which might be of independent interest. The practical merits of our proposed inference procedure are demonstrated on both simulated and real-world data examples.

Ronak Mehta, Hyunwoo J. Kim, Shulei Wang et al.

Recent results in coupled or temporal graphical models offer schemes for estimating the relationship structure between features when the data come from related (but distinct) longitudinal sources. A novel application of these ideas is for analyzing group-level differences, i.e., in identifying if trends of estimated objects (e.g., covariance or precision matrices) are different across disparate conditions (e.g., gender or disease). Often, poor effect sizes make detecting the differential signal over the full set of features difficult: for example, dependencies between only a subset of features may manifest differently across groups. In this work, we first give a parametric model for estimating trends in the space of SPD matrices as a function of one or more covariates. We then generalize scan statistics to graph structures, to search over distinct subsets of features (graph partitions) whose temporal dependency structure may show statistically significant group-wise differences. We theoretically analyze the Family Wise Error Rate (FWER) and bounds on Type 1 and Type 2 error. On a cohort of individuals with risk factors for Alzheimer's disease (but otherwise cognitively healthy), we find scientifically interesting group differences where the default analysis, i.e., models estimated on the full graph, do not survive reasonable significance thresholds.

Dong Xia, Ming Yuan, Cun-Hui Zhang

In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of noisy tensor completion. Much of the attention has been focused on the fundamental computational challenges often associated with problems involving higher order tensors, yet very little is known about their statistical performance. To fill in this void, in this article, we characterize the fundamental statistical limits of noisy tensor completion by establishing minimax optimal rates of convergence for estimating a $k$th order low rank tensor under the general $\ell_p$ ($1\le p\le 2$) norm which suggest significant room for improvement over the existing approaches. Furthermore, we propose a polynomial-time computable estimating procedure based upon power iteration and a second-order spectral initialization that achieves the optimal rates of convergence. Our method is fairly easy to implement and numerical experiments are presented to further demonstrate the practical merits of our estimator.

Dong Xia, Ming Yuan

In this paper, we investigate effective sketching schemes via sparsification for high dimensional multilinear arrays or tensors. More specifically, we propose a novel tensor sparsification algorithm that retains a subset of the entries of a tensor in a judicious way, and prove that it can attain a given level of approximation accuracy in terms of tensor spectral norm with a much smaller sample complexity when compared with existing approaches. In particular, we show that for a $k$th order $d\times\cdots\times d$ cubic tensor of {\it stable rank} $r_s$, the sample size requirement for achieving a relative error $\varepsilon$ is, up to a logarithmic factor, of the order $r_s^{1/2} d^{k/2} /\varepsilon$ when $\varepsilon$ is relatively large, and $r_s d /\varepsilon^2$ and essentially optimal when $\varepsilon$ is sufficiently small. It is especially noteworthy that the sample size requirement for achieving a high accuracy is of an order independent of $k$. To further demonstrate the utility of our techniques, we also study how higher order singular value decomposition (HOSVD) of large tensors can be efficiently approximated via sparsification.

Krishnakumar Balasubramanian, Tong Li, Ming Yuan

The reproducing kernel Hilbert space (RKHS) embedding of distributions offers a general and flexible framework for testing problems in arbitrary domains and has attracted considerable amount of attention in recent years. To gain insights into their operating characteristics, we study here the statistical performance of such approaches within a minimax framework. Focusing on the case of goodness-of-fit tests, our analyses show that a vanilla version of the kernel-embedding based test could be suboptimal, and suggest a simple remedy by moderating the embedding. We prove that the moderated approach provides optimal tests for a wide range of deviations from the null and can also be made adaptive over a large collection of interpolation spaces. Numerical experiments are presented to further demonstrate the merits of our approach.

Dong Xia, Ming Yuan

In this paper, we investigate the sample size requirement for exact recovery of a high order tensor of low rank from a subset of its entries. We show that a gradient descent algorithm with initial value obtained from a spectral method can, in particular, reconstruct a ${d\times d\times d}$ tensor of multilinear ranks $(r,r,r)$ with high probability from as few as $O(r^{7/2}d^{3/2}\log^{7/2}d+r^7d\log^6d)$ entries. In the case when the ranks $r=O(1)$, our sample size requirement matches those for nuclear norm minimization (Yuan and Zhang, 2016a), or alternating least squares assuming orthogonal decomposability (Jain and Oh, 2014). Unlike these earlier approaches, however, our method is efficient to compute, easy to implement, and does not impose extra structures on the tensor. Numerical results are presented to further demonstrate the merits of the proposed approach.

Han Chen, Garvesh Raskutti, Ming Yuan

In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polynomial-time algorithms they are often slow in practice. On the other hand, limited theoretical guarantees exist for non-convex methods. In this paper we provide a general framework that provides theoretical guarantees for learning high-dimensional tensor regression models under different low-rank structural assumptions using the projected gradient descent algorithm applied to a potentially non-convex constraint set $Θ$ in terms of its \emph{localized Gaussian width}. We juxtapose our theoretical results for non-convex projected gradient descent algorithms with previous results on regularized convex approaches. The two main differences between the convex and non-convex approach are: (i) from a computational perspective whether the non-convex projection operator is computable and whether the projection has desirable contraction properties and (ii) from a statistical upper bound perspective, the non-convex approach has a superior rate for a number of examples. We provide three concrete examples of low-dimensional structure which address these issues and explain the pros and cons for the non-convex and convex approaches. We supplement our theoretical results with simulations which show that, under several common settings of generalized low rank tensor regression, the projected gradient descent approach is superior both in terms of statistical error and run-time provided the step-sizes of the projected descent algorithm are suitably chosen.

Ming Yuan, Cun-Hui Zhang

In this paper, we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion. We introduce a class of tensor norms by allowing for different levels of coherence, which allows us to leverage the incoherence of a tensor. In particular, we show that a $k$th order tensor of rank $r$ and dimension $d\times\cdots\times d$ can be recovered perfectly from as few as $O((r^{(k-1)/2}d^{3/2}+r^{k-1}d)(\log(d))^2)$ uniformly sampled entries through an appropriate incoherent nuclear norm minimization. Our results demonstrate some key differences between completing a matrix and a higher order tensor: They not only point to potential room for improvement over the usual nuclear norm minimization but also highlight the importance of explicitly accounting for incoherence, when dealing with higher order tensors.

Ming Yuan, Ding-Xuan Zhou

We establish minimax optimal rates of convergence for estimation in a high dimensional additive model assuming that it is approximately sparse. Our results reveal an interesting phase transition behavior universal to this class of high dimensional problems. In the {\it sparse regime} when the components are sufficiently smooth or the dimensionality is sufficiently large, the optimal rates are identical to those for high dimensional linear regression, and therefore there is no additional cost to entertain a nonparametric model. Otherwise, in the so-called {\it smooth regime}, the rates coincide with the optimal rates for estimating a univariate function, and therefore they are immune to the "curse of dimensionality".

Luwan Zhang, Grace Wahba, Ming Yuan

Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the so-called regularized kernel estimate. We show that such an estimate can be characterized as simply applying a constant amount of shrinkage to all observed pairwise distances. This fact allows us to establish risk bounds for the estimate implying that the true distances can be estimated consistently in an average sense as the number of objects increases. In addition, such a characterization suggests an efficient algorithm to compute the distance matrix estimator, as an alternative to the usual second order cone programming known not to scale well for large problems. Numerical experiments and an application in visualizing the diversity of Vpu protein sequences from a recent HIV-1 study further demonstrate the practical merits of the proposed method.

T. Tony Cai, Ming Yuan

Motivated by a range of applications in engineering and genomics, we consider in this paper detection of very short signal segments in three settings: signals with known shape, arbitrary signals, and smooth signals. Optimal rates of detection are established for the three cases and rate-optimal detectors are constructed. The detectors are easily implementable and are based on scanning with linear and quadratic statistics. Our analysis reveals both similarities and differences in the strategy and fundamental difficulty of detection among these three settings.

Ming Yuan, Cun-Hui Zhang

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To overcome these difficulties, existing approaches often proceed by unfolding tensors into matrices and then apply techniques for matrix completion. We show here that such matricization fails to exploit the tensor structure and may lead to suboptimal procedure. More specifically, we investigate a convex optimization approach to tensor completion by directly minimizing a tensor nuclear norm and prove that this leads to an improved sample size requirement. To establish our results, we develop a series of algebraic and probabilistic techniques such as characterization of subdifferetial for tensor nuclear norm and concentration inequalities for tensor martingales, which may be of independent interests and could be useful in other tensor related problems.

Han Liu, Fang Han, Ming Yuan et al.

We propose a semiparametric approach, named nonparanormal skeptic, for estimating high dimensional undirected graphical models. In terms of modeling, we consider the nonparanormal family proposed by Liu et al (2009). In terms of estimation, we exploit nonparametric rank-based correlation coefficient estimators including the Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This result suggests that the nonparanormal graphical models are a safe replacement of the Gaussian graphical models, even when the data are Gaussian.

Han Liu, Fang Han, Ming Yuan et al.

In this paper, we propose a semiparametric approach, named nonparanormal skeptic, for efficiently and robustly estimating high dimensional undirected graphical models. To achieve modeling flexibility, we consider Gaussian Copula graphical models (or the nonparanormal) as proposed by Liu et al. (2009). To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This celebrating result suggests that the Gaussian copula graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare different estimators for their graph recovery performance under both ideal and noisy settings. The proposed methods are then applied on a large-scale genomic dataset to illustrate their empirical usefulness. The R language software package huge implementing the proposed methods is available on the Comprehensive R Archive Network: http://cran. r-project.org/.