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45
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
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Highdimensional covariance estimation by minimizing ℓ1penalized logdeterminant divergence
, 2008
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Highdimensional covariance estimation by minimizing 1penalized logdeterminant divergence
 Electron. J. Stat
, 2011
"... Given i.i.d. observations of a random vector X ∈ Rp, we study the problem of estimating both its covariance matrix Σ∗, and its inverse covariance or concentration matrix Θ ∗ = (Σ∗)−1. When X is multivariate Gaussian, the nonzero structure of Θ ∗ is specified by the graph of an associated Gaussian ..."
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Cited by 54 (6 self)
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Given i.i.d. observations of a random vector X ∈ Rp, we study the problem of estimating both its covariance matrix Σ∗, and its inverse covariance or concentration matrix Θ ∗ = (Σ∗)−1. When X is multivariate Gaussian, the nonzero structure of Θ ∗ is specified by the graph of an associated Gaussian Markov random field; and a popular estimator for such sparse Θ ∗ is the `1regularized Gaussian MLE. This estimator is sensible even for for nonGaussian X, since it corresponds to minimizing an `1penalized logdeterminant Bregman divergence. We analyze its performance under highdimensional scaling, in which the number of nodes in the graph p, the number of edges s, and the maximum node degree d, are allowed to grow as a function of the sample size n. In addition to the parameters (p, s, d), our analysis identifies other key quantities that control rates: (a) the `∞operator norm of the true covariance matrix Σ∗; and (b) the ` ∞ operator norm of the submatrix Γ∗SS, where S indexes the graph edges, and Γ ∗ = (Θ∗)−1 ⊗ (Θ∗)−1; and (c) a mutual incoherence or irrepresentability measure on the matrix Γ ∗ and (d) the rate of decay 1/f(n, δ) on the probabilities {Σ̂nij − Σ∗ij > δ}, where Σ̂n is the sample covariance based on n samples. Our first result establishes consistency of our estimate Θ ̂ in the elementwise maximumnorm. This in turn allows us to derive convergence rates in Frobenius and spectral norms, with improvements upon existing results for graphs with maximum node degrees d = o( s). In our second result, we show that with probability converging to one, the estimate Θ ̂ correctly speci
A constrained ℓ1minimization approach to sparse precision matrix estimation
 J. Amer. Statist. Assoc
, 2011
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GraphValued Regression
"... Undirected graphical models encode in a graph G the dependency structure of a random vector Y. In many applications, it is of interest to model Y given another random vector X as input. We refer to the problem of estimating the graph G(x) of Y conditioned on X = x as “graphvalued regression”. In th ..."
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Cited by 13 (4 self)
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Undirected graphical models encode in a graph G the dependency structure of a random vector Y. In many applications, it is of interest to model Y given another random vector X as input. We refer to the problem of estimating the graph G(x) of Y conditioned on X = x as “graphvalued regression”. In this paper, we propose a semiparametric method for estimating G(x) that builds a tree on the X space just as in CART (classification and regression trees), but at each leaf of the tree estimates a graph. We call the method “Graphoptimized CART”, or GoCART. We study the theoretical properties of GoCART using dyadic partitioning trees, establishing oracle inequalities on risk minimization and tree partition consistency. We also demonstrate the application of GoCART to a meteorological dataset, showing how graphvalued regression can provide a useful tool for analyzing complex data. 1
Sparse methods for biomedical data
 SIGKDD Explor. Newsl
, 2012
"... Following recent technological revolutions, the investigation of massive biomedical data with growing scale, diversity, and complexity has taken a center stage in modern data analysis. Although complex, the underlying representations of many biomedical data are often sparse. For example, for a certa ..."
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Cited by 11 (2 self)
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Following recent technological revolutions, the investigation of massive biomedical data with growing scale, diversity, and complexity has taken a center stage in modern data analysis. Although complex, the underlying representations of many biomedical data are often sparse. For example, for a certain disease such as leukemia, even though humans have tens of thousands of genes, only a few genes are relevant to the disease; a gene network is sparse since a regulatory pathway involves only a small number of genes; many biomedical signals are sparse or compressible in the sense that they have concise representations when expressed in a proper basis. Therefore, finding sparse representations is fundamentally important for scientific discovery. Sparse methods based on the ℓ1 norm have attracted a great amount of research efforts in the past decade due to its sparsityinducing property, convenient convexity, and strong theoretical guarantees. They have achieved great success in various applications such as biomarker selection, biological network construction, and magnetic resonance imaging. In this paper, we review stateoftheart sparse methods and their applications to biomedical data.
ContinuousTime Regression Models for Longitudinal Networks
"... The development of statistical models for continuoustime longitudinal network data is of increasing interest in machine learning and social science. Leveraging ideas from survival and event history analysis, we introduce a continuoustime regression modeling framework for network event data that ca ..."
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Cited by 10 (2 self)
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The development of statistical models for continuoustime longitudinal network data is of increasing interest in machine learning and social science. Leveraging ideas from survival and event history analysis, we introduce a continuoustime regression modeling framework for network event data that can incorporate both timedependent network statistics and timevarying regression coefficients. We also develop an efficient inference scheme that allows our approach to scale to large networks. On synthetic and realworld data, empirical results demonstrate that the proposed inference approach can accurately estimate the coefficients of the regression model, which is useful for interpreting the evolution of the network; furthermore, the learned model has systematically better predictive performance compared to standard baseline methods. 1
Differential Privacy with Compression
"... Abstract—This work studies formal utility and privacy guarantees for a simple multiplicative database transformation, where the data are compressed by a random linear or affine transformation, reducing the number of data records substantially, while preserving the number of original input variables. ..."
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Cited by 9 (0 self)
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Abstract—This work studies formal utility and privacy guarantees for a simple multiplicative database transformation, where the data are compressed by a random linear or affine transformation, reducing the number of data records substantially, while preserving the number of original input variables. We provide an analysis framework inspired by a recent concept known as differential privacy. Our goal is to show that, despite the general difficulty of achieving the differential privacy guarantee, it is possible to publish synthetic data that are useful for a number of common statistical learning applications. This includes high dimensional sparse regression [24], principal component analysis (PCA), and other statistical measures [16] based on the covariance of the initial data. I.
Sparse Group Lasso: Consistency and Climate Applications
"... The design of statistical predictive models for climate data gives rise to some unique challenges due to the high dimensionality and spatiotemporal nature of the datasets, which dictate that models should exhibit parsimony in variable selection. Recently, a class of methods which promote structured ..."
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Cited by 6 (2 self)
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The design of statistical predictive models for climate data gives rise to some unique challenges due to the high dimensionality and spatiotemporal nature of the datasets, which dictate that models should exhibit parsimony in variable selection. Recently, a class of methods which promote structured sparsity in the model have been developed, which is suitable for this task. In this paper, we prove theoretical statistical consistency of estimators with treestructured norm regularizers. We consider one particular model, the Sparse Group Lasso (SGL), to construct predictors of land climate using ocean climate variables. Our experimental results demonstrate that the SGL model provides better predictive performance than the current stateoftheart, remains climatologically interpretable, and is robust in its variable selection.