Adaptive risk bounds in unimodal regression

Chatterjee, Sabyasachi; Lafferty, John

doi:10.3150/16-bej922

Cited by 24 publications

(24 citation statements)

References 36 publications

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“…This bound was also shown to be optimal in a minimax sense [CGS15b,BT15]. Unlike its monotone counterpart, unimodal regression where θ * ∈ U has received sporadic attention [SZ01, KBI14,CL15]. This state of affairs is all the more surprising given that unimodal density estimation has been the subject of much more research [BF96, Bir97, EL00, DDS12, DDS + 13, TG14].…”

Section: Related Workmentioning

confidence: 99%

“…It was recently shown in [CL15] that the LS estimator also adapts to V (θ * ) and k(θ * ) for unimodal regression:…”

Section: Related Workmentioning

confidence: 99%

“…with probability at least 1 − n −α for some α > 0. The exponent 3/2 in the second term was improved to 1 in the new version of [CL15] after the first version of our current paper was posted. Note that the exponents in (2.6) are different from the isotonic case.…”

Section: Related Workmentioning

confidence: 99%

“…Note that the first term is related to the Gaussian width (see, e.g., [CRPW12]) of Θ defined by IE[sup θ∈Θ θ, z ], whose connection to the statistical dimension was studied in [ALMT14]. The variational formula was first proposed for convex regression [Cha14], and later exploited in several different settings, including matrix estimation with shape constraints [CGS15a] and unimodal regression [CL15]. Similar ideas have appeared in other works, for example, analysis of empirical risk minimization [Men15], ranking from pairwise comparison [SBGW15] and isotonic regression [Bel15].…”

Section: Appendix A: Proof Of the Upper Boundsmentioning

confidence: 99%

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Optimal rates of statistical seriation

Flammarion¹,

Mao²,

Rigollet³

2019

Bernoulli

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Given a matrix, the seriation problem consists in permuting its rows in such way that all its columns have the same shape, for example, they are monotone increasing. We propose a statistical approach to this problem where the matrix of interest is observed with noise and study the corresponding minimax rate of estimation of the matrices. Specifically, when the columns are either unimodal or monotone, we show that the least squares estimator is optimal up to logarithmic factors and adapts to matrices with a certain natural structure. Finally, we propose a computationally efficient estimator in the monotonic case and study its performance both theoretically and experimentally. Our work is at the intersection of shape constrained estimation and recent work that involves permutation learning, such as graph denoising and ranking.

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Section: Related Workmentioning

confidence: 99%

“…It was recently shown in [CL15] that the LS estimator also adapts to V (θ * ) and k(θ * ) for unimodal regression:…”

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Appendix A: Proof Of the Upper Boundsmentioning

confidence: 99%

See 2 more Smart Citations

Optimal rates of statistical seriation

Flammarion¹,

Mao²,

Rigollet³

2019

Bernoulli

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show abstract

“…has not yet been studies in the literature. Next we follow Chatterjee and Lafferty (2015), which establishes the adaptive risk of the unimodal least square estimator, to obtain the asymptotic property of the shape constrained estimatorθ kl from Step 2. Finally, we show that the error of the fused and shape constrained estimator is no greater than that of the shape constrained estimator using the property of our hard thresholding fusion operator.…”

Section: Theorymentioning

confidence: 99%

Mixed-Effect Time-Varying Network Model and Application in Brain Connectivity Analysis

Zhang

Sun

2019

Journal of the American Statistical Association

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Time-varying networks are fast emerging in a wide range of scientific and business disciplines. Most existing dynamic network models are limited to a single-subject and discrete-time setting. In this article, we propose a mixed-effect multi-subject continuous-time stochastic blockmodel that characterizes the time-varying behavior of the network at the population level, meanwhile taking into account individual subject variability. We develop a multi-step optimization procedure for a constrained stochastic blockmodel estimation, and derive the asymptotic property of the estimator. We demonstrate the effectiveness of our method through both simulations and an application to a study of brain development in youth.KEY WORDS: brain connectivity analysis; fused lasso; generalized linear mixed-effect model; stochastic blockmodel; time-varying network IntroductionThe study of networks has recently attracted enormous attention, as they provide a natural characterization of many complex social, physical and biological systems. A variety 1 arXiv:1806.03829v1 [stat.ME] ; Zhao et al., 2017, among many others). See also Kolaczyk (2009) for a review. To date, much research, however, has focused on static networks, where a single snapshot of the network is observed and modeled. Dynamic networks, where the data consist of a sequence of snapshots of the network that evolves over time, are fast emerging. Examples include brain connectivity networks, gene regulatory networks, protein signaling networks, and terrorist networks. Modeling of dynamic networks has appeared only recently (Xu and Hero, 2014;Matias and Miele, 2017;Pensky, 2016;Zhang and Cao, 2017). Most existing dynamic network models, however, considered a discrete-time and single-subject setting, in which the network observed at multiple time points is based upon the same study subject; the snapshots of the dynamic network are observed on a finite and typically small number of discrete time points. Methodology for modeling dynamic networks in a multi-subject and continuous-time setting remains largely missing. In this article, we develop a new network Stochastic blockmodels have been intensively studied in the networks literature (Nowicki and Snijders, 2001;Bickel and Chen, 2009;Rohe et al., 2011;Zhao et al., 2012), and there have been some recent studies of dynamic stochastic blockmodels (Xu and Hero, 2014;Matias and Miele, 2017;Zhang and Cao, 2017). However, the existing models assume a single-subject and discrete-time setting, and thus are not directly applicable to our data problem. In contrast, by introducing a mixed-effect term in the stochastic blockmodel, our new proposal adapts to the multi-subject setting. It both characterizes the time-varying behavior of the network at the population level, and accounts for subject-specific variability. Moreover, by modeling the connecting probabilities as functions of the time variable, our method works for the continuous-time setting. In addition, we introduce a shape constraint and a fusion constraint to further regularize an...

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Statistical and Computational Efficiency for Smooth Tensor Estimation with Unknown Permutations

Lee,

Wang

2024

Journal of the American Statistical Association

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Adaptive risk bounds in unimodal regression

Cited by 24 publications

References 36 publications

Optimal rates of statistical seriation

Optimal rates of statistical seriation

Mixed-Effect Time-Varying Network Model and Application in Brain Connectivity Analysis

Statistical and Computational Efficiency for Smooth Tensor Estimation with Unknown Permutations

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