A review on minimax rates in change point detection and localisation

Yu, Yi

doi:10.48550/arxiv.2011.01857

Cited by 6 publications

(8 citation statements)

References 155 publications

(182 reference statements)

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“…These two types of control share many similarities but with some subtle differences: controlling the overall false alarm probability is more preferable in theoretical analysis and controlling the average run length is handier in practice. We refer readers to Yu et al (2020) for detailed discussions on this matter and stick with controlling the overall false alarm probability in this paper.…”

Section: Problem Setupmentioning

confidence: 99%

“…On top of the matrix estimation, we also have the change point analysis procedure. In 3, for notational simplicity, we present a rather expensive procedure exploiting all integer pairs s < t. In fact, without any loss of statistical accuracy, one can instead exploit a dyadic grid s ∈ {(t − 2 j ) ∨ 1} j∈N * , for any t ≥ 2, see Yu et al (2020) for detailed discussions. This means, for every newly collected time point t ≥ 2, the additional cost of estimating the change point is of order O{log(t)Q}, where Q is the cost of 1.…”

Section: The Change Point Detection Algorithmmentioning

confidence: 99%

“…Change point analysis on its own is an area with rich literature, and has been studied in univariate, multivariate, matrix, functional, manifold types of data, from both online (detecting change points while collecting data) and offline (estimating change points retrospectively with all data already available) perspectives, concerning testing and estimating goals. We refer to Yu (2020) and Aminikhanghahi and Cook (2017) for recent reviews on theoretical and com-putational change point analysis.…”

Section: Introductionmentioning

confidence: 99%

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Online network change point detection with missing values

Dubey¹,

Xu²,

Yu³

2021

Preprint

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In this paper we study online change point detection in dynamic networks with heterogeneous missing pattern across the networks and the time course. The missingness probabilities, the networks' entrywise sparsity, the rank of the networks and the jump size in terms of the Frobenius norm, are all allowed to vary as functions of the pre-change sample size. To the best of our knowledge, such general framework has not been rigorously studied before in the literature. We propose a polynomial-time change point detection algorithm, with a version of soft-impute algorithm (e.g. Mazumder et al. 2010;Klopp 2015) as the imputation sub-routine. We investigate the fundamental limits of this problem and show that the detection delay of our algorithm is nearly optimal, saving for logarithmic factors, in some low-rank regimes, with a pre-specified tolerance on the probability of false alarms. Extensive numerical experiments are conducted demonstrating the outstanding performances of our proposed method in practice.

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Section: Problem Setupmentioning

confidence: 99%

Section: The Change Point Detection Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Online network change point detection with missing values

Dubey¹,

Xu²,

Yu³

2021

Preprint

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“…When V = V * , we see that the localisation rate is, aside from a poly-logarithmic factor, of order σ 2 κ −2 p,n s. This type of guarantee is consistent with analogous results established in array of change point detection problems and implies the nearoptimality of our results. Indeed, in change point detection problems, minimax lower bounds on the localisation errors are usually of the form of σ 2 κ −2 and their upper bounds obtained by polynomialtime algorithms are usually in the form of σ 2 κ −2 × a sparsity parameter × a logarithmic factor, see Yu (2020).…”

Section: The Localisation Errormentioning

confidence: 99%

Denoising and change point localisation in piecewise-constant high-dimensional regression coefficients

Wang¹,

Padilla²,

Yu³

et al. 2021

Preprint

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We study the theoretical properties of the fused lasso procedure originally proposed by Tibshirani et al. (2005) in the context of a linear regression model in which the regression coefficient are totally ordered and assumed to be sparse and piecewise constant. Despite its popularity, to the best of our knowledge, estimation error bounds in high-dimensional settings have only been obtained for the simple case in which the design matrix is the identity matrix. We formulate a novel restricted isometry condition on the design matrix that is tailored to the fused lasso estimator and derive estimation bounds for both the constrained version of the fused lasso assuming dense coefficients and for its penalised version. We observe that the estimation error can be dominated by either the lasso or the fused lasso rate, depending on whether the number of non-zero coefficient is larger than the number of piece-wise constant segments. Finally, we devise a post-processing procedure to recover the piecewise-constant pattern of the coefficients. Extensive numerical experiments support our theoretical findings.

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“…There has been a vast body of literature discussing the detection boundary and optimal estimation in the offline change point analysis (e.g. Verzelen et al, 2020;Yu, 2020). Their counterparts in online change point analysis are relatively scarce and existing work includes univariate mean change (e.g.…”

Section: Summary Of Contributions and Related Literaturementioning

confidence: 99%

Locally private online change point detection

Berrett¹,

Yu²

2021

Preprint

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We study online change point detection problems under the constraint of local differential privacy (LDP) where, in particular, the statistician does not have access to the raw data. As a concrete problem, we study a multivariate nonparametric regression problem. At each time point t, the raw data are assumed to be of the form (X t , Y t ), where X t is a d-dimensional feature vector and Y t is a response variable. Our primary aim is to detect changes in the regression function m t (x) = E(Y t |X t = x) as soon as the change occurs. We provide algorithms which respect the LDP constraint, which control the false alarm probability, and which detect changes with a minimal (minimax rate-optimal) delay. To quantify the cost of privacy, we also present the optimal rate in the benchmark, non-private setting. These non-private results are also new to the literature and thus are interesting per se. In addition, we study the univariate mean online change point detection problem, under privacy constraints. This serves as the blueprint of studying more complicated private change point detection problems.

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A review on minimax rates in change point detection and localisation

Cited by 6 publications

References 155 publications

Online network change point detection with missing values

Online network change point detection with missing values

Denoising and change point localisation in piecewise-constant high-dimensional regression coefficients

Locally private online change point detection

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