On nonparametric estimation of density level sets

By Alexandre B. Tsybakov. 1997

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Citations in the corpus

Security analysis of online centroid anomaly detection (2010, methodology): “Hence several direct methods have been proposed for anomaly detection, e.g., oneclass SVM (Schölkopf et al., 2001), support vector data description (SVDD) (Tax and Duin, 1999a,b), and density level set estimation (Polonik, 1995; Tsybakov, 1997; Steinwart et al., 2005).”
To Trust Or Not To Trust A Classifier (2018, background): “The latter term matches the lower bound for level-set estimation up to log factors [39].”
Asymptotic normality of plug-in level set estimates. (2009, background): “Relevant published work (not yet mentioned above) include Hartigan (1987), Polonik (1995), Cavalier (1997), Tsybakov (1997), Walther (1997), Baíllo, Cuevas and Justel (2000), Baíllo, Cuestas-Albertos and Cuevas (2001), Cuevas, Febrero and Fraiman (2000), Baíllo (2003), Tsybakov (2004), Steinwart,…”
Theory of classification : a survey of some recent advances (2005, background): “(27) In the extreme case of the Mammen-Tsybakov noise condition, that is, when sup x |2η(x) − 1| ≥ h for some positive h, we have seen that the rate can be improved and that we essentially have, when g n is the empirical error minimizer, E (L(g n ) − L * ) ≤ C V n ∧ V log n nh , no matter what L * i”
PLUG‐IN ESTIMATION OF GENERAL LEVEL SETS (2006, background): “…of level sets L(c) = {x : f (x) ≥ c} (henceforth denoted {f ≥ c}), where f is an unknown density function on Rd and c > 0 a given constant has received some attention in the literature; see for example Hartigan (1987), Polonik (1995), Cuevas & Fraiman (1997), Tsybakov (1997), Walther (1997).”
Finding and Leveraging Structure in Learning Problems (2012, background): “See for example the papers [52, 53, 135, 155, 159, 160, 161, 172, 193, 200], and references therein.”
Asymptotic theory for density ridges (2014, background): “…the Hausdorff distance appear in estimating density level sets [Cuevas, González-Manteiga and Rodríguez-Casal (2006), Rinaldo and Wasserman (2010), Tsybakov (1997), Walther (1997)], curves [Cheng et al. (2005), Lee (2000)], filaments [Genovese et al. (2012a, 2014)] and manifolds [Genovese et al.…”
A tutorial on kernel density estimation and recent advances (2017, unknown): “Given a level λ, the λ-level set [80,81] is Lλ 1⁄4 fx : p x ð Þ λg:”
Topological Consistency via Kernel Estimation (2014, methodology): “s used for measuringeﬃciency). Various solutions have been proposed to the level set estimation problem. Standard solutions include the plug-in estimator [4, 5, 27, 51, 52], the excess mass estimator [42, 53, 54, 60, 63, 68], and the “naive” estimator [28, 32, 72]. The distance measure used to evalu”
Distribution Free Prediction Sets (2013, methodology): “The bottom right plot shows the estimator and prediction set based on the bandwidth whose corresponding conformal prediction set has the minimal Lebesgue measure.”
Kernel estimation of density level sets (2005, background): “Mainly, there are two families of su h estimators : the plug-in estimators and the estimators onstru ted by an ex ess mass approa h. Assume that an estimator fn of the density f is available.”
Total Error in a Plug-in Estimator of Level Sets (2003, unknown): “The statement Lebff = cg = 0 just means that f is not at at level c, an unavoidable condition in nonparametric estimation of level sets ff > cg (see, for example, Tsybakov 1997, Polonik 1995, 1997).”
A nonparametric approach to the estimation of lengths and surface areas (2007, background): “In a way, the length and surface area estimation problem can be seen as a further, more difficult, stage in set estimation theory, after the early developments concerned with the estimation of volume (associated with the L1 (measure) distance; see [8]), “visual” shape (associated with the Hausdorff ”
Estimating the Support of a High-Dimensional Distribution (2001, unknown): “Tsybakov (1997) has studied an estimator based on piecewise polynomial approximation of C.fi/ and has shown it attains the asymptotically minimax rate for certain classes of densities p. Polonik (1997) has studied the estimation of C.fi/ by C‘.fi/.”