The multivariate L1-median and associated data depth.

By Y. Vardi, Cun-Hui Zhang. Proceedings of the National Academy of Sciences of the United States of America, 2000

483 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/f6751c74e31b4ff675405a8136c05132ecce1f73

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Citations in the corpus

Uncertainty-Driven 6D Pose Estimation of Objects and Scenes from a Single RGB Image (2016, methodology): “To calculate g Y(c,pi) we use the iterative algorithm of [30].”
Geometric median in nearly linear time (2016, methodology): “There have been many proposed modifications to Weiszfeld’s algorithm [8, 25, 23, 3, 27, 16] that in general give non-asymptotic runtime guarantees.”
On data depth in infinite dimensional spaces (2014, methodology): “epth from Rd into inﬁnite dimensional spaces. Spatial depth of x ∈ Rd with respect to the probability distribution of a random vector X∈ Rd is deﬁned as SD(x) = 1 − ||E{(x− X)/||x− X||}|| (see, e.g., Vardi and Zhang (2000) and Serﬂing (2002)). It has been widely used for various statistical procedur”
Robust Covariance Matrix Estimation via Matrix Depth (2015, background): “Zuo and Serfling [52] advocated the notion of statistical depth function that satisfies the four properties in [29] and verified that Tukey’s depth indeed satisfies all these properties while many other depth functions [29, 37, 38, 44] do not.”
Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm (2011, methodology): “Table 1 presents basic statistics for the estimation errors (ﬁrst quartile Q 1 , median and third quartile Q 3 ), according to criterion (14), for the algorithm by Vardi and Zhang (2000) and our averaged procedure considering different values for c γ ∈ { 0.2, 0.6, 1, 2, 5, 10, 15, 25, 50, 75 .”
The spatial distribution in infinite dimensional spaces and related quantiles and depths (2014, methodology): “In the finite dimensional setup, the spatial distribution has been used to define the spatial depth (see [29] and [35]).”
Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation (2010, background): “Möttönen and Oja (1995) apply the function RS(x, F ) in hypothesis testing, and Vardi and Zhang (2000) treat the related spatial depth function, DS(x, F ) = 1 − ‖RS(x, F )‖, whose inverse OS(x, F ) = ‖RS(x, F )‖(= ‖u‖ relative to Equation (5)) may be interpreted as a spatial outlyingness function.”
Robust covariance and scatter matrix estimation under Huber’s contamination model (2015, background): “Zuo and Serfling [52] advocated the notion of statistical depth function that satisfies the four properties in [29] and verified that Tukey’s depth indeed satisfies all these properties while many other depth functions [29, 37, 38, 44] do not.”
Outlier Detection (2008, background): “The KSD is a generalization of the spatial depth [52, 10, 56].”
On Maximum Depth and Related Classifiers (2005, methodology): “• The notion of spatial depth (SPD) or L1 depth (Vardi & Zhang, 2000; Serfling, 2002) follows from the work of Chaudhuri (1996) and Koltchinskii (1997) on spatial quantiles.”
Deep Image Retrieval: Learning Global Representations for Image Search (2016, methodology): “In this case, the initial bounding box is the geometric median of all boxes(3), efficiently computed with [53].”
End-to-End Learning of Deep Visual Representations for Image Retrieval (2016, methodology): “In this case, the initial bounding box is the geometric median of all boxes, efficiently computed as in Vardi and Zhang (2004).”
Nonparametric Phase-II control charts for monitoring high-dimensional processes with unknown parameters (2020, methodology): “The computational algorithm is available in Vardi and Zhang (2000) and Serfling (2002).”