The Concentration of Fractional Distances

By Damien François, Vincent Wertz, Michel Verleysen. IEEE Transactions on Knowledge and Data Engineering, 2007

324 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/9397f53b1b136ad6f8767601d4184426fe1062bf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Computer Vision and Pattern Recognition

In Signal Processing

Citations in the corpus

Type 1 and 2 mixtures of Kullback-Leibler divergences as cost functions in dimensionality reduction based on similarity preservation (2013, background): “Due to the presence of cross-entropies, it can easily be seen that DkKLt1ðri99siÞ ranges from 0 to 1.”
Distance geometry and data science (2019, background): “that smallest values of plead to more stable norms; in particular, quasinorms with 0 <p<1 are considered. Some counterexamples are given against a generalization of this claim for quasinorms in [73]. In [64], the converse of Thm. 8.1 is proved, namely that Eq. (48) follows from Eq. (49): from ”
Shift-invariant similarities circumvent distance concentration in stochastic neighbor embedding and variants (2011, background): “The second curve (solid black line) corresponds to the complementary CDF (CCDF) of this distribution, which is arguably the optimal similarity in terms of contrast for the considered distance distribution [30, 25].”
A survey on unsupervised outlier detection in high‐dimensional numerical data (2012, background): “12, 25; however, Durrant and Kab´an [57] show that a low intrinsic dimensionality alone does not sufﬁce to avoid the concentration, but essentially the correlation between attributes needs to be strong enough.”
Hubs in Space: Popular Nearest Neighbors in High-Dimensional Data (2010, methodology): “On the other hand, for lp-norm distances with p> 2, the standard deviation would tend to 0 (François et al., 2007).”
Minkowski metric, feature weighting and anomalous cluster initializing in K-Means clustering (2012, methodology): “Rudin [20] and Francois et al. [21] point to the differences in effects of data concentration at different p and possibilities of utilizing them for effectively tackling data analysis problems; the latter paper cites a number of useful mathematical properties of Minkowski metric.”