# CFOF: A Concentration Free Measure for Anomaly Detection

## TL;DR

The paper introduces CFOF, a new outlier measure that remains effective in high dimensions, is theoretically immune to concentration effects, and is computationally efficient for large datasets.

## Contribution

It formalizes CFOF as a concentration-free outlier measure, provides its distribution in high dimensions, and introduces a scalable algorithm for outlier detection.

## Key findings

- CFOF does not concentrate in high dimensions.
- CFOF scores depend on squared norm and kurtosis.
- The fast-CFOF algorithm is efficient and scalable.

## Abstract

We present a novel notion of outlier, called the Concentration Free Outlier Factor, or CFOF. As a main contribution, we formalize the notion of concentration of outlier scores and theoretically prove that CFOF does not concentrate in the Euclidean space for any arbitrary large dimensionality. To the best of our knowledge, there are no other proposals of data analysis measures related to the Euclidean distance for which it has been provided theoretical evidence that they are immune to the concentration effect. We determine the closed form of the distribution of CFOF scores in arbitrarily large dimensionalities and show that the CFOF score of a point depends on its squared norm standard score and on the kurtosis of the data distribution, thus providing a clear and statistically founded characterization of this notion. Moreover, we leverage this closed form to provide evidence that the definition does not suffer of the hubness problem affecting other measures. We prove that the number of CFOF outliers coming from each cluster is proportional to cluster size and kurtosis, a property that we call semi-locality. We determine that semi-locality characterizes existing reverse nearest neighbor-based outlier definitions, thus clarifying the exact nature of their observed local behavior. We also formally prove that classical distance-based and density-based outliers concentrate both for bounded and unbounded sample sizes and for fixed and variable values of the neighborhood parameter. We introduce the fast-CFOF algorithm for detecting outliers in large high-dimensional dataset. The algorithm has linear cost, supports multi-resolution analysis, and is embarrassingly parallel. Experiments highlight that the technique is able to efficiently process huge datasets and to deal even with large values of the neighborhood parameter, to avoid concentration, and to obtain excellent accuracy.