Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Independent component analysis by general nonlinear Hebbian-like learning rules
Signal Processing - Special issue on neural networks
On the geometry of similarity search: dimensionality curse and concentration of measure
Information Processing Letters
An elementary proof of a theorem of Johnson and Lindenstrauss
Random Structures & Algorithms
When Is ''Nearest Neighbor'' Meaningful?
ICDT '99 Proceedings of the 7th International Conference on Database Theory
Learning Mixtures of Gaussians
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
The Concentration of Fractional Distances
IEEE Transactions on Knowledge and Data Engineering
On the Design and Applicability of Distance Functions in High-Dimensional Data Space
IEEE Transactions on Knowledge and Data Engineering
When is 'nearest neighbour' meaningful: A converse theorem and implications
Journal of Complexity
New instability results for high-dimensional nearest neighbor search
Information Processing Letters
Compressed fisher linear discriminant analysis: classification of randomly projected data
Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining
Non-parametric detection of meaningless distances in high dimensional data
Statistics and Computing
A survey on unsupervised outlier detection in high-dimensional numerical data
Statistical Analysis and Data Mining
| Hi-index | 0.01 |
We make a first investigation into a recently raised concern about the suitability of existing data analysis techniques when faced with the counter-intuitive properties of high dimensional data spaces, such as the phenomenon of distance concentration. Under the structural assumption of a generic linear model with a latent variable and an additive unstructured noise, we find that dimension reduction that explicitly guards against distance concentration recovers the well-known techniques of Fisher's linear discriminant analysis, Fisher's discriminant ratio and a variant of projection pursuit. Extrapolation to regression uncovers a close link to sure independence screening, which is a recently proposed technique for variable selection in ultra-high dimensional feature spaces. Hence, these techniques may be seen as distance concentration aware, despite they have not been explicitly designed to have this property. Throughout our analysis, other than the dependency structure implied by the mentioned linear model, we make no assumptions about the distributions of the variables involved.