We explore the effect of dimensionality on the nearest neighbor problem. We show that under a broad set of conditions (much broader than independent and identically distributed dimensions), as dimensionality increases, the distance to the nearest data point approaches the distance to the farthest data point. To provide a practical perspective, we present empirical results on both real and synthetic data sets that demonstrate that this effect can occur for as few as 10-15 dimensions. These results should not be interpreted to mean that high-dimensional indexing is never meaningful; we illustrate this point by identifying some high-dimensional workloads for which this effect does not occur. However, our results do emphasize that the methodology used almost universally in the database literature to evaluate high-dimensional indexing techniques is flawed, and should be modified. In particular, most such techniques proposed in the literature are not evaluated versus simple linear scan, and are evaluated over workloads for which nearest neighbor is not meaningful. Often, even the reported experiments, when analyzed carefully, show that linear scan would outperform the techniques being proposed on the workloads studied in high (10-15) dimensionality!.

When is nearest neighbor meaningful

Exploring and identifying structure is even more important for multivariate data than univariate data, given the difficulties in graphically presenting multivariate data and the comparative lack of parametric models to represent it. Unfortunately, such exploration is also inherently more difficult.

Multivariate Density Estimation

Studies in Fuzziness and Soft Computing

Theory of rings

New results on overlap and grouping functions

In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference.

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Fuzzy Implications: Some Recently Solved Problems

In this work, we discuss the law of α-migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to pexider-type generalisations of the law of importation, viz., the generalised law of importation I(C(α,x),y) = I(x,J(α,y)) (GLI) and the generalised cross-law of importation I(C(α,x),y)=J(x,I(α,y)) (CLI), where C is a generalised conjunction. We firstly show that the satisfaction of law of importation by the pairs (C,I) and/or (C,J) does not necessarily lead to the satisfaction of (GLI). Hence, we study the conditions under which these three laws may be related.

On alpha-Migrativity of Fuzzy Implications and the Generalised Laws of Importation

The compositional rule of inference (CRI) introduced by Zadeh is widely used in approximate reasoning schemes using fuzzy sets. In this work we show that the Bandler-Kohout subproduct does possess all the important properties such as equivalent and rea- sonable conditions for their solvability, their interpolative properties and the preservation of the indistinguishability that may be inherent in the input fuzzy sets, as cited in favor of using CRI. Keywords— Bandler-Kohout subproduct, compositional rule of inference, correctness and continuity of inference, fuzzy relation equations.

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Balasubramaniam Jayaram

Papers

Fuzzy Implications: Some Recently Solved Problems

On alpha-Migrativity of Fuzzy Implications and the Generalised Laws of Importation

The Bandler-Kohout Subproduct as a Suitable Inference Mechanism

Bijective transformations of fuzzy implications An algebraic perspective

On the Order-Compatibility of Fuzzy Logic Connectives on the Generated Clifford Poset