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

And how this book will influence you to do better future? It will relate to how the readers will get the lessons that are coming. As known, commonly many people will believe that reading can be an entrance to enter the new perception. The perception will influence how you step you life. Even that is difficult enough; people with high sprit may not feel bored or give up realizing that concept. It's what fuzzy implications will give the thoughts for you.

Fuzzy Implications

https://www.iith.ac.in/~jbala/publications/fss/sn_r_imp_survey.pdf

(S, N)- and R-implications: A state-of-the-art survey

https://www.iith.ac.in/~jbala/publications/fss/fss-2007-a.pdf

On the characterizations of (S,N)-implications

The law of importation, given by the equivalence (x Lambda y) rarr z equiv (xrarr (y rarr z)), is a tautology in classical logic. In A-implications defined by Turksen et aL, the above equivalence is taken as an axiom. In this paper, we investigate the general form of the law of importation J(T(x, y), z) = J(x, J(y, z)), where T is a t-norm and J is a fuzzy implication, for the three main classes of fuzzy implications, i.e., R-, S- and QL-implications and also for the recently proposed Yager's classes of fuzzy implications, i.e., f- and g-implications. We give necessary and sufficient conditions under which the law of importation holds for R-, S-, f- and g-implications. In the case of QL-implications, we investigate some specific families of QL-implications. Also, we investigate the general form of the law of importation in the more general setting of uninorms and t-operators for the above classes of fuzzy implications. Following this, we propose a novel modified scheme of compositional rule of inference (CRI) inferencing called the hierarchical CRI, which has some advantages over the classical CRI. Following this, we give some sufficient conditions on the operators employed under which the inference obtained from the classical CRI and the hierarchical CRI become identical, highlighting the significant role played by the law of importation.

On the Law of Importation $(x \wedge y) \longrightarrow z \equiv (x \longrightarrow (y \longrightarrow z))$ in Fuzzy Logic

Fuzzy relational inference (FRI) systems form an important part of approximate reasoning schemes using fuzzy sets. The compositional rule of inference (CRI), which was introduced by Zadeh, has attracted the most attention so far. In this paper, we show that the FRI scheme that is based on the Bandler-Kohout (BK) subproduct, along with a suitable realization of the fuzzy rules, possesses all the important properties that are cited in favor of using CRI, viz., equivalent and reasonable conditions for their solvability, their interpolative properties, and the preservation of the indistinguishability that may be inherent in the input fuzzy sets. Moreover, we show that under certain conditions, the equivalence of first-infer-then-aggregate (FITA) and first-aggregate-then-infer (FATI) inference strategies can be shown for the BK subproduct, much like in the case of CRI. Finally, by addressing the computational complexity that may exist in the BK subproduct, we suggest a hierarchical inferencing scheme. Thus, this paper shows that the BK-subproduct-based FRI is as effective and efficient as the CRI itself.

/pdf/on-the-suitability-of-the-bandler-kohout-subproduct-as-an-20nmf8ibwe.pdf

Balasubramaniam Jayaram

Papers

Fuzzy Implications

(S, N)- and R-implications: A state-of-the-art survey

On the characterizations of (S,N)-implications

On the Law of Importation $(x \wedge y) \longrightarrow z \equiv (x \longrightarrow (y \longrightarrow z))$ in Fuzzy Logic

On the Suitability of the Bandler–Kohout Subproduct as an Inference Mechanism