Fast PSD Matrix Estimation by Column Reductions

Abstract

Efficient evaluation of the similarity measures, e.g., correlations and kernels, among objects is one of the most important tasks required by major data mining techniques. However, the complexity of direct computations of the similarity among the n objects is at least O(n2) which is practically intractable for the large number n if the computations are expensive due to high dimensionality and/or highly complexs tructure of the objects. Moreover, direct similarity observations among all objects are often prohibitively expensive in some scientific fields. The objective of this paper is to propose techniques called "Column Reduction" and "Range Limited Column Reduction" to efficiently estimate the similarity measures among the objects by using the limited number of the directly computed and/or observed similarity measures. This technique effectively uses the property of the similarity matrix named "Positive Semi-Definiteness (PSD)." The superior performance of our approach in both effi-ciency and accuracy is demonstrated though the evaluation based on artificial and real world data sets.