Modified Principal Points: A Flexible and Differentiable Approach for Data Summarization
Abstract
Principal points∗are a small set of characteristic locations which minimize the average squared Euclidian distance from the data points, and should be more informative about the data’s structure than simple features such as mean and variance. However it is also non-differentiable w.r.t point collapse due to minimum operation in (2.8) and weak-sparse in defining point spread penalty. In this paper we define the generalized principal points as the Gaussian weighted mean of distances. It results in a differentiable objective and has a tuning parameter for point closeness adjustment. When the bandwidth approaches to zero, modified points tend towards classical points from a real direction and when it tends to the infinity they define the mean. Finally, the simulation studies are reported and reveal higher robustness of our proposed methods against outliers, non-normality, and small sample sizes. Empirical studies on real statistical data also confirm that lower sensitivity to extremes is better
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