Centerless Clustering

Although lots of clustering models have been proposed recently, <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq1-3150981.gif"/></alternatives></inline-formula>-means and the family of spectral clustering methods are both still drawing a lot of attention due to their simplicity and efficacy. We first reviewed the unified framework of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq2-3150981.gif"/></alternatives></inline-formula>-means and graph cut models, and then proposed a clustering method called <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq3-3150981.gif"/></alternatives></inline-formula>-sums where a <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq4-3150981.gif"/></alternatives></inline-formula>-nearest neighbor (<inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq5-3150981.gif"/></alternatives></inline-formula>-NN) graph is adopted. The main idea of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq6-3150981.gif"/></alternatives></inline-formula>-sums is to minimize directly the sum of the distances between points in the same cluster. To deal with the situation where the graph is unavailable, we proposed <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq7-3150981.gif"/></alternatives></inline-formula>-sums-x that takes features as input. The computational and memory overhead of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq8-3150981.gif"/></alternatives></inline-formula>-sums are both <inline-formula><tex-math notation="LaTeX">$O(nk)$</tex-math><alternatives><mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="pei-ieq9-3150981.gif"/></alternatives></inline-formula>, indicating that it can scale linearly w.r. t. the number of objects to group. Moreover, the costs of computational and memory are Irrelevant to the product of the number of points and clusters. The computational and memory complexity of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq10-3150981.gif"/></alternatives></inline-formula>-sums-x are both linear w.r. t. the number of points. To validate the advantage of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq11-3150981.gif"/></alternatives></inline-formula>-sums and <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq12-3150981.gif"/></alternatives></inline-formula>-sums-x on facial datasets, extensive experiments have been conducted on 10 synthetic datasets and 17 benchmark datasets. While having a low time complexity, the performance of <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="pei-ieq13-3150981.gif"/></alternatives></inline-formula>-sums is comparable with several state-of-the-art clustering methods.