Second-order Democratic Aggregation

Tsung-Yu Lin, Subhransu Maji, Piotr Koniusz; Proceedings of the European Conference on Computer Vision (ECCV), 2018, pp. 620-636

Aggregated second-order features extracted from deep convolutional networks have been shown to be effective for texture generation, fine-grained recognition, material classification, and scene understanding. In this paper we study a class of orderless aggregation functions designed to minimize emph{interference} or equalize emph{contributions} in the context of second-order features and show that they can be computed just as efficiently as their first-order counterparts and have favorable properties over aggregation by summation. Another line of work has shown that matrix power normalization after aggregation can significantly improve the generalization of second-order representations. We show that matrix power normalization implicitly equalizes contributions during aggregation thus establishing a connection between matrix normalization techniques and prior work on minimizing interference. Based on the analysis we present $gamma$-democratic aggregators that interpolate between sum ($gamma$=1) and democratic pooling ($gamma$=0) outperforming both on several classification tasks. Moreover unlike power normalization the $gamma$-democratic aggregations can be computed in a low dimensional space using sketching allowing the use of very high-dimensional second-order features. This results in a state-of-the-art performance on several datasets.