# Second-order Democratic Aggregation

arXiv: https://arxiv.org/abs/1808.07503 · HTML: https://mlsanity.com/paper/1808.07503/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Aug 2018.

## Built on by (most cited first, 12 of 12)

- [Spectral Feature Augmentation for Graph Contrastive Learning and Beyond](https://mlsanity.com/paper/2212.01026/) (2022)
- [3Mformer: Multi-order Multi-mode Transformer for Skeletal Action Recognition](https://mlsanity.com/paper/2303.14474/) (2023)
- [Multi-level Second-order Few-shot Learning](https://mlsanity.com/paper/2201.05916/) (2022)
- [Traffic Forecasting on New Roads Using Spatial Contrastive Pre-Training (SCPT)](https://mlsanity.com/paper/2305.05237/) (2023)
- [Meet JEANIE: a Similarity Measure for 3D Skeleton Sequences via Temporal-Viewpoint Alignment](https://mlsanity.com/paper/2402.04599/) (2024)
- [Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2407.10484/) (2024)
- [How well does CLIP understand texture?](https://mlsanity.com/paper/2203.11449/) (2022)
- [Set-to-Sequence Methods in Machine Learning: a Review](https://mlsanity.com/paper/2103.09656/) (2021)
- [Learning Partial Correlation based Deep Visual Representation for Image Classification](https://mlsanity.com/paper/2304.11597/) (2023)
- [Learning Log-Determinant Divergences for Positive Definite Matrices](https://mlsanity.com/paper/2104.06461/) (2021)
- [Exploiting Field Dependencies for Learning on Categorical Data](https://mlsanity.com/paper/2307.09321/) (2023)
- [A Deeper Look into Second-Order Feature Aggregation for LiDAR Place Recognition](https://mlsanity.com/paper/2409.15919/) (2024)

## Abstract (arXiv)

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 interference or equalize contributions in the context of second-order features and we show that they can be computed just as efficiently as their first-order counterparts and they 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 γ-democratic aggregators that interpolate between sum (γ=1) and democratic pooling (γ=0) outperforming both on several classification tasks. Moreover, unlike power normalization, the γ-democratic aggregations can be computed in a low dimensional space by sketching that allows the use of very high-dimensional second-order features. This results in a state-of-the-art performance on several datasets.