# Geometric Mean Pooling for Equal-Weight Multiplicative Coarse-Graining

arXiv: https://arxiv.org/abs/2609.21876 · HTML: https://mlsanity.com/paper/2609.21876/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 56 references that are in the map.
- First posted on arXiv 18 Sep 2026.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Fine-tuning CNN Image Retrieval with No Human Annotation](https://mlsanity.com/paper/1711.02512/) (2017)
- [Analyzing Learned Molecular Representations for Property Prediction](https://mlsanity.com/paper/1904.01561/) (2019)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Primal-Dual Mesh Convolutional Neural Networks](https://mlsanity.com/paper/2010.12455/) (2020)
- [Learned-Norm Pooling for Deep Feedforward and Recurrent Neural Networks](https://mlsanity.com/paper/1311.1780/) (2013)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [Neural Arithmetic Units](https://mlsanity.com/paper/2001.05016/) (2020)
- [Learning scale-variant and scale-invariant features for deep image classification](https://mlsanity.com/paper/1602.01255/) (2016)
- [Attention-Aware Generalized Mean Pooling for Image Retrieval](https://mlsanity.com/paper/1811.00202/) (2018)
- [Geometric Convolutional Neural Network for Analyzing Surface-Based Neuroimaging Data](https://mlsanity.com/paper/1708.00587/) (2017)
- [Neural Power Units](https://mlsanity.com/paper/2006.01681/) (2020)
- [Generalised f-Mean Aggregation for Graph Neural Networks](https://mlsanity.com/paper/2306.13826/) (2023)
- [Group Generalized Mean Pooling for Vision Transformer](https://mlsanity.com/paper/2212.04114/) (2022)
- [Compact Neural Networks based on the Multiscale Entanglement Renormalization Ansatz](https://mlsanity.com/paper/1711.03357/) (2017)
- [Alpha-Integration Pooling for Convolutional Neural Networks](https://mlsanity.com/paper/1811.03436/) (2018)
- [Torch Geometric Pool: the PyTorch library for pooling in Graph Neural Networks](https://mlsanity.com/paper/2512.12642/) (2025)

## Abstract (arXiv)

As an alternative to the additive and extremal biases of average and max pooling, we introduce Geometric Mean Pooling (GMP), a signed pooling operator that combines the product of feature signs with the geometric mean of feature magnitudes. Motivated by local-to-global composition in quantum many-body physics, GMP retains both joint sign information and a characteristic multiplicative scale without introducing learnable pooling parameters. We show that non-overlapping hierarchical GMP preserves the corresponding global multiplicative statistic and evaluate it on synthetic sequence tasks, iterative coarse-graining, image classification, and molecular lipophilicity regression. On the synthetic tasks, GMP recovers product-based signals more accurately than average and max pooling and maintains predictive performance under the tested levels of multiplicative input noise. On image and molecular data, however, its effectiveness depends on the representation, target parameterization, and placement of local and global pooling. These results position GMP as a complementary, regime-dependent inductive bias for tasks in which equal-weight multiplicative composition is plausible, rather than as a universal replacement for standard pooling operators.