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Stacking Deep Set Networks and Pooling by Quantiles
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Proposes Quantile Pooling, a permutation-invariant pooling operator that interpolates between max and average pooling via a relaxed quantile density, and Stacked Deep Sets, a residual multi-layer deep-set architecture, evaluated on set regression, sorting, envelope nesting, mixture of Gaussians, and point cloud classification.
Problem
Pooling for set-structured data involves trade-offs: max pooling is robust but ignores subtle features, average pooling captures distribution but dilutes salient features; learned pooling is computationally expensive. The paper seeks an efficient pooling operator that balances both and a deeper deep
Approach
Quantile pooling aggregates a relaxed band of quantiles (q±ε, e.g., q=0.95, ε=0.05) of each feature dimension, spanning max pooling (q→1) to average pooling (wide band); combined with residual stacking of deep-set layers that concatenate the pooled feature with per-element features.
Key results
- On ModelNet40, Stacked Deep Sets with quantile pooling reaches 0.9071 ± 0.0022 accuracy, versus 0.9030 for max and 0.8997 for average pooling in Stacked Deep Sets.
- On sorting clusters, Stacked Deep Sets with average and quantile pooling surpass max pooling and Set Transformer; quantile pooling is slightly better than average (0.7419 vs 0.7336 in the mix-pooling table).
- On mixture of Gaussians, Set Transformer PMA attains -1.4867 log likelihood, with Stacked Deep Sets quantile pooling at -1.4937, ahead of max (-1.5186) and average (-1.5501).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use quantile pooling instead of max or average pooling in Stacked Deep Sets | Stacked Deep Sets · ModelNet40 | accuracy: 0.9030 ± 0.0035 -> 0.9071 ± 0.0022; 0.8997 ± 0.0012 -> 0.9071 ± 0.0022 | Helped |
| use quantile pooling instead of max or average pooling | Mixture of Gaussians · Mixture of Gaussians | log likelihood: -1.5186 -> -1.4937; -1.5501 -> -1.4937 | Mixed |
| combine quantile and average pooling (ablation) | pooling method · Sorting clusters | accuracy: 0.7419 ± 0.0034 -> 0.8089 ± 0.0049 | Helped |
| combine quantile and max pooling (ablation) | pooling method · Sorting clusters | accuracy: 0.7419 ± 0.0034 -> 0.7170 ± 0.0042 | Hurt |
| combine quantile and average pooling (ablation) | pooling method · Russian-dollable envelopes | accuracy: 0.4624 ± 0.0021 -> 0.4583 ± 0.0024 | Hurt |
| combine quantile and max pooling (ablation) | pooling method · Russian-dollable envelopes | accuracy: 0.4624 ± 0.0021 -> 0.4651 ± 0.0034 | Helped |
| combine quantile and average pooling (ablation) | pooling method · Point cloud classification | accuracy: 0.9369 ± 0.0012 -> 0.9338 ± 0.0023 | Hurt |
| combine quantile and max pooling (ablation) | pooling method · Point cloud classification | accuracy: 0.9369 ± 0.0012 -> 0.9366 ± 0.0005 | No clear effect |
| combine average and max pooling (ablation) | pooling method · Sorting clusters | accuracy: 0.7336 ± 0.0056 -> 0.7270 ± 0.0100 | No clear effect |
| combine average and max pooling (ablation) | pooling method · Russian-dollable envelopes | accuracy: 0.4527 ± 0.0021 -> 0.4593 ± 0.0028 | Helped |
| combine average and max pooling (ablation) | pooling method · Point cloud classification | accuracy: 0.9345 ± 0.0010 -> 0.9352 ± 0.0012 | Helped |
| learn the quantile level q instead of fixing it (ablation) | quantile pooling · Sorting clusters | accuracy | No clear effect |
Surprising findings
- Learning the quantile level q end-to-end gave values that barely moved from initialization, so the authors kept q fixed.
- Batch normalization was detrimental on the mixture of Gaussians task, and adding layers did not help there.
Limitations
- Quantile pooling with tuned q and epsilon is sensitive; performance approaches max pooling as q approaches 1 and sorting accuracy plunges.
- Set Transformer with PMA is best on sample mean regression and mixture of Gaussians, so the method does not dominate all tasks.
- Inference time may increase depending on the choice of k, and the authors note implementation enhancements could mitigate this.
Experimental setup
Datasets: ModelNet40 synthetic sample statistics synthetic sorting clusters synthetic Russian-dollable envelopes synthetic mixture of Gaussians
Benchmarks: ModelNet40 sample statistic regression sorting clusters Russian-dollable envelopes mixture of Gaussians
Metrics: accuracy mean squared error log likelihood inference time
Baselines: Deep Sets with max pooling Deep Sets with average pooling Set Transformer (PMA) PointNet++ PointMLP with max pooling
Builds on: Deep Sets Set Transformer PointMLP ResNet
Closest papers by meaning
- On the Representation Power of Set Pooling Networks (NeurIPS 2021 · similarity 0.84)
- TI-Pooling: Transformation-Invariant Pooling for Feature Learning in Convolutional Neural Networks (CVPR 2016 · similarity 0.84)
- Inductive Bias of Deep Convolutional Networks through Pooling Geometry (ICLR 2017 · similarity 0.84)
- Pooling by Sliced-Wasserstein Embedding (NeurIPS 2021 · similarity 0.83)
- FSPool: Learning Set Representations with Featurewise Sort Pooling (ICLR 2020 · similarity 0.83)
- Geometry-Aware Edge Pooling for Graph Neural Networks (NeurIPS 2025 · similarity 0.82)
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