# Task-Relevant Null-Space Residuals for Non-Injective Neural Mappings

arXiv: https://arxiv.org/abs/2609.37272 · HTML: https://mlsanity.com/paper/2609.37272/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 33 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Placed in

- Efficient LLM inference and serving

## Builds on (most cited first, 25 of 25)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [The Cityscapes Dataset for Semantic Urban Scene Understanding](https://mlsanity.com/paper/1604.01685/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Token Merging: Your ViT But Faster](https://mlsanity.com/paper/2210.09461/) (2022)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [A critical look at the evaluation of GNNs under heterophily: Are we really making progress?](https://mlsanity.com/paper/2302.11640/) (2023)
- [i-RevNet: Deep Invertible Networks](https://mlsanity.com/paper/1802.07088/) (2018)
- [Excessive Invariance Causes Adversarial Vulnerability](https://mlsanity.com/paper/1811.00401/) (2018)
- [Characterizing Graph Datasets for Node Classification: Homophily-Heterophily Dichotomy and Beyond](https://mlsanity.com/paper/2209.06177/) (2022)
- [Accelerating Transformers with Spectrum-Preserving Token Merging](https://mlsanity.com/paper/2405.16148/) (2024)
- [ALGM: Adaptive Local-then-Global Token Merging for Efficient Semantic Segmentation with Plain Vision Transformers](https://mlsanity.com/paper/2406.09936/) (2024)
- [Learning to Merge Tokens via Decoupled Embedding for Efficient Vision Transformers](https://mlsanity.com/paper/2412.10569/) (2024)
- [LiftPool: Bidirectional ConvNet Pooling](https://mlsanity.com/paper/2104.00996/) (2021)
- [MPM: Mutual Pair Merging for Efficient Vision Transformers](https://mlsanity.com/paper/2604.05718/) (2026)

## Abstract (arXiv)

Non-injective mappings in neural networks map distinct inputs to the same representation, thereby implicitly inducing equivalence relations in the input space. However, the input differences eliminated by these mappings may still be required by downstream tasks, creating a mismatch between operator-induced indistinguishability and task-required distinctions. For non-injective linear operators realized in the current forward pass, their null spaces exactly characterize these invisible input variations. We propose Task-Relevant Null-Space Residuals (NSR), a general residual framework for non-injective linear mappings. NSR combines null-space component extraction from pre-mapping representations, member-level encoding and gating, and application-specific integration to exploit potentially task-relevant information under downstream supervision while preserving the original aggregation or merging rules. We evaluate NSR in two structurally different settings: token merging and graph aggregation. In token merging, NSR achieves higher semantic segmentation performance than the corresponding compressed baselines in 34 out of 36 evaluated configurations, with a maximum observed gain of 31.51 mIoU points under strong compression. In graph aggregation, NSR achieves 100% training accuracy on Tree-NeighborsMatch at depths d=2--6 across three backbones, alongside gains on heterophilic node classification and molecular graph regression. Together, these results support null-space residuals as a practical complement to non-injective linear mappings, enabling downstream models to learn from input distinctions invisible in the original operator's output.