# Training Networks in Null Space of Feature Covariance for Continual Learning

arXiv: https://arxiv.org/abs/2103.07113 · HTML: https://mlsanity.com/paper/2103.07113/

## Key facts

- Cited by 143 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 56 references that are in the map.
- First posted on arXiv 12 Mar 2021.

## Builds on (most cited first, 30 of 37)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Learning without Forgetting](https://mlsanity.com/paper/1606.09282/) (2016)
- [iCaRL: Incremental Classifier and Representation Learning](https://mlsanity.com/paper/1611.07725/) (2016)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Gradient Episodic Memory for Continual Learning](https://mlsanity.com/paper/1706.08840/) (2017)
- [Continual Learning with Deep Generative Replay](https://mlsanity.com/paper/1705.08690/) (2017)
- [Memory Aware Synapses: Learning what (not) to forget](https://mlsanity.com/paper/1711.09601/) (2017)
- [Efficient Lifelong Learning with A-GEM](https://mlsanity.com/paper/1812.00420/) (2018)
- [Riemannian Walk for Incremental Learning: Understanding Forgetting and Intransigence](https://mlsanity.com/paper/1801.10112/) (2018)
- [Large Scale Incremental Learning](https://mlsanity.com/paper/1905.13260/) (2019)
- [Lifelong Learning with Dynamically Expandable Networks](https://mlsanity.com/paper/1708.01547/) (2017)
- [Overcoming catastrophic forgetting with hard attention to the task](https://mlsanity.com/paper/1801.01423/) (2018)
- [Gradient based sample selection for online continual learning](https://mlsanity.com/paper/1903.08671/) (2019)
- [Variational Continual Learning](https://mlsanity.com/paper/1710.10628/) (2017)
- [Learn to Grow: A Continual Structure Learning Framework for Overcoming Catastrophic Forgetting](https://mlsanity.com/paper/1904.00310/) (2019)
- [Maintaining Discrimination and Fairness in Class Incremental Learning](https://mlsanity.com/paper/1911.07053/) (2019)
- [Learning without Memorizing](https://mlsanity.com/paper/1811.08051/) (2018)
- [Overcoming Catastrophic Forgetting by Incremental Moment Matching](https://mlsanity.com/paper/1703.08475/) (2017)
- [Re-evaluating Continual Learning Scenarios: A Categorization and Case for Strong Baselines](https://mlsanity.com/paper/1810.12488/) (2018)
- [Mnemonics Training: Multi-Class Incremental Learning without Forgetting](https://mlsanity.com/paper/2002.10211/) (2020)
- [Compacting, Picking and Growing for Unforgetting Continual Learning](https://mlsanity.com/paper/1910.06562/) (2019)
- [Learning to Remember: A Synaptic Plasticity Driven Framework for Continual Learning](https://mlsanity.com/paper/1904.03137/) (2019)
- [Continual Learning of Context-dependent Processing in Neural Networks](https://mlsanity.com/paper/1810.01256/) (2018)
- [FearNet: Brain-Inspired Model for Incremental Learning](https://mlsanity.com/paper/1711.10563/) (2017)

## Built on by (most cited first, 30 of 143)

- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [S-Prompts Learning with Pre-trained Transformers: An Occam's Razor for Domain Incremental Learning](https://mlsanity.com/paper/2207.12819/) (2022)
- [AlphaEdit: Null-Space Constrained Knowledge Editing for Language Models](https://mlsanity.com/paper/2410.02355/) (2024)
- [Class-Incremental Learning: A Survey](https://mlsanity.com/paper/2302.03648/) (2023)
- [Continual Learning of Natural Language Processing Tasks: A Survey](https://mlsanity.com/paper/2211.12701/) (2022)
- [Recent Advances of Continual Learning in Computer Vision: An Overview](https://mlsanity.com/paper/2109.11369/) (2021)
- [Mimicking the Oracle: An Initial Phase Decorrelation Approach for Class Incremental Learning](https://mlsanity.com/paper/2112.04731/) (2021)
- [AlphaSteer: Learning Refusal Steering with Principled Null-Space Constraint](https://mlsanity.com/paper/2506.07022/) (2025)
- [A Continual Deepfake Detection Benchmark: Dataset, Methods, and Essentials](https://mlsanity.com/paper/2205.05467/) (2022)
- [Dense Network Expansion for Class Incremental Learning](https://mlsanity.com/paper/2303.12696/) (2023)
- [Learning Federated Visual Prompt in Null Space for MRI Reconstruction](https://mlsanity.com/paper/2303.16181/) (2023)
- [SPEED: Scalable, Precise, and Efficient Concept Erasure for Diffusion Models](https://mlsanity.com/paper/2503.07392/) (2025)
- [The Ideal Continual Learner: An Agent That Never Forgets](https://mlsanity.com/paper/2305.00316/) (2023)
- [Towards Better Plasticity-Stability Trade-off in Incremental Learning: A Simple Linear Connector](https://mlsanity.com/paper/2110.07905/) (2021)
- [Continual Learning with Scaled Gradient Projection](https://mlsanity.com/paper/2302.01386/) (2023)
- [Visual Prompt Tuning in Null Space for Continual Learning](https://mlsanity.com/paper/2406.05658/) (2024)
- [Balancing Stability and Plasticity through Advanced Null Space in Continual Learning](https://mlsanity.com/paper/2207.12061/) (2022)
- [Continual Learning with Bayesian Model based on a Fixed Pre-trained Feature Extractor](https://mlsanity.com/paper/2204.13349/) (2022)
- [GradMA: A Gradient-Memory-based Accelerated Federated Learning with Alleviated Catastrophic Forgetting](https://mlsanity.com/paper/2302.14307/) (2023)
- [Machine Unlearning via Null Space Calibration](https://mlsanity.com/paper/2404.13588/) (2024)
- [HiDe-PET: Continual Learning via Hierarchical Decomposition of Parameter-Efficient Tuning](https://mlsanity.com/paper/2407.05229/) (2024)
- [Continual Multimodal Contrastive Learning](https://mlsanity.com/paper/2503.14963/) (2025)
- [Sparsity and Heterogeneous Dropout for Continual Learning in the Null Space of Neural Activations](https://mlsanity.com/paper/2203.06514/) (2022)
- [SLIM: Let LLM Learn More and Forget Less with Soft LoRA and Identity Mixture](https://mlsanity.com/paper/2410.07739/) (2024)
- [LoRA Subtraction for Drift-Resistant Space in Exemplar-Free Continual Learning](https://mlsanity.com/paper/2503.18985/) (2025)
- [Upweighting Easy Samples in Fine-Tuning Mitigates Forgetting](https://mlsanity.com/paper/2502.02797/) (2025)
- [Class-Incremental Learning with Cross-Space Clustering and Controlled Transfer](https://mlsanity.com/paper/2208.03767/) (2022)
- [Exploring Data Geometry for Continual Learning](https://mlsanity.com/paper/2304.03931/) (2023)
- [Continual All-in-One Adverse Weather Removal with Knowledge Replay on a Unified Network Structure](https://mlsanity.com/paper/2403.07292/) (2024)
- [Gated Integration of Low-Rank Adaptation for Continual Learning of Large Language Models](https://mlsanity.com/paper/2505.15424/) (2025)

## Abstract (arXiv)

In the setting of continual learning, a network is trained on a sequence of tasks, and suffers from catastrophic forgetting. To balance plasticity and stability of network in continual learning, in this paper, we propose a novel network training algorithm called Adam-NSCL, which sequentially optimizes network parameters in the null space of previous tasks. We first propose two mathematical conditions respectively for achieving network stability and plasticity in continual learning. Based on them, the network training for sequential tasks can be simply achieved by projecting the candidate parameter update into the approximate null space of all previous tasks in the network training process, where the candidate parameter update can be generated by Adam. The approximate null space can be derived by applying singular value decomposition to the uncentered covariance matrix of all input features of previous tasks for each linear layer. For efficiency, the uncentered covariance matrix can be incrementally computed after learning each task. We also empirically verify the rationality of the approximate null space at each linear layer. We apply our approach to training networks for continual learning on benchmark datasets of CIFAR-100 and TinyImageNet, and the results suggest that the proposed approach outperforms or matches the state-ot-the-art continual learning approaches.