# Learning from the Kernel and the Range Space

arXiv: https://arxiv.org/abs/1810.09071 · HTML: https://mlsanity.com/paper/1810.09071/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2018.

## Built on by (most cited first, 9 of 9)

- [ACIL: Analytic Class-Incremental Learning with Absolute Memorization and Privacy Protection](https://mlsanity.com/paper/2205.14922/) (2022)
- [AFL: A Single-Round Analytic Approach for Federated Learning with Pre-trained Models](https://mlsanity.com/paper/2405.16240/) (2024)
- [Towards Efficient and Exact Forgetting Services in Pre-Trained-Model-based Continual Learning](https://mlsanity.com/paper/2505.12239/) (2025)
- [AFCL: Analytic Federated Continual Learning for Spatio-Temporal Invariance of Non-IID Data](https://mlsanity.com/paper/2505.12245/) (2025)
- [Analytic Learning of Convolutional Neural Network For Pattern Recognition](https://mlsanity.com/paper/2202.06504/) (2022)
- [DeepAFL: Deep Analytic Federated Learning](https://mlsanity.com/paper/2603.00579/) (2026)
- [Analytic Personalized Federated Meta-Learning](https://mlsanity.com/paper/2502.06915/) (2025)
- [FedHiP: Heterogeneity-Invariant Personalized Federated Learning Through Closed-Form Solutions](https://mlsanity.com/paper/2508.04470/) (2025)
- [One Round Is All You Need: Analytic Federated Learning for Task-Heterogeneous Multi-Label Medical Image Classification](https://mlsanity.com/paper/2607.20641/) (2026)

## Abstract (arXiv)

In this article, a novel approach to learning a complex function which can be written as the system of linear equations is introduced. This learning is grounded upon the observation that solving the system of linear equations by a manipulation in the kernel and the range space boils down to an estimation based on the least squares error approximation. The learning approach is applied to learn a deep feedforward network with full weight connections. The numerical experiments on network learning of synthetic and benchmark data not only show feasibility of the proposed learning approach but also provide insights into the mechanism of data representation.