# The Weight Gram Matrix Captures Sequential Feature Linearization in Deep Networks

arXiv: https://arxiv.org/abs/2605.06258 · HTML: https://mlsanity.com/paper/2605.06258/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 52 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 30 of 35)

- [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)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Neural Networks can Learn Representations with Gradient Descent](https://mlsanity.com/paper/2206.15144/) (2022)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Neural collapse with unconstrained features](https://mlsanity.com/paper/2011.11619/) (2020)
- [The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks](https://mlsanity.com/paper/2202.08658/) (2022)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [Toward universal steering and monitoring of AI models](https://mlsanity.com/paper/2502.03708/) (2025)
- [Average gradient outer product as a mechanism for deep neural collapse](https://mlsanity.com/paper/2402.13728/) (2024)
- [Provable Guarantees for Nonlinear Feature Learning in Three-Layer Neural Networks](https://mlsanity.com/paper/2305.06986/) (2023)

## Built on by (most cited first, 5 of 5)

- [Deep Learning as Neural Low-Degree Filtering: A Spectral Theory of Hierarchical Feature Learning](https://mlsanity.com/paper/2605.13612/) (2026)
- [A Commutator Framework for Selective Spectral Alignment in Deep Neural Networks](https://mlsanity.com/paper/2608.22910/) (2026)
- [Learning Dynamics Reveal a Hierarchy of Weight-Induced Layerwise Gram Metrics](https://mlsanity.com/paper/2606.09744/) (2026)
- [Dual-Manifold Geometry Guided Representation Learning: Adaptive Coupling between Kernel and Data Spaces](https://mlsanity.com/paper/2608.12737/) (2026)
- [Refusal geometry reflects refusal training: diverse refusal prefixes can raise stable rank and weaken refusal vector ablation attacks](https://mlsanity.com/paper/2608.25390/) (2026)

## Abstract (arXiv)

Understanding how deep neural networks learn representations remains a central challenge in machine learning theory. In this work, we propose a feature-centric framework for analyzing neural network training by relating weight updates to feature evolution. We introduce a simple identity, the Feature Learning Equation, which identifies the weight Gram matrix as the key object capturing feature dynamics. This enables us to interpret gradient descent as implicitly inducing a hypothetical evolution of features, whose covariance structure - termed the Virtual Covariance - characterizes how representations evolve during training. Building on this perspective, we introduce Target Linearity, a measure quantifying the linear alignment between features and targets. By analyzing the training and layer-wise dynamics, we show that deep networks learn to sequentially transform representations toward target-linear structure. This linearization perspective provides a unified interpretation of several empirical phenomena, including Neural Collapse and linear interpolation in generative models.