# A Rainbow in Deep Network Black Boxes

arXiv: https://arxiv.org/abs/2305.18512 · HTML: https://mlsanity.com/paper/2305.18512/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 69 of its 129 references that are in the map.
- First posted on arXiv 29 May 2023.

## Builds on (most cited first, 30 of 69)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [Insights on representational similarity in neural networks with canonical correlation](https://mlsanity.com/paper/1806.05759/) (2018)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Exploiting Linear Structure Within Convolutional Networks for Efficient Evaluation](https://mlsanity.com/paper/1404.0736/) (2014)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer](https://mlsanity.com/paper/2203.03466/) (2022)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Understanding the Role of Individual Units in a Deep Neural Network](https://mlsanity.com/paper/2009.05041/) (2020)

## Built on by (most cited first, 18 of 18)

- [How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model](https://mlsanity.com/paper/2307.02129/) (2023)
- [Scaling and renormalization in high-dimensional regression](https://mlsanity.com/paper/2405.00592/) (2024)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [The Universal Weight Subspace Hypothesis](https://mlsanity.com/paper/2512.05117/) (2025)
- [Small Singular Values Matter: A Random Matrix Analysis of Transformer Models](https://mlsanity.com/paper/2410.17770/) (2024)
- [Asymptotics of Learning with Deep Structured (Random) Features](https://mlsanity.com/paper/2402.13999/) (2024)
- [Large Language Models and Emergence: A Complex Systems Perspective](https://mlsanity.com/paper/2506.11135/) (2025)
- [On the universality of neural encodings in CNNs](https://mlsanity.com/paper/2409.19460/) (2024)
- [Global Convergence and Rich Feature Learning in L-Layer Infinite-Width Neural Networks under μP Parametrization](https://mlsanity.com/paper/2503.09565/) (2025)
- [Connecting Neural Models Latent Geometries with Relative Geodesic Representations](https://mlsanity.com/paper/2506.01599/) (2025)
- [Mitigating the Curse of Detail: Scaling Arguments for Feature Learning and Sample Complexity](https://mlsanity.com/paper/2512.04165/) (2025)
- [Spring-block theory of feature learning in deep neural networks](https://mlsanity.com/paper/2407.19353/) (2024)
- [The Effect of Label Noise on the Information Content of Neural Representations](https://mlsanity.com/paper/2510.06401/) (2025)
- [On the Neural Feature Ansatz for Deep Neural Networks](https://mlsanity.com/paper/2510.15563/) (2025)
- [Hamiltonian Mechanics of Feature Learning: Bottleneck Structure in Leaky ResNets](https://mlsanity.com/paper/2405.17573/) (2024)
- [Latent Functional Maps: a spectral framework for representation alignment](https://mlsanity.com/paper/2406.14183/) (2024)
- [Characterizing the Generalization Error of Random Feature Regression with Arbitrary Data-Augmentation](https://mlsanity.com/paper/2605.10290/) (2026)
- [Localizing Transfer Between Memorization Tasks](https://mlsanity.com/paper/2610.00771/) (2026)

## Abstract (arXiv)

A central question in deep learning is to understand the functions learned by deep networks. What is their approximation class? Do the learned weights and representations depend on initialization? Previous empirical work has evidenced that kernels defined by network activations are similar across initializations. For shallow networks, this has been theoretically studied with random feature models, but an extension to deep networks has remained elusive. Here, we provide a deep extension of such random feature models, which we call the rainbow model. We prove that rainbow networks define deterministic (hierarchical) kernels in the infinite-width limit. The resulting functions thus belong to a data-dependent RKHS which does not depend on the weight randomness. We also verify numerically our modeling assumptions on deep CNNs trained on image classification tasks, and show that the trained networks approximately satisfy the rainbow hypothesis. In particular, rainbow networks sampled from the corresponding random feature model achieve similar performance as the trained networks. Our results highlight the central role played by the covariances of network weights at each layer, which are observed to be low-rank as a result of feature learning.