# On Learning Gaussian Multi-index Models with Gradient Flow

arXiv: https://arxiv.org/abs/2310.19793 · HTML: https://mlsanity.com/paper/2310.19793/

## Key facts

- Cited by 57 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 79 references that are in the map.
- First posted on arXiv 30 Oct 2023.

## Builds on (most cited first, 22 of 22)

- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Towards Understanding the Spectral Bias of Deep Learning](https://mlsanity.com/paper/1912.01198/) (2019)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics](https://mlsanity.com/paper/2302.11055/) (2023)
- [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)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [The Landscape of Empirical Risk for Non-convex Losses](https://mlsanity.com/paper/1607.06534/) (2016)
- [Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity](https://mlsanity.com/paper/2106.15933/) (2021)
- [High-dimensional limit theorems for SGD: Effective dynamics and critical scaling](https://mlsanity.com/paper/2206.04030/) (2022)
- [How Two-Layer Neural Networks Learn, One (Giant) Step at a Time](https://mlsanity.com/paper/2305.18270/) (2023)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [Learning time-scales in two-layers neural networks](https://mlsanity.com/paper/2303.00055/) (2023)
- [Saddle-to-Saddle Dynamics in Diagonal Linear Networks](https://mlsanity.com/paper/2304.00488/) (2023)
- [From high-dimensional & mean-field dynamics to dimensionless ODEs: A unifying approach to SGD in two-layers networks](https://mlsanity.com/paper/2302.05882/) (2023)
- [SGD Finds then Tunes Features in Two-Layer Neural Networks with near-Optimal Sample Complexity: A Case Study in the XOR problem](https://mlsanity.com/paper/2309.15111/) (2023)
- [Beyond NTK with Vanilla Gradient Descent: A Mean-Field Analysis of Neural Networks with Polynomial Width, Samples, and Time](https://mlsanity.com/paper/2306.16361/) (2023)
- [Leveraging the two timescale regime to demonstrate convergence of neural networks](https://mlsanity.com/paper/2304.09576/) (2023)
- [On Single Index Models beyond Gaussian Data](https://mlsanity.com/paper/2307.15804/) (2023)
- [Learning Narrow One-Hidden-Layer ReLU Networks](https://mlsanity.com/paper/2304.10524/) (2023)
- [Nonparametric Linear Feature Learning in Regression Through Regularisation](https://mlsanity.com/paper/2307.12754/) (2023)
- [Symmetric Single Index Learning](https://mlsanity.com/paper/2310.02117/) (2023)

## Built on by (most cited first, 30 of 57)

- [The Benefits of Reusing Batches for Gradient Descent in Two-Layer Networks: Breaking the Curse of Information and Leap Exponents](https://mlsanity.com/paper/2402.03220/) (2024)
- [Transformers Learn Nonlinear Features In Context: Nonconvex Mean-field Dynamics on the Attention Landscape](https://mlsanity.com/paper/2402.01258/) (2024)
- [Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit](https://mlsanity.com/paper/2406.01581/) (2024)
- [Computational-Statistical Gaps in Gaussian Single-Index Models](https://mlsanity.com/paper/2403.05529/) (2024)
- [Learning Mixtures of Gaussians Using Diffusion Models](https://mlsanity.com/paper/2404.18869/) (2024)
- [Repetita Iuvant: Data Repetition Allows SGD to Learn High-Dimensional Multi-Index Functions](https://mlsanity.com/paper/2405.15459/) (2024)
- [Pretrained transformer efficiently learns low-dimensional target functions in-context](https://mlsanity.com/paper/2411.02544/) (2024)
- [Bayes-optimal learning of an extensive-width neural network from quadratically many samples](https://mlsanity.com/paper/2408.03733/) (2024)
- [Fundamental computational limits of weak learnability in high-dimensional multi-index models](https://mlsanity.com/paper/2405.15480/) (2024)
- [A Random Matrix Theory Perspective on the Spectrum of Learned Features and Asymptotic Generalization Capabilities](https://mlsanity.com/paper/2410.18938/) (2024)
- [Towards a theory of model distillation](https://mlsanity.com/paper/2403.09053/) (2024)
- [On the Complexity of Learning Sparse Functions with Statistical and Gradient Queries](https://mlsanity.com/paper/2407.05622/) (2024)
- [Learning Multi-Index Models with Neural Networks via Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2408.07254/) (2024)
- [Survey on Algorithms for multi-index models](https://mlsanity.com/paper/2504.05426/) (2025)
- [Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks](https://mlsanity.com/paper/2506.06489/) (2025)
- [Mean-field Analysis on Two-layer Neural Networks from a Kernel Perspective](https://mlsanity.com/paper/2403.14917/) (2024)
- [Learning sum of diverse features: computational hardness and efficient gradient-based training for ridge combinations](https://mlsanity.com/paper/2406.11828/) (2024)
- [Sliding down the stairs: how correlated latent variables accelerate learning with neural networks](https://mlsanity.com/paper/2404.08602/) (2024)
- [The Nuclear Route: Sharp Asymptotics of ERM in Overparameterized Quadratic Networks](https://mlsanity.com/paper/2505.17958/) (2025)
- [Learning Gaussian Multi-Index Models with Gradient Flow: Time Complexity and Directional Convergence](https://mlsanity.com/paper/2411.08798/) (2024)
- [Asymptotics of SGD in Sequence-Single Index Models and Single-Layer Attention Networks](https://mlsanity.com/paper/2506.02651/) (2025)
- [The Generative Leap: Sharp Sample Complexity for Efficiently Learning Gaussian Multi-Index Models](https://mlsanity.com/paper/2506.05500/) (2025)
- [Should Under-parameterized Student Networks Copy or Average Teacher Weights?](https://mlsanity.com/paper/2311.01644/) (2023)
- [Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks](https://mlsanity.com/paper/2404.18769/) (2024)
- [How Transformers Get Rich: Approximation and Dynamics Analysis](https://mlsanity.com/paper/2410.11474/) (2024)
- [Spectral Estimators for Multi-Index Models: Precise Asymptotics and Optimal Weak Recovery](https://mlsanity.com/paper/2502.01583/) (2025)
- [Low-dimensional Functions are Efficiently Learnable under Randomly Biased Distributions](https://mlsanity.com/paper/2502.06443/) (2025)
- [Optimal Spectral Transitions in High-Dimensional Multi-Index Models](https://mlsanity.com/paper/2502.02545/) (2025)
- [Understanding the Mechanisms of Fast Hyperparameter Transfer](https://mlsanity.com/paper/2512.22768/) (2025)
- [Provable Benefits of Unsupervised Pre-training and Transfer Learning via Single-Index Models](https://mlsanity.com/paper/2502.16849/) (2025)

## Abstract (arXiv)

We study gradient flow on the multi-index regression problem for high-dimensional Gaussian data. Multi-index functions consist of a composition of an unknown low-rank linear projection and an arbitrary unknown, low-dimensional link function. As such, they constitute a natural template for feature learning in neural networks. We consider a two-timescale algorithm, whereby the low-dimensional link function is learnt with a non-parametric model infinitely faster than the subspace parametrizing the low-rank projection. By appropriately exploiting the matrix semigroup structure arising over the subspace correlation matrices, we establish global convergence of the resulting Grassmannian population gradient flow dynamics, and provide a quantitative description of its associated `saddle-to-saddle' dynamics. Notably, the timescales associated with each saddle can be explicitly characterized in terms of an appropriate Hermite decomposition of the target link function. In contrast with these positive results, we also show that the related planted problem, where the link function is known and fixed, in fact has a rough optimization landscape, in which gradient flow dynamics might get trapped with high probability.