# Symmetric Single Index Learning

arXiv: https://arxiv.org/abs/2310.02117 · HTML: https://mlsanity.com/paper/2310.02117/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 3 Oct 2023.

## Builds on (most cited first, 15 of 15)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [A simple neural network module for relational reasoning](https://mlsanity.com/paper/1706.01427/) (2017)
- [Neural Networks can Learn Representations with Gradient Descent](https://mlsanity.com/paper/2206.15144/) (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)
- [Online stochastic gradient descent on non-convex losses from high-dimensional inference](https://mlsanity.com/paper/2003.10409/) (2020)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [High-dimensional limit theorems for SGD: Effective dynamics and critical scaling](https://mlsanity.com/paper/2206.04030/) (2022)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [Learning a Single Neuron with Gradient Methods](https://mlsanity.com/paper/2001.05205/) (2020)
- [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)
- [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)
- [On Single Index Models beyond Gaussian Data](https://mlsanity.com/paper/2307.15804/) (2023)
- [Exponential Separations in Symmetric Neural Networks](https://mlsanity.com/paper/2206.01266/) (2022)

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

- [On Learning Gaussian Multi-index Models with Gradient Flow](https://mlsanity.com/paper/2310.19793/) (2023)
- [A Random Matrix Theory Perspective on the Spectrum of Learned Features and Asymptotic Generalization Capabilities](https://mlsanity.com/paper/2410.18938/) (2024)
- [Asymptotics of SGD in Sequence-Single Index Models and Single-Layer Attention Networks](https://mlsanity.com/paper/2506.02651/) (2025)
- [Online Learning and Information Exponents: On The Importance of Batch size, and Time/Complexity Tradeoffs](https://mlsanity.com/paper/2406.02157/) (2024)
- [Joint Learning in the Gaussian Single Index Model](https://mlsanity.com/paper/2505.21336/) (2025)

## Abstract (arXiv)

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar link function. Learning this model with SGD is relatively well-understood, whereby the so-called information exponent of the link function governs a polynomial sample complexity rate. However, extending this analysis to deeper or more complicated architectures remains challenging. In this work, we consider single index learning in the setting of symmetric neural networks. Under analytic assumptions on the activation and maximum degree assumptions on the link function, we prove that gradient flow recovers the hidden planted direction, represented as a finitely supported vector in the feature space of power sum polynomials. We characterize a notion of information exponent adapted to our setting that controls the efficiency of learning.