# Spectral Estimators for Multi-Index Models: Precise Asymptotics and Optimal Weak Recovery

arXiv: https://arxiv.org/abs/2502.01583 · HTML: https://mlsanity.com/paper/2502.01583/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 53 references that are in the map.
- First posted on arXiv 3 Feb 2025.

## Builds on (most cited first, 26 of 26)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (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)
- [Online stochastic gradient descent on non-convex losses from high-dimensional inference](https://mlsanity.com/paper/2003.10409/) (2020)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [The committee machine: Computational to statistical gaps in learning a two-layers neural network](https://mlsanity.com/paper/1806.05451/) (2018)
- [On Learning Gaussian Multi-index Models with Gradient Flow](https://mlsanity.com/paper/2310.19793/) (2023)
- [Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems](https://mlsanity.com/paper/1505.05114/) (2015)
- [Computational-Statistical Gaps in Gaussian Single-Index Models](https://mlsanity.com/paper/2403.05529/) (2024)
- [Estimation of Low-Rank Matrices via Approximate Message Passing](https://mlsanity.com/paper/1711.01682/) (2017)
- [Fundamental Limits of Weak Recovery with Applications to Phase Retrieval](https://mlsanity.com/paper/1708.05932/) (2017)
- [Phase retrieval in high dimensions: Statistical and computational phase transitions](https://mlsanity.com/paper/2006.05228/) (2020)
- [Learning Polynomials of Few Relevant Dimensions](https://mlsanity.com/paper/2004.13748/) (2020)
- [The estimation error of general first order methods](https://mlsanity.com/paper/2002.12903/) (2020)
- [Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow](https://mlsanity.com/paper/1605.08285/) (2016)
- [Phase Transitions of Spectral Initialization for High-Dimensional Nonconvex Estimation](https://mlsanity.com/paper/1702.06435/) (2017)
- [Hitting the High-Dimensional Notes: An ODE for SGD learning dynamics on GLMs and multi-index models](https://mlsanity.com/paper/2308.08977/) (2023)
- [Approximate Message Passing with Spectral Initialization for Generalized Linear Models](https://mlsanity.com/paper/2010.03460/) (2020)
- [Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2112.04330/) (2021)
- [Optimal Combination of Linear and Spectral Estimators for Generalized Linear Models](https://mlsanity.com/paper/2008.03326/) (2020)
- [Learning sum of diverse features: computational hardness and efficient gradient-based training for ridge combinations](https://mlsanity.com/paper/2406.11828/) (2024)
- [Learning Orthogonal Multi-Index Models: A Fine-Grained Information Exponent Analysis](https://mlsanity.com/paper/2410.09678/) (2024)
- [Spectral Estimators for Structured Generalized Linear Models via Approximate Message Passing](https://mlsanity.com/paper/2308.14507/) (2023)

## Built on by (most cited first, 8 of 8)

- [Survey on Algorithms for multi-index models](https://mlsanity.com/paper/2504.05426/) (2025)
- [The Generative Leap: Sharp Sample Complexity for Efficiently Learning Gaussian Multi-Index Models](https://mlsanity.com/paper/2506.05500/) (2025)
- [Optimal scaling laws in learning hierarchical multi-index models](https://mlsanity.com/paper/2602.05846/) (2026)
- [Optimal Spectral Transitions in High-Dimensional Multi-Index Models](https://mlsanity.com/paper/2502.02545/) (2025)
- [Deep Learning of Compositional Targets with Hierarchical Spectral Methods](https://mlsanity.com/paper/2602.10867/) (2026)
- [A Noise Sensitivity Exponent Controls Large Statistical-to-Computational Gaps in Single- and Multi-Index Models](https://mlsanity.com/paper/2603.17896/) (2026)
- [Scaling Laws from Sequential Feature Recovery: A Solvable Hierarchical Model](https://mlsanity.com/paper/2605.14567/) (2026)
- [Approximate Message Passing with Random Initialization for Phase Retrieval](https://mlsanity.com/paper/2608.01654/) (2026)

## Abstract (arXiv)

Multi-index models provide a popular framework to investigate the learnability of functions with low-dimensional structure and, also due to their connections with neural networks, they have been object of recent intensive study. In this paper, we focus on recovering the subspace spanned by the signals via spectral estimators -- a family of methods routinely used in practice, often as a warm-start for iterative algorithms. Our main technical contribution is a precise asymptotic characterization of the performance of spectral methods, when sample size and input dimension grow proportionally and the dimension p of the space to recover is fixed. Specifically, we locate the top-p eigenvalues of the spectral matrix and establish the overlaps between the corresponding eigenvectors (which give the spectral estimators) and a basis of the signal subspace. Our analysis unveils a phase transition phenomenon in which, as the sample complexity grows, eigenvalues escape from the bulk of the spectrum and, when that happens, eigenvectors recover directions of the desired subspace. The precise characterization we put forward enables the optimization of the data preprocessing, thus allowing to identify the spectral estimator that requires the minimal sample size for weak recovery.