# Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval

arXiv: https://arxiv.org/abs/2602.17779 · HTML: https://mlsanity.com/paper/2602.17779/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 53 references that are in the map.
- First posted on arXiv 19 Feb 2026.

## Builds on (most cited first, 27 of 27)

- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [The Power of Interpolation: Understanding the Effectiveness of SGD in Modern Over-parametrized Learning](https://mlsanity.com/paper/1712.06559/) (2017)
- [High-dimensional limit theorems for SGD: Effective dynamics and critical scaling](https://mlsanity.com/paper/2206.04030/) (2022)
- [On Learning Gaussian Multi-index Models with Gradient Flow](https://mlsanity.com/paper/2310.19793/) (2023)
- [Optimization and Generalization of Shallow Neural Networks with Quadratic Activation Functions](https://mlsanity.com/paper/2006.15459/) (2020)
- [Bad Global Minima Exist and SGD Can Reach Them](https://mlsanity.com/paper/1906.02613/) (2019)
- [Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit](https://mlsanity.com/paper/2406.01581/) (2024)
- [Comparing Dynamics: Deep Neural Networks versus Glassy Systems](https://mlsanity.com/paper/1803.06969/) (2018)
- [The landscape of the spiked tensor model](https://mlsanity.com/paper/1711.05424/) (2017)
- [No bad local minima: Data independent training error guarantees for multilayer neural networks](https://mlsanity.com/paper/1605.08361/) (2016)
- [Passed & Spurious: Descent Algorithms and Local Minima in Spiked Matrix-Tensor Models](https://mlsanity.com/paper/1902.00139/) (2019)
- [Hitting the High-Dimensional Notes: An ODE for SGD learning dynamics on GLMs and multi-index models](https://mlsanity.com/paper/2308.08977/) (2023)
- [Who is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Model](https://mlsanity.com/paper/1907.08226/) (2019)
- [Marvels and Pitfalls of the Langevin Algorithm in Noisy High-dimensional Inference](https://mlsanity.com/paper/1812.09066/) (2018)
- [Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions](https://mlsanity.com/paper/1804.02686/) (2018)
- [On the Optimization Landscape of Tensor Decompositions](https://mlsanity.com/paper/1706.05598/) (2017)
- [On the Impact of Overparameterization on the Training of a Shallow Neural Network in High Dimensions](https://mlsanity.com/paper/2311.03794/) (2023)
- [Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrieval](https://mlsanity.com/paper/2006.06997/) (2020)
- [Landscape Complexity for the Empirical Risk of Generalized Linear Models](https://mlsanity.com/paper/1912.02143/) (2019)
- [Escaping mediocrity: how two-layer networks learn hard generalized linear models with SGD](https://mlsanity.com/paper/2305.18502/) (2023)
- [Phase Transitions for Feature Learning in Neural Networks](https://mlsanity.com/paper/2602.01434/) (2026)
- [Solving phase retrieval with random initial guess is nearly as good as by spectral initialization](https://mlsanity.com/paper/2101.03540/) (2021)
- [The star-shaped space of solutions of the spherical negative perceptron](https://mlsanity.com/paper/2305.10623/) (2023)
- [The Role of the Time-Dependent Hessian in High-Dimensional Optimization](https://mlsanity.com/paper/2403.02418/) (2024)
- [Local minima of the empirical risk in high dimension: General theorems and convex examples](https://mlsanity.com/paper/2502.01953/) (2025)
- [Landscape Complexity for the Empirical Risk of Generalized Linear Models: Discrimination between Structured Data](https://mlsanity.com/paper/2503.14403/) (2025)

## Built on by (most cited first, 1 of 1)

- [Replica Symmetry Breaking and Algorithmic Thresholds in Empirical Risk Minimization under Multi-Index Model](https://mlsanity.com/paper/2606.28573/) (2026)

## Abstract (arXiv)

We consider the landscape of empirical risk minimization for high-dimensional Gaussian single-index models (generalized linear models). The objective is to recover an unknown signal boldsymbolθ^star ∈ ℝd (where d gg 1) from a loss function hatR(boldsymbolθ) that depends on pairs of labels (xi · boldsymbolθ, xi · boldsymbolθ^star)i=1n, with xi ~ N(0, Id), in the proportional asymptotic regime n asymp d. Using the Kac-Rice formula, we analyze different complexities of the landscape -- defined as the expected number of critical points -- corresponding to various types of critical points, including local minima. We first show that some variational formulas previously established in the literature for these complexities can be drastically simplified, reducing to explicit variational problems over a finite number of scalar parameters that we can efficiently solve numerically. Our framework also provides detailed predictions for properties of the critical points, including the spectral properties of the Hessian and the joint distribution of labels. We apply our analysis to the real phase retrieval problem for which we derive complete topological phase diagrams of the loss landscape, characterizing notably BBP-type transitions where the Hessian at local minima (as predicted by the Kac-Rice formula) becomes unstable in the direction of the signal. We test the predictive power of our analysis to characterize gradient flow dynamics, finding excellent agreement with finite-size simulations of local optimization algorithms, and capturing fine-grained details such as the empirical distribution of labels. Overall, our results open new avenues for the asymptotic study of loss landscapes and topological trivialization phenomena in high-dimensional statistical models.