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[2602.17779] Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval
Summary
This paper explores the topological properties of high-dimensional empirical risk landscapes, focusing on phase retrieval applications and the complexities of critical points in loss functions.
Why It Matters
Understanding the topological characteristics of empirical risk landscapes is crucial for improving optimization techniques in machine learning. This research provides insights into the behavior of critical points, which can enhance the efficiency of algorithms used in high-dimensional statistical models.
Key Takeaways
- The paper simplifies variational formulas for analyzing critical points in high-dimensional landscapes.
- It provides detailed predictions about the spectral properties of Hessians at local minima.
- The study applies its findings to real-world phase retrieval problems, illustrating the practical implications of the theoretical framework.
- The research captures the dynamics of gradient flow and aligns well with finite-size simulations.
- New avenues for studying loss landscapes and topological phenomena in statistical models are opened.
Statistics > Machine Learning arXiv:2602.17779 (stat) [Submitted on 19 Feb 2026] Title:Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval Authors:Antoine Maillard, Tony Bonnaire, Giulio Biroli View a PDF of the paper titled Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval, by Antoine Maillard and 2 other authors View PDF HTML (experimental) Abstract: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 $\boldsymbol{\theta}^\star \in \mathbb{R}^d$ (where $d \gg 1$) from a loss function $\hat{R}(\boldsymbol{\theta})$ that depends on pairs of labels $(\mathbf{x}_i \cdot \boldsymbol{\theta}, \mathbf{x}_i \cdot \boldsymbol{\theta}^\star)_{i=1}^n$, with $\mathbf{x}_i \sim \mathcal{N}(0, I_d)$, 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 numeri...
