# Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions

arXiv: https://arxiv.org/abs/1804.02686 · HTML: https://mlsanity.com/paper/1804.02686/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Apr 2018.

## Built on by (most cited first, 23 of 23)

- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [Bayes-optimal learning of an extensive-width neural network from quadratically many samples](https://mlsanity.com/paper/2408.03733/) (2024)
- [Learning a Single Neuron for Non-monotonic Activation Functions](https://mlsanity.com/paper/2202.08064/) (2022)
- [Symmetry Breaking in Symmetric Tensor Decomposition](https://mlsanity.com/paper/2103.06234/) (2021)
- [Long Random Matrices and Tensor Unfolding](https://mlsanity.com/paper/2110.10210/) (2021)
- [Exploring Neural Network Landscapes: Star-Shaped and Geodesic Connectivity](https://mlsanity.com/paper/2404.06391/) (2024)
- [Appearance of Random Matrix Theory in Deep Learning](https://mlsanity.com/paper/2102.06740/) (2021)
- [A spin-glass model for the loss surfaces of generative adversarial networks](https://mlsanity.com/paper/2101.02524/) (2021)
- [Optimal learning rate schedules in high-dimensional non-convex optimization problems](https://mlsanity.com/paper/2202.04509/) (2022)
- [Universal characteristics of deep neural network loss surfaces from random matrix theory](https://mlsanity.com/paper/2205.08601/) (2022)
- [Asymptotic Dynamics of Alternating Minimization for Bilinear Regression](https://mlsanity.com/paper/2402.04751/) (2024)
- [Learning with latent group sparsity via heat flow dynamics on networks](https://mlsanity.com/paper/2201.08326/) (2022)
- [Overparametrization bends the landscape: BBP transitions at initialization in simple Neural Networks](https://mlsanity.com/paper/2510.18435/) (2025)
- [Sharp description of local minima in the loss landscape of high-dimensional two-layer ReLU neural networks](https://mlsanity.com/paper/2604.09412/) (2026)
- [Topological Exploration of High-Dimensional Empirical Risk Landscapes: general approach, and applications to phase retrieval](https://mlsanity.com/paper/2602.17779/) (2026)
- [Selective Multiple Power Iteration: from Tensor PCA to gradient-based exploration of landscapes](https://mlsanity.com/paper/2112.12306/) (2021)
- [Large-N dynamics of the spiked tensor model with random initial conditions](https://mlsanity.com/paper/2208.12586/) (2022)
- [Topological complexity of spiked random polynomials and finite-rank spherical integrals](https://mlsanity.com/paper/2312.12323/) (2023)
- [Permutation recovery of spikes in noisy high-dimensional tensor estimation](https://mlsanity.com/paper/2412.14650/) (2024)
- [Learning under Latent Group Sparsity via Diffusion on Networks](https://mlsanity.com/paper/2507.15097/) (2025)
- [On the Effect of Regularization on Nonparametric Mean-Variance Regression](https://mlsanity.com/paper/2511.22004/) (2025)
- [High-Dimensional Latents Should Be Diagnosed Through Phase Structure](https://mlsanity.com/paper/2606.02600/) (2026)

## Abstract (arXiv)

We study rough high-dimensional landscapes in which an increasingly stronger preference for a given configuration emerges. Such energy landscapes arise in glass physics and inference. In particular we focus on random Gaussian functions, and on the spiked-tensor model and generalizations. We thoroughly analyze the statistical properties of the corresponding landscapes and characterize the associated geometrical phase transitions. In order to perform our study, we develop a framework based on the Kac-Rice method that allows to compute the complexity of the landscape, i.e. the logarithm of the typical number of stationary points and their Hessian. This approach generalizes the one used to compute rigorously the annealed complexity of mean-field glass models. We discuss its advantages with respect to previous frameworks, in particular the thermodynamical replica method which is shown to lead to partially incorrect predictions.