# Visualizing high-dimensional loss landscapes with Hessian directions

arXiv: https://arxiv.org/abs/2208.13219 · HTML: https://mlsanity.com/paper/2208.13219/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 74 references that are in the map.
- First posted on arXiv 28 Aug 2022.

## Builds on (most cited first, 30 of 30)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [ResMLP: Feedforward networks for image classification with data-efficient training](https://mlsanity.com/paper/2105.03404/) (2021)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [Qualitatively characterizing neural network optimization problems](https://mlsanity.com/paper/1412.6544/) (2014)
- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [On Empirical Comparisons of Optimizers for Deep Learning](https://mlsanity.com/paper/1910.05446/) (2019)
- [Minimum Width for Universal Approximation](https://mlsanity.com/paper/2006.08859/) (2020)
- [Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approach](https://mlsanity.com/paper/1806.01316/) (2018)
- [Hessian-based Analysis of Large Batch Training and Robustness to Adversaries](https://mlsanity.com/paper/1802.08241/) (2018)
- [Visualizing and Understanding the Effectiveness of BERT](https://mlsanity.com/paper/1908.05620/) (2019)
- [The gap between theory and practice in function approximation with deep neural networks](https://mlsanity.com/paper/2001.07523/) (2020)
- [Shaping the learning landscape in neural networks around wide flat minima](https://mlsanity.com/paper/1905.07833/) (2019)
- [Hessian Eigenspectra of More Realistic Nonlinear Models](https://mlsanity.com/paper/2103.01519/) (2021)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Gradient Descent Converges to Minimizers](https://mlsanity.com/paper/1602.04915/) (2016)
- [Entropic gradient descent algorithms and wide flat minima](https://mlsanity.com/paper/2006.07897/) (2020)
- [Neural Ordinary Differential Equation Control of Dynamics on Graphs](https://mlsanity.com/paper/2006.09773/) (2020)
- [Negative eigenvalues of the Hessian in deep neural networks](https://mlsanity.com/paper/1902.02366/) (2019)
- [Exploring the Geometry and Topology of Neural Network Loss Landscapes](https://mlsanity.com/paper/2102.00485/) (2021)

## Built on by (most cited first, 16 of 16)

- [Gradient-free training of neural ODEs for system identification and control using ensemble Kalman inversion](https://mlsanity.com/paper/2307.07882/) (2023)
- [Explore the Effect of Data Selection on Poison Efficiency in Backdoor Attacks](https://mlsanity.com/paper/2310.09744/) (2023)
- [Unification of Symmetries Inside Neural Networks: Transformer, Feedforward and Neural ODE](https://mlsanity.com/paper/2402.02362/) (2024)
- [Bringing the Discussion of Minima Sharpness to the Audio Domain: a Filter-Normalised Evaluation for Acoustic Scene Classification](https://mlsanity.com/paper/2309.16369/) (2023)
- [TRACE for Tracking the Emergence of Semantic Representations in Transformers](https://mlsanity.com/paper/2505.17998/) (2025)
- [Tractable Sharpness-Aware Learning of Probabilistic Circuits](https://mlsanity.com/paper/2508.05537/) (2025)
- [When Active Learning Fails, Uncalibrated Out of Distribution Uncertainty Quantification Might Be the Problem](https://mlsanity.com/paper/2511.17760/) (2025)
- [The effects of Hessian eigenvalue spectral density type on the applicability of Hessian analysis to generalization capability assessment of neural networks](https://mlsanity.com/paper/2504.17618/) (2025)
- [On the failure of ReLU activation for physics-informed machine learning](https://mlsanity.com/paper/2512.11184/) (2025)
- [Visualizing the loss landscapes of physics-informed neural networks](https://mlsanity.com/paper/2602.05849/) (2026)
- [SANE: The phases of gradient descent through Sharpness Adjusted Number of Effective parameters](https://mlsanity.com/paper/2305.18490/) (2023)
- [Statistical Mechanics and Artificial Neural Networks: Principles, Models, and Applications](https://mlsanity.com/paper/2405.10957/) (2024)
- [Deep Learning Evidence for Global Optimality of Gerver's Sofa](https://mlsanity.com/paper/2407.11106/) (2024)
- [Deterministic Bounds and Random Estimates of Metric Tensors on Neuromanifolds](https://mlsanity.com/paper/2505.13614/) (2025)
- [Contrast transfer functions help quantify neural network out-of-distribution generalization in HRTEM](https://mlsanity.com/paper/2512.09067/) (2025)
- [LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks](https://mlsanity.com/paper/2607.14233/) (2026)

## Abstract (arXiv)

Analyzing geometric properties of high-dimensional loss functions, such as local curvature and the existence of other optima around a certain point in loss space, can help provide a better understanding of the interplay between neural network structure, implementation attributes, and learning performance. In this work, we combine concepts from high-dimensional probability and differential geometry to study how curvature properties in lower-dimensional loss representations depend on those in the original loss space. We show that saddle points in the original space are rarely correctly identified as such in expected lower-dimensional representations if random projections are used. The principal curvature in the expected lower-dimensional representation is proportional to the mean curvature in the original loss space. Hence, the mean curvature in the original loss space determines if saddle points appear, on average, as either minima, maxima, or almost flat regions. We use the connection between expected curvature in random projections and mean curvature in the original space (i.e., the normalized Hessian trace) to compute Hutchinson-type trace estimates without calculating Hessian-vector products as in the original Hutchinson method. Because random projections are not suitable to correctly identify saddle information, we propose to study projections along dominant Hessian directions that are associated with the largest and smallest principal curvatures. We connect our findings to the ongoing debate on loss landscape flatness and generalizability. Finally, for different common image classifiers and a function approximator, we show and compare random and Hessian projections of loss landscapes with up to about 7× 106 parameters.