# Deep Learning Evidence for Global Optimality of Gerver's Sofa

arXiv: https://arxiv.org/abs/2407.11106 · HTML: https://mlsanity.com/paper/2407.11106/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 15 Jul 2024.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
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- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Visualizing high-dimensional loss landscapes with Hessian directions](https://mlsanity.com/paper/2208.13219/) (2022)
- [Searching for ribbons with machine learning](https://mlsanity.com/paper/2304.09304/) (2023)
- [Generalized Correntropy for Robust Adaptive Filtering](https://mlsanity.com/paper/1504.02931/) (2015)
- [Piecewise Linear Neural Networks and Deep Learning](https://mlsanity.com/paper/2206.09149/) (2022)
- [Improved Imaging by Invex Regularizers with Global Optima Guarantees](https://mlsanity.com/paper/2211.10112/) (2022)
- [Fair Sparse Regression with Clustering: An Invex Relaxation for a Combinatorial Problem](https://mlsanity.com/paper/2102.09704/) (2021)
- [On the Compatibility between Neural Networks and Partial Differential Equations for Physics-informed Learning](https://mlsanity.com/paper/2212.00270/) (2022)
- [Differential equations and exact solutions in the moving sofa problem](https://mlsanity.com/paper/1606.08111/) (2016)
- [Improved upper bounds in the moving sofa problem](https://mlsanity.com/paper/1706.06630/) (2017)

## Abstract (arXiv)

The Moving Sofa Problem, formally proposed by Leo Moser in 1966, seeks to determine the largest area of a two-dimensional shape that can navigate through an L-shaped corridor with unit width. The current best lower bound is about 2.2195, achieved by Joseph Gerver in 1992, though its global optimality remains unproven. In this paper, we investigate this problem by leveraging the universal approximation strength and computational efficiency of neural networks. We report two approaches, both supporting Gerver's conjecture that his shape is the unique global maximum. Our first approach is continuous function learning. We drop Gerver's assumptions that i) the rotation of the corridor is monotonic and symmetric and, ii) the trajectory of its corner as a function of rotation is continuously differentiable. We parameterize rotation and trajectory by independent piecewise linear neural networks (with input being some pseudo time), allowing for rich movements such as backward rotation and pure translation. We then compute the sofa area as a differentiable function of rotation and trajectory using our "waterfall" algorithm. Our final loss function includes differential terms and initial conditions, leveraging the principles of physics-informed machine learning. Under such settings, extensive training starting from diverse function initialization and hyperparameters is conducted, unexceptionally showing rapid convergence to Gerver's solution. Our second approach is via discrete optimization of the Kallus-Romik upper bound, which converges to the maximum sofa area from above as the number of rotation angles increases. We uplift this number to 10000 to reveal its asymptotic behavior. It turns out that the upper bound yielded by our models does converge to Gerver's area (within an error of 0.01% when the number of angles reaches 2100). We also improve their five-angle upper bound from 2.37 to 2.3337.