# Matricial Free Energy as a Gaussianizing Regularizer: Enhancing Autoencoders for Gaussian Code Generation

arXiv: https://arxiv.org/abs/2510.17120 · HTML: https://mlsanity.com/paper/2510.17120/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 68 references that are in the map.
- First posted on arXiv 20 Oct 2025.

## Builds on (most cited first, 30 of 33)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Conformer: Convolution-augmented Transformer for Speech Recognition](https://mlsanity.com/paper/2005.08100/) (2020)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [EfficientViT: Memory Efficient Vision Transformer with Cascaded Group Attention](https://mlsanity.com/paper/2305.07027/) (2023)
- [High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification](https://mlsanity.com/paper/1507.03003/) (2015)
- [4+3 Phases of Compute-Optimal Neural Scaling Laws](https://mlsanity.com/paper/2405.15074/) (2024)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [Learning Rates as a Function of Batch Size: A Random Matrix Theory Approach to Neural Network Training](https://mlsanity.com/paper/2006.09092/) (2020)
- [Sliced-Wasserstein Autoencoder: An Embarrassingly Simple Generative Model](https://mlsanity.com/paper/1804.01947/) (2018)
- [Exact expressions for double descent and implicit regularization via surrogate random design](https://mlsanity.com/paper/1912.04533/) (2019)
- [Hessian Eigenspectra of More Realistic Nonlinear Models](https://mlsanity.com/paper/2103.01519/) (2021)
- [Iterative Gaussianization: from ICA to Random Rotations](https://mlsanity.com/paper/1602.00229/) (2016)
- [Free-form Flows: Make Any Architecture a Normalizing Flow](https://mlsanity.com/paper/2310.16624/) (2023)
- [Precise Learning Curves and Higher-Order Scaling Limits for Dot Product Kernel Regression](https://mlsanity.com/paper/2205.14846/) (2022)
- [Double Descent and Overfitting under Noisy Inputs and Distribution Shift for Linear Denoisers](https://mlsanity.com/paper/2305.17297/) (2023)
- [Spectral alignment of stochastic gradient descent for high-dimensional classification tasks](https://mlsanity.com/paper/2310.03010/) (2023)
- [Near-Interpolators: Rapid Norm Growth and the Trade-Off between Interpolation and Generalization](https://mlsanity.com/paper/2403.07264/) (2024)
- [Appearance of Random Matrix Theory in Deep Learning](https://mlsanity.com/paper/2102.06740/) (2021)
- [On the Convergence Rate of Gaussianization with Random Rotations](https://mlsanity.com/paper/2306.13520/) (2023)

## Abstract (arXiv)

We introduce a novel regularization scheme for autoencoders based on matricial free energy. Our approach defines a differentiable loss function in terms of the singular values of the code matrix (code dimension x batch size). From the standpoint of free probability an d random matrix theory, this loss achieves its minimum when the singular value distribution of the code matrix coincides with that of an appropriately sculpted random metric with i.i.d. Gaussian entries. Empirical simulations demonstrate that minimizing the negative matricial free energy through standard stochastic gradient-based training yields Gaussian-like codes that generalize across training and test sets. Building on this foundation, we propose a matricidal free energy maximizing autoencoder that reliably produces Gaussian codes and show its application to underdetermined inverse problems.