# Evaluating the Efficiency of Latent Spaces via the Coupling-Matrix

arXiv: https://arxiv.org/abs/2509.06314 · HTML: https://mlsanity.com/paper/2509.06314/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 8 Sep 2025.

## Builds on (most cited first, 9 of 9)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [On Mutual Information Maximization for Representation Learning](https://mlsanity.com/paper/1907.13625/) (2019)
- [Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networks](https://mlsanity.com/paper/1805.12076/) (2018)

## Built on by (most cited first, 2 of 2)

- [Redundancy as a Structural Information Principle for Learning and Generalization](https://mlsanity.com/paper/2510.10938/) (2025)
- [Multivariate Variational Autoencoder](https://mlsanity.com/paper/2511.07472/) (2025)

## Abstract (arXiv)

A central challenge in representation learning is constructing latent embeddings that are both expressive and efficient. In practice, deep networks often produce redundant latent spaces where multiple coordinates encode overlapping information, reducing effective capacity and hindering generalization. Standard metrics such as accuracy or reconstruction loss provide only indirect evidence of such redundancy and cannot isolate it as a failure mode. We introduce a redundancy index, denoted rho(C), that directly quantifies inter-dimensional dependencies by analyzing coupling matrices derived from latent representations and comparing their off-diagonal statistics against a normal distribution via energy distance. The result is a compact, interpretable, and statistically grounded measure of representational quality. We validate rho(C) across discriminative and generative settings on MNIST variants, Fashion-MNIST, CIFAR-10, and CIFAR-100, spanning multiple architectures and hyperparameter optimization strategies. Empirically, low rho(C) reliably predicts high classification accuracy or low reconstruction error, while elevated redundancy is associated with performance collapse. Estimator reliability grows with latent dimension, yielding natural lower bounds for reliable analysis. We further show that Tree-structured Parzen Estimators (TPE) preferentially explore low-rho regions, suggesting that rho(C) can guide neural architecture search and serve as a redundancy-aware regularization target. By exposing redundancy as a universal bottleneck across models and tasks, rho(C) offers both a theoretical lens and a practical tool for evaluating and improving the efficiency of learned representations.