Evaluating the Efficiency of Latent Spaces via the Coupling-Matrix

2025-09-07

AI Summary

Latent space redundancy in representation learning degrades effective capacity and hinders generalization, yet existing evaluation metrics lack direct interpretability for quantifying this issue. Method: We propose a redundancy quantification framework grounded in coupling matrices and energy distance, defining an interpretable, statistically robust redundancy index ρ(C) that explicitly measures statistical dependencies among latent dimensions. Our approach constructs a coupling matrix from latent representations and uses energy distance to quantify the deviation of its off-diagonal structure from normality, with hyperparameters optimized via Tree-structured Parzen Estimator (TPE). Results: Evaluated on MNIST, Fashion-MNIST, and CIFAR, ρ(C) exhibits strong negative correlation with classification accuracy and reconstruction error, reliably predicting performance collapse; estimation fidelity improves with increasing latent dimensionality. The metric enables decoupled, cross-model and cross-task assessment of representation quality, providing differentiable, theoretically grounded guidance for neural architecture search and regularization design.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Representation Learning for Vision
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Quantifying redundancy in latent embeddings
• Measuring inter-dimensional dependencies in representations
• Evaluating efficiency of learned representations

Innovation

• Introduces redundancy index rho(C) via coupling matrices
• Quantifies inter-dimensional dependencies using energy distance
• Validated across discriminative and generative settings