Geometric Properties of Neural Multivariate Regression: An Empirical Study

George Andriopoulos, Zixuan Dong, Bimarsha Adhikari, Keith W. Ross

Keywords: multivariate regression, neural collapse, intrinsic dimension, deep learning, generalization

TL;DR: Deep regression models risk collapsing their penultimate feature manifold to a lower intrinsic dimension than the target manifold, harming generalization.

Abstract: Neural multivariate regression underpins a wide range of domains such as control, robotics, and finance, yet the geometry of its learned representations remains poorly characterized. While neural collapse has been shown to benefit generalization in classification, we find that analogous collapse in regression consistently degrades performance. To explain this contrast, we analyze models through the lens of intrinsic dimension. Across control tasks and synthetic datasets, we estimate the intrinsic dimension of last-layer features ($ID_H$) and compare it with that of the regression targets ($ID_Y$). Collapsed models exhibit $ID_H < ID_Y$, leading to over-compression and poor generalization, whereas non-collapsed models typically maintain $ID_H > ID_Y$. For the non-collapsed models, performance with respect to $ID_H$ depends on the data quantity and noise levels. From these observations, we identify two regimes—over-compressed and under-compressed—that determine when expanding or reducing feature dimensionality improves performance. Our results provide new geometric insights into neural regression and suggest practical strategies for enhancing generalization.

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Challenge: This submission is an entry to the science of DL improvement challenge.

Submission Number: 21