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Poster
Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data
Hien Dang · Tho Tran Huu · Stanley Osher · Hung Tran-The · Nhat Ho · TAN NGUYEN

Tue Jul 25 02:00 PM -- 04:30 PM (PDT) @ Exhibit Hall 1 #205
in Poster Session 1 »

Modern deep neural networks have achieved impressive performance on tasks from image classification to natural language processing. Surprisingly, these complex systems with massive amounts of parameters exhibit the same structural properties in their last-layer features and classifiers across canonical datasets when training until convergence. In particular, it has been observed that the last-layer features collapse to their class-means, and those class-means are the vertices of a simplex Equiangular Tight Frame (ETF). This phenomenon is known as Neural Collapse (NC). Recent papers have theoretically shown that NC emerges in the global minimizers of training problems with the simplified ``unconstrained feature model''. In this context, we take a step further and prove the NC occurrences in deep linear networks for the popular mean squared error (MSE) and cross entropy (CE) losses, showing that global solutions exhibit NC properties across the linear layers. Furthermore, we extend our study to imbalanced data for MSE loss and present the first geometric analysis of NC under bias-free setting. Our results demonstrate the convergence of the last-layer features and classifiers to a geometry consisting of orthogonal vectors, whose lengths depend on the amount of data in their corresponding classes. Finally, we empirically validate our theoretical analyses on synthetic and practical network architectures with both balanced and imbalanced scenarios.

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Author Information

Hien Dang (FPT Software Company Limited, FPT Cau Giay building, Duy Tan street, Dich Vong Hau ward, Cau Giay district, Hanoi)
Tho Tran Huu (FPT Software Company Limited, FPT Cau Giay building, Duy Tan street, Dich Vong Hau ward, Cau Giay district, Hanoi)
Stanley Osher (UCLA)
Hung Tran-The (Deakin University)
Nhat Ho (University of Texas at Austin)
TAN NGUYEN (UCLA)

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