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The rank of neural networks measures information flowing across layers. It is an instance of a key structural condition that applies across broad domains of machine learning. In particular, the assumption of low-rank feature representations led to algorithmic developments in many architectures. For neural networks, however, the intrinsic mechanism that yields low-rank structures remains vague and unclear. To fill this gap, we perform a rigorous study on the behavior of network rank, focusing particularly on the notion of rank deficiency. We theoretically establish a universal monotone decreasing property of network ranks from the basic rules of differential and algebraic composition, and uncover rank deficiency of network blocks and deep function coupling. By virtue of our numerical tools, we provide the first empirical analysis of the per-layer behavior of network ranks in realistic settings, \ieno, ResNets, deep MLPs, and Transformers on ImageNet. These empirical results are in direct accord with our theory. Furthermore, we reveal a novel phenomenon of independence deficit caused by the rank deficiency of deep networks, where classification confidence of a given category can be linearly decided by the confidence of a handful of other categories. The theoretical results of this work, together with the empirical findings, may advance understanding of the inherent principles of deep neural networks. Code to detect the rank behavior of networks can be found in https://github.com/RuiLiFeng/Rank-Diminishing-in-Deep-Neural-Networks.
Author Information
Ruili Feng (University of Science and Technology of China)
Kecheng Zheng (Zhejiang University)
Yukun Huang (University of Science and Technology of China)
Deli Zhao (Xiaomi AI Lab)
Michael Jordan (UC Berkeley)
Zheng-Jun Zha (University of Science and Technology of China)
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XuanLong Nguyen · Martin J Wainwright · Michael Jordan -
2007 Poster: Resampling Methods for Protein Structure Prediction with Rosetta »
Ben Blum · David Baker · Michael Jordan · Philip Bradley · Rhiju Das · David Kim -
2007 Poster: Estimating divergence functionals and the likelihood ratio by penalized convex risk minimization »
XuanLong Nguyen · Martin J Wainwright · Michael Jordan -
2006 Poster: Distributed PCA and Network Anomaly Detection »
Ling Huang · XuanLong Nguyen · Minos Garofalakis · Michael Jordan · Anthony D Joseph · Nina Taft