Rank Diminishing in Deep Neural Networks

Abstract

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 leads 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 monotonic decreasing property of network rank 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 rank in practical settings, i.e., 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.Preprint. Under review.

High-performing neural network models of visual cortex benefit from high latent dimensionality

“…To the extent that the previous literature has considered latent dimensionality, it has predominantly argued that both biological and artificial vision systems gain computational benefits by representing stimuli in low-dimensional manifolds (Ansuini et al, 2019; Cohen et al, 2020; Lehky et al, 2014). For instance, it has been hypothesized that dimensionality reduction along the visual hierarchy confers robustness to incidental image features (Amsaleg et al, 2017; Feng et al, 2022; I. Fischer & Alemi, 2020; I.…”

“…The results, illustrated in Figure 4b, show a striking benefit of high-dimensionality for this task. Even though high-dimensional representations have traditionally been thought to be undesirable for object classification (Chung et al, 2018; Cohen et al, 2020; Feng et al, 2022; I. Fischer & Alemi, 2020; I.…”

“…The results, illustrated in Figure 5b, show a striking benefit of high-dimensionality for this task. Even though high-dimensional representations have traditionally been thought to be undesirable for object classification (Chung et al, 2018; Feng et al, 2022; I. Fischer & Alemi, 2020; I. S. Fischer, 2020; Lee et al, 2021; Recanatesi et al, 2019), they proved to be extremely effective in separating novel categories.…”

“…Our results speak to a fundamental question about the dimensionality of neural population codes. Empirical estimates have consistently shown that the latent dimensionality of both DNNs and neural systems is orders of magnitude lower than their ambient dimensionality (i.e., the number of neurons they contain) (Ansuini et al, 2019; Churchland et al, 2012; Cohen et al, 2020; Feng et al, 2022;Gallego et al, 2018; Gao & Ganguli, 2015; Gao et al, 2017; Gong et al, 2019; Lehky et al, 2014; Nieh et al, 2021; Op de Beeck et al, 2001). Furthermore, there are compelling theoretical arguments for the computational benefits of low-dimensional codes, which may promote robustness to noise (Amsaleg et al, 2017; I.…”

“…Ansuini, Laio, Macke, & Zoccolan, 2019). It is widely believed that DNNs benefit from representing stimuli on manifolds that are as low-dimensional as possible, and it is argued that low dimensionality improves a network’s generalization performance, its robustness to noise, and its ability to separate stimuli into meaningful categories (Amsaleg et al, 2017; Ansuini et al, 2019; Cohen et al, 2020; Feng et al, 2022; I. Fischer & Alemi, 2020; I.…”