On Nullspace of Vision Transformers and What Does it Tell Us?

Aditya Singh, Haohan Wang

Keywords: nullspace, vision transformers, robustness, watermarking, fooling interpretations, fooling models

TL;DR: Our work highlights and discusses the concept of nullspace wrt vision transformers.

Abstract: Nullspace of a linear mapping is the subspace which is mapped to the zero vector. For a linear map, adding an element of the nullspace to its input has no effect on the output of the mapping. We position this work as an exposition towards answering one simple question, ``Does a vision transformer have a non-trivial nullspace?" If TRUE, this would imply that adding elements from this non-trivial nullspace to an input will have no effect on the output of the network. This finding can eventually lead us closer to understanding the generalization properties of vision transformers. In this paper, we first demonstrate that provably a non-trivial nullspace exists for a particular class of vision transformers. This proof is drawn by simply computing the nullspace of the patch embedding matrices. We extend this idea to the non-linear layers of the vision transformer and show that it is possible to learn a non-linear counterpart to the nullspace via simple optimisations for any vision transformer. Subsequently, we perform studies to understand robustness properties of ViTs under nullspace noise. Under robustness, we investigate prediction stability, and (network and interpretation) fooling properties of the noise. Lastly, we provide image watermarking as an application of nullspace noise.

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