We repeat the theorems presented in Sec. 3 and provide their proofs below. The theorems hold for Neumann boundary conditions, which we use in our implementation--this is achieved by the construction of the differential operators. The proofs follow the ones presented in [22]. If the activation function σ() is monotonically non-decreasing and sign-preserving, then the forward propagation through the diffusive PDE in (1) for t [0,) yields a non-increasing feature norm, that is, t kfk2 0. Proof. Let us examine the following inner product following Eq.

We explore the capability of plain Vision Transformers (ViTs) for semantic segmentation and propose the SegViT. Previous ViT-based segmentation networks usually learn a pixel-level representation from the output of the ViT. Differently, we make use of the fundamental component--attention mechanism, to generate masks for semantic segmentation. Specifically, we propose the Attention-to-Mask (ATM) module, in which the similarity maps between a set of learnable class tokens and the spatial feature maps are transferred to the segmentation masks. Experiments show that our proposed SegViT using the ATM module outperforms its counterparts using the plain ViT backbone on the ADE20K dataset and achieves new state-of-the-art performance on COCO-Stuff-10K and PASCAL-Context datasets. Furthermore, to reduce the computational cost of the ViT backbone, we propose query-based down-sampling (QD) and query-based up-sampling (QU) to build a Shrunk structure. With the proposed Shrunk structure, the model can save up to 40%computations while maintaining competitive performance.

In this section, we provide further intuition about the proposed AdaQN method. As shown in Figure 5, in AdaQN we need to ensure that the approximate solution of the ERM problem with m samples denoted by wm is within the superlinear convergence neighborhood of BFGS for the ERM problem with n = 2msamples. Here, w m and w n are the optimal solutions of the risks Rm and Rn corresponding to the sets Sm and Sn with mand nsamples, respectively, where Sm Sn. The statistical accuracy region of Rm is denoted by a blue circle, the statistical accuracy region of Rn is denoted by a red circle, and the superlinear convergence neighborhood of BFGS for Rn is denoted by a dotted purple circle. As we observe, any point within the statistical accuracy of w m is within the superlinear convergence neighborhood of BFGS for Rn.