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Appendix 446 A Proof of Proposition 1 in Section 2 447 Proof

Neural Information Processing Systems

ReLU (T (v u) + b) = ReLU( Tv + b), where u = 0, that is, ReLU (T + b) is not injective. By injectivity of T, we finally get a = b . Remark 2. An example that satisfies (3.1) is the neural operator whose This construction is given by the combination of "Pairs of projections" discussed in Kato [2013, Section I.4.6] with the idea presented in [Puthawala et al., 2022b, Lemma 29]. R. We write operator null G by Thus, in both cases, H is injective. Remark 4. W e make the following observations using Theorem 1: Leaky ReLU is one of example that satisfies (ii) in Theorem 1. Puthawala et al. [2022a, Theorem 15] assumes that We first revisit layerwise injectivity and bijectivity in the case of the finite rank approximation.




Appendix for " Residual Alignment: Uncovering the Mechanisms of Residual Networks " Anonymous Author(s) Affiliation Address email

Neural Information Processing Systems

We start by providing motivation for the unconstrained Jacobians problem introduced in the main text. We will continue our proof using contradiction. Figure 1: Fully-connected ResNet34 (Type 1 model) trained on MNIST.Figure 2: Fully-connected ResNet34 (Type 1 model) trained on FashionMNIST. Figure 10: Fully-connected ResNet34 (Type 1 model) trained on MNIST. Figure 24: Fully-connected ResNet34 (Type 1 model) trained on MNIST.





Supplemental Material For GenAI Arena Dongfu Jiang Max Ku Tianle Li

Neural Information Processing Systems

For what purpose was the dataset created? To foster the research in aligning diffusion models further and analyze the user preferences. Who created the dataset (e.g., which team, research group) and on behalf of which entity Who funded the creation of the dataset? What do the instances that comprise the dataset represent (e.g., documents, photos, people, How many instances are there in total (of each type, if appropriate)? What data does each instance consist of?