For iterative algorithms, implicit differentiation alleviates this issue but requires custom implementation of Jacobian evaluation. In this paper, we study one-step differentiation, also known as Jacobian-free backpropagation, a method as easy as automatic differentiation and as efficient as implicit differentiation for fast algorithms (e.g., superlinear

Unlike distance metric learning where the subsequent tasks utilizing the estimated distance metric is the usual focus, the proposal focuses on the estimated metric characterizing the geometry structure. Despite the illustrated taxi and MNIST examples, it is still open to finding more compelling applications that target the data space geometry. Interpreting mathematical concepts such as Riemannian metric and geodesic in the context of potential application (e.g., cognition and perception research where similarity measures are common) could be inspiring. Our proposal requires sufficiently dense data, which could be demanding, especially for high-dimensional data due to the curse of dimensionality. Dimensional reduction (e.g., manifold embedding as in the MNIST example) can substantially alleviate the curse of dimensionality, and the dense data requirement will more likely hold true.

A.1.1 V ariance Preserving SDE as a potential function We can re-express the widely used V ariance Preserving (VP-SDE) (DDPM): dY

Step 2. The deduction in Step 1 can be iterated repeatedly due to the symmetry of

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.

Neural operators [Kovachki et al., 2021a,b] are neural networks that take a

Learning continuous-time point processes is essential to many discrete event forecasting tasks.