Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces

Diffusion models is a popular computational tool to generate new data samples. It utilizes a forward diffusion process that add noise to the data distribution and then use a reverse process to remove noises to produce samples from the data distribution. However, when the empirical data distribution consists of $n$ data point, using the empirical diffusion model will necessarily produce one of the existing data points. This is often referred to as the memorization effect, which is usually resolved by sophisticated machine learning procedures in the current literature. This work shows that the memorization problem can be resolved by a simple inertia update step at the end of the empirical diffusion model simulation. Our inertial diffusion model requires only the empirical diffusion model score function and it does not require any further training. We show that choosing the inertia diffusion model sample distribution is an $O\left(n^{-\frac{2}{d+4}}\right)$ Wasserstein-1 approximation of a data distribution lying on a $C^2$ manifold of dimension $d$. Since this estimate is significant smaller the Wasserstein1 distance between population and empirical distributions, it rigorously shows the inertial diffusion model produces new data samples. Remarkably, this upper bound is completely free of the ambient space dimension, since there is no training involved. Our analysis utilizes the fact that the inertial diffusion model samples are approximately distributed as the Gaussian kernel density estimator on the manifold. This reveals an interesting connection between diffusion model and manifold learning.

GIF: A General Graph Unlearning Strategy via Influence Function

With the greater emphasis on privacy and security in our society, the problem of graph unlearning -- revoking the influence of specific data on the trained GNN model, is drawing increasing attention. However, ranging from machine unlearning to recently emerged graph unlearning methods, existing efforts either resort to retraining paradigm, or perform approximate erasure that fails to consider the inter-dependency between connected neighbors or imposes constraints on GNN structure, therefore hard to achieve satisfying performance-complexity trade-offs. In this work, we explore the influence function tailored for graph unlearning, so as to improve the unlearning efficacy and efficiency for graph unlearning. We first present a unified problem formulation of diverse graph unlearning tasks \wrt node, edge, and feature. Then, we recognize the crux to the inability of traditional influence function for graph unlearning, and devise Graph Influence Function (GIF), a model-agnostic unlearning method that can efficiently and accurately estimate parameter changes in response to a $\epsilon$-mass perturbation in deleted data. The idea is to supplement the objective of the traditional influence function with an additional loss term of the influenced neighbors due to the structural dependency. Further deductions on the closed-form solution of parameter changes provide a better understanding of the unlearning mechanism. We conduct extensive experiments on four representative GNN models and three benchmark datasets to justify the superiority of GIF for diverse graph unlearning tasks in terms of unlearning efficacy, model utility, and unlearning efficiency. Our implementations are available at \url{https://github.com/wujcan/GIF-torch/}.