Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equations

The diffusion model has shown remarkable success in computer vision, but it remains unclear whether ODE-based probability flow or SDE-based diffusion models are superior and under what circumstances. Comparing the two is challenging due to dependencies on data distribution, score training, and other numerical factors. In this paper, we examine the problem mathematically by examining two limiting scenarios: the ODE case and the large diffusion case. We first introduce a pulse-shape error to perturb the score function and analyze error accumulation, with a generalization to arbitrary error. Our findings indicate that when the perturbation occurs at the end of the generative process, the ODE model outperforms the SDE model (with a large diffusion coefficient). However, when the perturbation occurs earlier, the SDE model outperforms the ODE model, and we demonstrate that the error of sample generation due to pulse-shape error can be exponentially suppressed as the diffusion term's magnitude increases to infinity. Numerical validation of this phenomenon is provided using toy models such as Gaussian, Gaussian mixture models, and Swiss roll. Finally, we experiment with MNIST and observe that varying the diffusion coefficient can improve sample quality even when the score function is not well trained.

Geometric Operator Convolutional Neural Network

The Convolutional Neural Network (CNN) has been successfully applied in many fields during recent decades; however it lacks the ability to utilize prior domain knowledge when dealing with many realistic problems. We present a framework called Geometric Operator Convolutional Neural Network (GO-CNN) that uses domain knowledge, wherein the kernel of the first convolutional layer is replaced with a kernel generated by a geometric operator function. This framework integrates many conventional geometric operators, which allows it to adapt to a diverse range of problems. Under certain conditions, we theoretically analyze the convergence and the bound of the generalization errors between GO-CNNs and common CNNs. Although the geometric operator convolution kernels have fewer trainable parameters than common convolution kernels, the experimental results indicate that GO-CNN performs more accurately than common CNN on CIFAR-10/100. Furthermore, GO-CNN reduces dependence on the amount of training examples and enhances adversarial stability. In the practical task of medically diagnosing bone fractures, GO-CNN obtains 3% improvement in terms of the recall.

Random Forests and VGG-NET: An Algorithm for the ISIC 2017 Skin Lesion Classification Challenge

This manuscript briefly describes an algorithm developed for the ISIC 2017 Skin Lesion Classification Competition. In this task, participants are asked to complete two independent binary image classification tasks that involve three unique diagnoses of skin lesions (melanoma, nevus, and seborrheic keratosis). In the first binary classification task, participants are asked to distinguish between (a) melanoma and (b) nevus and seborrheic keratosis. In the second binary classification task, participants are asked to distinguish between (a) seborrheic keratosis and (b) nevus and melanoma. The other phases of the competition are not considered. Our proposed algorithm consists of three steps: preprocessing, classification using VGG-NET and Random Forests, and calculation of a final score.