Abstract
In this work, we propose a novel generative learning paradigm, K-Flow, an algorithm that flows along the -amplitude. Here, is a scaling parameter that organizes frequency bands (or projected coefficients), and amplitude describes the norm of such projected coefficients. By incorporating the -amplitude decomposition, K-Flow enables flow matching across the scaling parameter as time. We discuss three venues and six properties of K-Flow, from theoretical foundations, energy and temporal dynamics, and practical applications, respectively. Specifically, from the practical usage perspective, K-Flow allows steerable generation by controlling the information at different scales. To demonstrate the effectiveness of K-Flow, we conduct experiments on unconditional image generation, class-conditional image generation, and molecule assembly generation. Additionally, we conduct three ablation studies to demonstrate how K-Flow steers scaling parameter to effectively control the resolution of image generation.