Abstract: In this work, we propose a novel generative learning paradigm, K-Flow, an algorithm that flows along the $K$-amplitude. In physics, $k$ is a measure to organize the frequency bands of objects, and the amplitude is the norm of projected coefficients. By incorporating the $K$-amplitude decomposition, K-Flow enables flow matching across scaling as time. We discuss three venues of 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 and class-conditional image generation. Additionally, we conduct three ablation studies to demonstrate how K-Flow steers scaling to effectively control the resolution of image generation.