Flow Along the K-Amplitude for Generative Modeling (2504.19353v1)

Abstract: In this work, we propose a novel generative learning paradigm, K-Flow, an algorithm that flows along the $K$-amplitude. Here, $k$ is a scaling parameter that organizes frequency bands (or projected coefficients), and amplitude describes the norm of such projected coefficients. By incorporating the $K$-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.

Flow Along the $K$-Amplitude for Generative Modeling

The paper introduces a novel generative modeling approach termed "K-Flow" based on flow matching techniques. This method leverages the $K$-amplitude space, where $k$ serves as a scaling parameter organizing frequency bands or projected coefficients, and the amplitude denotes the norm of these projected coefficients. By decomposing data into $K$-amplitude components, K-Flow proposes to perform flow matching across multiple scales, drawing inspiration from continuous normalizing flows and diffusion processes.

Key Concepts and Methodology

In the K-Flow framework, the scaling parameter $k$ and amplitude are central to modeling the inherent hierarchical structure of data. The $K$-amplitude decomposition is explored through three transformations: Fourier, Wavelet, and PCA. The approach involves projecting data into the $K$-amplitude space, learning a time-dependent velocity field in this space, and mapping it back to the spatial domain for generative processes. The methodology is augmented by intra-scaling interpolation, allowing the model to operate over continuous scaling parameters—a feature that distinguishes K-Flow from other flow-based models.

The paper postulates six properties of K-Flow, highlighting its ability to organize scaling parameters, enable multi-scale modeling, support explicit notions of energy, reinterpret scaling as a function of time, fuse intra- and inter-scaling modeling, and provide explicit steerability in generative outputs.

Experimental Results

The paper demonstrates the effectiveness of K-Flow in unconditional image generation tasks on datasets like CelebA-HQ and LSUN Church, as well as class-conditioned generation on ImageNet. Quantitatively, K-Flow showcases competitive results, particularly in generating high-fidelity output with controlled attributes. Furthermore, K-Flow exhibits steerability, allowing fine-grained control over different scaling resolutions in generated images—a distinct advantage over more traditional generative models.

In molecular assembly tasks, K-Flow advances the state-of-the-art by utilizing spectral methods to decompose pairwise molecular distances, thereby enhancing packing matching performance.

Implications and Future Directions

K-Flow represents a sophisticated approach to generative modeling, with potential applications across multi-modal generation tasks, such as text-guided image creation, and scientific discovery domains. Its theoretical foundation opens avenues for integrating energy-based modeling within the generative process, a promising direction for further exploration. The adaptability of K-Flow to various neural architectures underscores its potential to become a pivotal framework, influencing future developments in AI generative methodologies.

Overall, K-Flow offers a robust framework for generative tasks, characterized by its innovative use of the $K$-amplitude space for multi-scale modeling, paving the way for more controllable and interpretable generative AI systems.