Aligning Latent Geometry for Spherical Flow Matching in Image Generation
Authors: Tuna Han Salih Meral, Kaan Oktay, Hidir Yesiltepe, Adil Kaan Akan, Pinar Yanardag
Abstract
Latent flow matching for image generation usually transports Gaussian noise to variational autoencoder latents along linear paths. Both endpoints, however, concentrate in thin spherical shells, and a Euclidean chord leaves those shells even when preprocessing aligns their radii. By decomposing each latent token into radial and angular components, we show through component-swap probes that decoded perceptual and semantic content is carried predominantly by direction, with radius contributing much less. We therefore project data latents onto a fixed token radius, use the radial projection of Gaussian noise as the spherical prior, finetune the decoder with the encoder frozen, and replace linear interpolation with spherical linear interpolation. The resulting geodesic paths stay on the sphere at every timestep, and their velocity targets are purely angular by construction. Under matched training, the method consistently improves class-conditional ImageNet-256 FID across different image tokenizers, leaves the diffusion architecture unchanged, and requires no auxiliary encoder or representation-alignment objective.
Pixel-space generative models bypass lossy latent compression, yet necessitate joint learning of global structure and fine-grained details in a high-dimensional space. Standard flow matching interpolates noise toward a fixed clean-image endpoint, leaving the spectral evolution to be learned implicitly. In this paper, we introduce Energy-Guided Flow Matching(EG-FM) that explicitly models a coarse-to-fine generative trajectory by moving endpoint. Specifically, EG-FM replaces the fixed endpoint with a heat-kernel-filtered endpoint that evolves smoothly from low-frequency image to clean image. The fraction of high-frequency signal in moving endpoint is released by an image-specific energy-guided scheduling, leading to the re-targeting of velocity in flow matching. Our framework requires no adaptation of the backbone and training data, bringing negligible cost on the training and inference stages. In our experiment, EG-FM consistently achieves lower FID on the ImageNet class-conditional image generation task at 256×256 with fewer epochs, reaching an FID of 1.55 at 200 epochs and 1.45 at 600 epochs. We continue training the generation task on the setting of 512×512 resolution, yielding a FID of 1.58 after only 40 high-resolution adaptation epochs. Furthermore, we transfer EG-FM on text-to-image generation and achieve 0.85 on GenEval score and 83.9 on DPG-Bench. Code is available at https://github.com/ysng123/EG-FM.
Most diffusion and flow-matching generators define the prior, probability path, and prediction target in the same representation space. Latent diffusion improves efficiency by moving this path into an autoencoder latent space, but the final sample is still produced by a separately trained decoder. This separation creates a mismatch: the generator is optimized for latent-space prediction, while final quality depends on how the decoder handles generated latents that may differ from clean encoder outputs. We introduce CrossFlow, a cross-space flow formulation that maps noisy latent inputs directly to pixel-space images. The key technical step is a velocity-free one-step objective: the latent trajectory defines the training path, but the supervised prediction is an image rather than a latent displacement. This lets one model act both as a one-step latent-to-pixel generator and as a decoder replacement for latent diffusion pipelines. On class-conditional ImageNet-1k at 256×256, CrossFlow-XL achieves 1.62 FID with one function evaluation. Ablations show that the latent encoder and pixel-space perceptual and adversarial losses are important for fidelity. These results indicate that cross-space flow objectives can combine the efficiency of latent representations with direct pixel-space supervision, without requiring a separate decoder at inference.
Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.