Safety mechanisms for diffusion and flow models have recently been developed along two distinct paths. In robot planning, control barrier functions are employed to guide generative trajectories away from obstacles at every denoising step by explicitly imposing geometric constraints. In parallel, recent data-driven, negative guidance approaches have been shown to suppress harmful content and promote diversity in generated samples. However, they rely on heuristics without clearly stating when safety guidance is actually necessary. In this paper, we first introduce a unified probabilistic framework using a Maximum Mean Discrepancy (MMD) potential for image generation tasks that recasts both Shielded Diffusion and Safe Denoiser as instances of our energy-based negative guidance against unsafe data samples. Furthermore, we leverage control-barrier functions analysis to justify the existence of a critical time window in which negative guidance must be strong; outside of this window, the guidance should decay to zero to ensure safe and high-quality generation. We evaluate our unified framework on several realistic safe generation scenarios, confirming that negative guidance should be applied in the early stages of the denoising process for successful safe generation.

We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of $\mathbf{x}_T$ as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime $\mathbf{z}=(\mathbf{x}_t,t)$ that indexes the family of denoising distributions $p(\mathbf{x}_0 | \mathbf{x}_t)$ across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.

Human mobility trajectory data supports a wide range of applications, including urban planning, intelligent transportation systems, and public safety monitoring. However, large-scale, high-quality mobility datasets are difficult to obtain due to privacy concerns. Raw trajectory data may reveal sensitive user information, such as home addresses, routines, or social relationships, making it crucial to develop privacy-preserving alternatives. Recent advances in deep generative modeling have enabled synthetic trajectory generation, but existing methods either lack formal privacy guarantees or suffer from reduced utility and scalability. Differential Privacy (DP) has emerged as a rigorous framework for data protection, and recent efforts such as PATE-GAN and \textsc{PateGail} integrate DP with generative adversarial learning. While promising, these methods struggle to generalize across diverse trajectory patterns and often incur significant utility degradation. In this work, we propose a new framework that builds on \textsc{PateGail\texttt{++}} by introducing a \emph{sensitivity-aware noise injection module} that dynamically adjusts privacy noise based on sample-level sensitivity. This design significantly improves trajectory fidelity, downstream task performance, and scalability under strong privacy guarantees. We further adapt our framework to the local differential privacy (LDP) setting, allowing individual-level protection without reliance on a trusted server. We evaluate our method on a real-world mobility dataset and demonstrate its superiority over state-of-the-art baselines in terms of privacy-utility trade-off.

Flow matching is a powerful approach for high-fidelity density estimation, but it often fails to capture the latent structure of complex data. Probabilistic models like variational autoencoders (VAEs), on the other hand, learn structured representations but underperform in sample quality. We propose Structured Flow Autoencoders (SFA), a family of probabilistic models that augments graphical models with conditional continuous normalizing flow (CNF) likelihoods, enabling flow-matching-based structured representation learning. At the core of SFA is a novel flow matching objective that explicitly accounts for latent variables, allowing joint learning of the CNF likelihood and posterior. SFA applies broadly to graphical models with continuous and mixture latents, as well as latent dynamical systems. Empirical studies across image, video, and RNA-seq data show that SFA consistently outperforms VAEs and their structured extensions in generation quality, representation utility, and scalability to large datasets. Compared to generative models like latent flow matching (LatentFM), SFA also produces more diverse samples, suggesting better coverage of the data distribution.

Diffusion models have revolutionized image synthesis but raise concerns around content provenance and authenticity. Digital watermarking offers a means of tracing generated media, yet traditional schemes often introduce distributional shifts and degrade visual quality. Recent lossless methods embed watermark bits directly into the latent Gaussian prior without modifying model weights, but still require per-image key storage or heavy cryptographic overhead. In this paper, we introduce Spherical Watermark, an encryption‐free and lossless watermarking framework that integrates seamlessly with diffusion architectures. First, our binary embedding module mixes repeated watermark bits with random padding to form a high-entropy code. Second, the spherical mapping module projects this code onto the unit sphere, applies an orthogonal rotation, and scales by a chi-square-distributed radius to recover exact multivariate Gaussian noise. We theoretically prove that the watermarked noise distribution preserves the target prior up to third-order moments, and empirically demonstrate that it is statistically indistinguishable from a standard multivariate normal distribution. Adopting Stable Diffusion, extensive experiments confirm that Spherical Watermark consistently preserves high visual fidelity while simultaneously improving traceability, computational efficiency, and robustness under attacks, thereby outperforming both lossy and lossless approaches.

We introduce the Latent Fourier Transform (LatentFT), a framework that provides novel frequency-domain controls for generative music models. LatentFT combines a diffusion autoencoder with a latent-space Fourier transform to separate musical patterns by timescale. By masking latents in the frequency domain during training, our method yields representations that can be manipulated coherently at inference. This allows us to generate musical variations and blends from reference examples while preserving characteristics at desired timescales, which are specified as frequencies in the latent space. LatentFT parallels the role of the equalizer in music production: while traditional equalizers operates on audible frequencies to shape timbre, LatentFT operates on latent-space frequencies to shape musical structure. Experiments and listening tests show that LatentFT improves condition adherence and quality compared to baselines. We also present a technique for hearing frequencies in the latent space in isolation, and show different musical attributes reside in different regions of the latent spectrum. Our results show how frequency-domain control in latent space provides an intuitive, continuous frequency axis for conditioning and blending, advancing us toward more interpretable and interactive generative music models.