Recent advancements in text-to-image (T2I) models have greatly benefited from large-scale datasets, but they also pose significant risks due to the potential generation of unsafe content. To mitigate this issue, researchers proposed unlearning techniques that attempt to induce the model to unlearn potentially harmful prompts. However, these methods are easily bypassed by adversarial attacks, making them unreliable for ensuring the safety of generated images. In this paper, we propose Direct Unlearning Optimization (DUO), a novel framework for removing NSFW content from T2I models while preserving their performance on unrelated topics. DUO employs a preference optimization approach using curated paired image data, ensuring that the model learns to remove unsafe visual concepts while retain unrelated features. Furthermore, we introduce an output-preserving regularization term to maintain the model's generative capabilities on safe content. Extensive experiments demonstrate that DUO can robustly defend against various state-of-the-art red teaming methods without significant performance degradation on unrelated topics, as measured by FID and CLIP scores. Our work contributes to the development of safer and more reliable T2I models, paving the way for their responsible deployment in both closed-source and open-source scenarios.

Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space $\mathbf{x}_t \in \mathcal{X}$, we analyze them from a geometrical perspective. Our approach involves deriving the local latent basis within $\mathcal{X}$ by leveraging the pullback metric associated with their encoding feature maps. Remarkably, our discovered local latent basis enables image editing capabilities by moving $\mathbf{x}_t$, the latent space of DMs, along the basis vector at specific timesteps. We further analyze how the geometric structure of DMs evolves over diffusion timesteps and differs across different text conditions. This confirms the known phenomenon of coarse-to-fine generation, as well as reveals novel insights such as the discrepancy between $\mathbf{x}_t$ across timesteps, the effect of dataset complexity, and the time-varying influence of text prompts. To the best of our knowledge, this paper is the first to present image editing through $\mathbf{x}$-space traversal, editing only once at specific timestep $t$ without any additional training, and providing thorough analyses of the latent structure of DMs. The code to reproduce our experiments can be found at the [link](https://github.com/enkeejunior1/Diffusion-Pullback).

Mirror descent is an elegant optimization technique that leverages a dual space of parametric models to perform gradient descent. While originally developed for convex optimization, it has increasingly been applied in the field of machine learning. In this study, we propose a novel approach for using mirror descent to initialize the parameters of neural networks. Specifically, we demonstrate that by using the Hopfield model as a prototype for neural networks, mirror descent can effectively train the model with significantly improved performance compared to traditional gradient descent methods that rely on random parameter initialization. Our findings highlight the potential of mirror descent as a promising initialization technique for enhancing the optimization of machine learning models.

Adipose tissue grows by two mechanisms: hyperplasia (cell number increase) and hypertrophy (cell size increase). Genetics and diet affect the relative contributions of these two mechanisms to the growth of adipose tissue in obesity. In this study, the size distributions of epididymal adipose cells from two mouse strains, obesity-resistant FVB/N and obesity-prone C57BL/6, were measured after 2, 4, and 12 weeks under regular and high-fat feeding conditions. The total cell number in the epididymal fat pad was estimated from the fat pad mass and the normalized cell-size distribution. The cell number and volume-weighted mean cell size increase as a function of fat pad mass. To address adipose tissue growth precisely, we developed a mathematical model describing the evolution of the adipose cell-size distributions as a function of the increasing fat pad mass, instead of the increasing chronological time. Our model describes the recruitment of new adipose cells and their subsequent development in different strains, and with different diet regimens, with common mechanisms, but with diet- and genetics-dependent model parameters. Compared to the FVB/N strain, the C57BL/6 strain has greater recruitment of small adipose cells. Hyperplasia is enhanced by high-fat diet in a strain-dependent way, suggesting a synergistic interaction between genetics and diet. Moreover, high-fat feeding increases the rate of adipose cell size growth, independent of strain, reflecting the increase in calories requiring storage. Additionally, high-fat diet leads to a dramatic spreading of the size distribution of adipose cells in both strains; this implies an increase in size fluctuations of adipose cells through lipid turnover.