Closed-Form Concept Erasure via Double Projections

While modern generative models such as diffusion-based architectures have enabled impressive creative capabilities, they also raise important safety and ethical risks. These concerns have led to growing interest in concept erasure, the process of removing unwanted concepts from model representations. Existing approaches often achieve strong erasure performance but rely on iterative optimization and may inadvertently distort unrelated concepts. In this work, we present a simple yet principled alternative: a linear transformation framework that achieves concept erasure analytically, without any training. Our method adapts a pretrained model through two sequential, closed-form steps: first, computing a proxy projection of the target concept, and second, applying a constrained transformation within the left null space of known concept directions. This design yields a deterministic and geometrically interpretable procedure for safe, efficient, and theory-grounded concept removal. Across a wide range of experiments, including object and style erasure on multiple Stable Diffusion variants and the flow-matching model (FLUX), our approach matches or surpasses the performance of state-of-the-art methods while preserving non-target concepts more faithfully. Requiring only a few seconds to apply, it offers a lightweight and drop-in tool for controlled model editing, advancing the goal of safer and more responsible generative models.

Machine Unlearning under Retain-Forget Entanglement

Forgetting a subset in machine unlearning is rarely an isolated task. Often, retained samples that are closely related to the forget set can be unintentionally affected, particularly when they share correlated features from pretraining or exhibit strong semantic similarities. To address this challenge, we propose a novel two-phase optimization framework specifically designed to handle such retai-forget entanglements. In the first phase, an augmented Lagrangian method increases the loss on the forget set while preserving accuracy on less-related retained samples. The second phase applies a gradient projection step, regularized by the Wasserstein-2 distance, to mitigate performance degradation on semantically related retained samples without compromising the unlearning objective. We validate our approach through comprehensive experiments on multiple unlearning tasks, standard benchmark datasets, and diverse neural architectures, demonstrating that it achieves effective and reliable unlearning while outperforming existing baselines in both accuracy retention and removal fidelity.

Deep learning and the rate of approximation by flows

We investigate the dependence of the approximation capacity of deep residual networks on its depth in a continuous dynamical systems setting. This can be formulated as the general problem of quantifying the minimal time-horizon required to approximate a diffeomorphism by flows driven by a given family $\mathcal F$ of vector fields. We show that this minimal time can be identified as a geodesic distance on a sub-Finsler manifold of diffeomorphisms, where the local geometry is characterised by a variational principle involving $\mathcal F$. This connects the learning efficiency of target relationships to their compatibility with the learning architectural choice. Further, the results suggest that the key approximation mechanism in deep learning, namely the approximation of functions by composition or dynamics, differs in a fundamental way from linear approximation theory, where linear spaces and norm-based rate estimates are replaced by manifolds and geodesic distances.

Interpolation, Approximation and Controllability of Deep Neural Networks

We investigate the expressive power of deep residual neural networks idealized as continuous dynamical systems through control theory. Specifically, we consider two properties that arise from supervised learning, namely universal interpolation - the ability to match arbitrary input and target training samples - and the closely related notion of universal approximation - the ability to approximate input-target functional relationships via flow maps. Under the assumption of affine invariance of the control family, we give a characterisation of universal interpolation, showing that it holds for essentially any architecture with non-linearity. Furthermore, we elucidate the relationship between universal interpolation and universal approximation in the context of general control systems, showing that the two properties cannot be deduced from each other. At the same time, we identify conditions on the control family and the target function that ensures the equivalence of the two notions.