Predicting When Random Low-Dimensional Reparameterizations Train Neural Networks

Neural networks can often be trained or fine-tuned through random low-dimensional reparameterization, where a small latent vector is mapped into a full parameter update by a frozen random map. This raises a practical question: how large must the latent search space be to reach a low-loss region? We first express the known accessibility transition in an equivalent conic form, centered for compact convex targets at the statistical dimension of the polar cone. Our main theoretical contribution is an orientation-resolved quadratic master formula that predicts the random-slice residual from both the curvature spectrum and the reference-to-solution displacement profile. It yields a self-consistent isotropic-orientation predictor and, in a conservative radius-only specialization, recovers the earlier Gaussian-width quadratic bound. Building on this analysis, we introduce Random Mapping Networks (RaMaN), which instantiate the predicted latent dimension using structured Hadamard or seed-regenerated Gaussian maps. These constructions avoid the O(dP) storage of dense random maps and reduce optimizer-state memory from O(P) to O(d). We also develop matrix-free curvature approximations and sweep-free dimension selection. Across controlled quadratic and neural-curvature experiments, the orientation-resolved predictor closely tracks measured transition locations and outperforms orientation-agnostic approximations when displacement direction matters. End-to-end experiments further show sharp, protocol-dependent training transitions across image and language models.

Discovering Geometric Biases in 3D Face Reconstruction: A Curvature-Aware Spectral Framework for Fairness Evaluation

3D Morphable Models (3DMMs) remain the standard parametric shape priors for many state-of-the-art 3D face reconstruction algorithms. However, as these models are derived from a finite number of 3D face samples, they inherit the morphological biases of their training data, potentially limiting their generalizability across diverse global populations. In this paper, we propose a novel framework to analyze 3DMM reconstructions through the lens of surface curvature, with the objective to discover, quantify and visualize biases. While standard evaluation metrics often rely on Euclidean distances, our reconstruction error captures subtle surface nuances such as local topology or undulations. To do so, we leverage the Laplace-Beltrami Operator (LBO) to generate high-resolution curvature error maps, providing a localized and geometrically meaningful visualization of discrepancies between ground truth faces and reconstructed meshes. We derive from it an error metric that we validated through a user study, observing a significantly higher correlation to human perception compared to traditional methods. Furthermore, we conduct extensive experiments across several 3DMM bases and fitting algorithms, uncovering systematic age-related biases and providing preliminary evidence of biases associated with gender and ethnicity. Our findings highlight the necessity of adopting curvature-aware evaluation protocols to ensure demographic fairness and geometric precision in future 3D face reconstruction research.

Curvature-Guided Module Localization for Low-Rank Detoxification of Backdoored Large Language Models

Backdoor attacks pose a serious threat to large language models (LLMs) by causing otherwise benign systems to produce attacker-specified malicious behavior when a hidden trigger is present. In this work, we study post hoc detoxification of backdoored LLMs in a practical setting where the defender has access to the poisoned model but does not wish to retrain the full network from scratch. We propose a mechanistically guided weight-space repair framework that first localizes modules involved in propagating trigger-induced behavior using activation patching and Fisher/K-FAC curvature analysis, and then applies targeted low-rank repair to only the most influential modules. We evaluate the method on poisoned variants of \texttt{Llama-3.2-1B-Instruct} with triggers inserted at the beginning, middle, and end of otherwise benign prompts. Results show that the proposed approach substantially suppresses trigger-conditioned malicious responses while preserving benign model behavior. These findings suggest that backdoor removal in LLMs can be formulated as a localized structural repair problem rather than only a broad behavioral alignment problem.