Low-rank adaptation (LoRA) has become a cornerstone of parameter-efficient fine-tuning for large pretrained models, augmenting frozen pretrained weights with a low-rank update formed by the product of two smaller matrices. Recent research indicates that scale mismatches between these two factors frequently induce unstable training dynamics, leading to suboptimal performance. Here, we reformulate low-rank adaptation by parameterizing the weight update as a single low-rank matrix multiplied by its own transpose. This formulation, which we term SingLoRA , eliminates inter-matrix scale conflicts by construction, thereby promoting stable optimization, while approximately halving the number of trainable parameters. Through the lens of the efficient feature learning framework, we show that SingLoRA is provably stable by design while retaining the expressive capacity of LoRA within the transformer architecture. Extensive experiments on multiple tasks validate these benefits. Extensive experiments on multiple tasks validate these benefits. In common sense reasoning, fine-tuning LLaMA-7B on MNLI with SingLoRA achieves 91.3% accuracy—surpassing LoRA (89.1%) and LoRA+ (90.2%)—while using only 60% of their parameter budget. In image generation, fine-tuning Stable Diffusion with SingLoRA significantly improves image fidelity on DreamBooth, achieving a DINO similarity score of 0.151, compared to scores of 0.148 and 0.143 for DoRA and LoRA, respectively.
Determining and analyzing geometric features are often an essential step in understanding and classifying three-dimensional objects. Among geometric features, the normal vector field provides a local and global approach to analyze the surface structure of objects. However, for digital surfaces, obtaining this field often requires parameter adaptation. In this article, we propose a novel parameter-free approach that structures the neighborhood of a point with a fan of local digital planar sections of the surface. This allows us to robustly capture local configurations and to adapt to the local flatness of digital surfaces. Experimental evaluations using multigrid approaches show that it is a reasonable proposal that maintains comparable accuracy to state-of-the-art methods. The experiments are reproducible as the source code is available online: https://github.com/amareche/local-fan-estimation .
Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric objects). The present paper provides a characterization of the class of homology and cohomology bases that are computed by standard algorithmic methods. The proof of this characterization relies on the Homological Discrete Vector Field, a combinatorial structure for computing homology, which encompasses several standard methods (persistent homology, tri-partitions, Smith Normal Form, discrete Morse theory). These results refine the combinatorial homology theory and provide novel ideas to gain more control over the computation of homology generators.
In mathematical morphology for grey-level functions, an image is analysed by another image named the structuring function. This structuring function is translated over the image domain and summed to the image. However, in an image presenting lighting variations, when the structuring function is added to this image, the amplitude of this translated structuring function should vary according to the image intensity. Such a property is not verified in mathematical morphology for grey-level functions, when the structuring function is summed to the image with the usual additive law. This issue has been addressed by defining a framework that uses an additive law, for which the amplitude of the structuring function added to the image varies according to the amplitude of the image itself. This additive law is chosen within the logarithmic image processing framework and models the lighting variations with a physical cause such as a change of light intensity. The framework is named logarithmic mathematical morphology (LMM) and allows the definition of operators which are robust to such lighting variations.
Partial differential equations (PDEs) are fundamental mathematical models in scientific computing and also play a central role in image processing tasks such as diffusion-based denoising, Poisson image editing, image inpainting, and variational reconstruction. In recent years, machine learning methods represented by the random feature method (RFM) have provided a mesh-free paradigm for solving PDEs on high-dimensional or geometrically complex domains. The adaptive feature capture method (AFCM) further improves RFM by using gradient information to concentrate collocation points and feature functions in critical regions. However, when the gradient of the solution is moderately varying or spatially widespread, the purely gradient-driven sampling strategy of AFCM can lead to excessive clustering of collocation points, numerical oscillations, and ill-conditioned least-squares systems. This paper proposes a stabilized adaptive feature capture method (S-AFCM) that improves the robustness of the adaptive loop through local density smoothing and diversity-enhanced feature generation. Specifically, we introduce a density-aware monitor function and normalize it into an adaptive sampling probability density function (PDF), thereby distinguishing the monitor from a supervised-learning loss and penalizing over-concentrated sampling. A greedy farthest point sampling (FPS) strategy we keep a comparable range of basis-function is also used to generate direction vectors for basis functions, improving directional diversity and reducing linear dependence. The required derivatives are evaluated from explicit derivative matrices of the random-feature expansion; unlike PINNs, S-AFCM does not train all network parameters by gradient descent but solves a linear least-squares system for the output coefficients after each adaptive reconstruction. Numerical experiments show that S-AFCM is effective not only for benchmark PDE solving, but can also be applied to image restoration problems with blurred or weak boundaries.
