Operator eigenvalue problems play a critical role in various scientific fields and engineering applications, yet numerical methods are hindered by the curse of dimensionality. Recent deep learning methods provide an efficient approach to address this challenge by iteratively updating neural networks. These methods' performance relies heavily on the spectral distribution of the given operator: larger gaps between the operator's eigenvalues will improve precision, thus tailored spectral transformations that leverage the spectral distribution can enhance their performance. Based on this observation, we propose the Spectral Transformation Network (STNet). During each iteration, STNet uses approximate eigenvalues and eigenfunctions to perform spectral transformations on the original operator, turning it into an equivalent but easier problem. Specifically, we employ deflation projection to exclude the subspace corresponding to already solved eigenfunctions, thereby reducing the search space and avoiding converging to existing eigenfunctions. Additionally, our filter transform magnifies eigenvalues in the desired region and suppresses those outside, further improving performance. Extensive experiments demonstrate that STNet consistently outperforms existing learning-based methods, achieving state-of-the-art performance in accuracy.

Low-Rank Adaptation (LoRA) has emerged as a prominent technique for fine-tuning large foundation models. Despite its successes, the substantial parameter redundancy, which limits the capacity and efficiency of LoRA, has been recognized as a bottleneck. In this work, we systematically investigate the impact of redundancy in fine-tuning LoRA and reveal that reducing density redundancy does not degrade expressiveness. Based on this insight, we introduce \underline{S}pectral-\underline{e}ncoding \underline{L}ow-\underline{R}ank \underline{A}daptation (SeLoRA), which harnesses the robust expressiveness of spectral bases to re-parameterize LoRA from a sparse spectral subspace. Designed with simplicity, SeLoRA enables seamless integration with various LoRA variants for performance boosting, serving as a scalable plug-and-play framework. Extensive experiments substantiate that SeLoRA achieves greater efficiency with fewer parameters, delivering superior performance enhancements over strong baselines on various downstream tasks, including commonsense reasoning, math reasoning, and code generation.

The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. However, as FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. We address this issue by proposing a novel method that leverages batch matrix multiplications to efficiently construct Vandermonde-structured matrices and compute forward and inverse transforms, on arbitrarily distributed points. An efficient implementation of such structured matrix methods is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows one to extend neural operators to very general point distributions with significant gains in training speed over baselines, while retaining or improving accuracy.

Whitening loss provides theoretical guarantee in avoiding feature collapse for self-supervised learning (SSL) using joint embedding architectures. One typical implementation of whitening loss is hard whitening that designs whitening transformation over embedding and imposes the loss on the whitened output. In this paper, we propose spectral transformation (ST) framework to map the spectrum of embedding to a desired distribution during forward pass, and to modulate the spectrum of embedding by implicit gradient update during backward pass. We show that whitening transformation is a special instance of ST by definition, and there exist other instances that can avoid collapse by our empirical investigation. Furthermore, we propose a new instance of ST, called IterNorm with trace loss (INTL). We theoretically prove that INTL can avoid collapse and modulate the spectrum of embedding towards an equal-eigenvalue distribution during the course of optimization. Moreover, INTL achieves 76.6% top-1 accuracy in linear evaluation on ImageNet using ResNet-50, which exceeds the performance of the supervised baseline, and this result is obtained by using a batch size of only 256. Comprehensive experiments show that INTL is a promising SSL method in practice. The code is available at https://github.com/winci-ai/intl.

We present an algorithm based on convex optimization for constructing kernels for semi-supervised learning. The kernel matrices are derived from the spectral decomposition of graph Laplacians, and combine labeled and unlabeled data in a systematic fashion. Unlike previous work using diffusion kernels and Gaussian random field kernels, a nonparametric kernel approach is presented that incorporates order constraints during optimization. This results in flexible kernels and avoids the need to choose among different parametric forms. Our approach relies on a quadratically constrained quadratic program (QCQP), and is computationally feasible for large datasets. We evaluate the kernels on real datasets using support vector machines, with encouraging results.

We present an algorithm based on convex optimization for constructing kernels for semi-supervised learning. The kernel matrices are derived from the spectral decomposition of graph Laplacians, and combine labeled and unlabeled data in a systematic fashion. Unlike previous work using diffusion kernels and Gaussian random field kernels, a nonparametric kernel approach is presented that incorporates order constraints during optimization. This results in flexible kernels and avoids the need to choose among different parametric forms. Our approach relies on a quadratically constrained quadratic program (QCQP), and is computationally feasible for large datasets. We evaluate the kernels on real datasets using support vector machines, with encouraging results.

We present an algorithm based on convex optimization for constructing kernels for semi-supervised learning. The kernel matrices are derived from the spectral decomposition of graph Laplacians, and combine labeled and unlabeled data in a systematic fashion. Unlike previous work using diffusion kernels and Gaussian random field kernels, a nonparametric kernel approach is presented that incorporates order constraints during optimization. This results in flexible kernels and avoids the need to choose among different parametric forms. Our approach relies on a quadratically constrained quadratic program (QCQP), and is computationally feasible for large datasets. We evaluate the kernels on real datasets using support vector machines, with encouraging results.