Low Tensor-Rank Adaptation of Kolmogorov--Arnold Networks

Kolmogorov--Arnold networks (KANs) have demonstrated their potential as an alternative to multi-layer perceptions (MLPs) in various domains, especially for science-related tasks. However, transfer learning of KANs remains a relatively unexplored area. In this paper, inspired by Tucker decomposition of tensors and evidence on the low tensor-rank structure in KAN parameter updates, we develop low tensor-rank adaptation (LoTRA) for fine-tuning KANs. We study the expressiveness of LoTRA based on Tucker decomposition approximations. Furthermore, we provide a theoretical analysis to select the learning rates for each LoTRA component to enable efficient training. Our analysis also shows that using identical learning rates across all components leads to inefficient training, highlighting the need for an adaptive learning rate strategy. Beyond theoretical insights, we explore the application of LoTRA for efficiently solving various partial differential equations (PDEs) by fine-tuning KANs. Additionally, we propose Slim KANs that incorporate the inherent low-tensor-rank properties of KAN parameter tensors to reduce model size while maintaining superior performance. Experimental results validate the efficacy of the proposed learning rate selection strategy and demonstrate the effectiveness of LoTRA for transfer learning of KANs in solving PDEs. Further evaluations on Slim KANs for function representation and image classification tasks highlight the expressiveness of LoTRA and the potential for parameter reduction through low tensor-rank decomposition.

Multisource Collaborative Domain Generalization for Cross-Scene Remote Sensing Image Classification

Cross-scene image classification aims to transfer prior knowledge of ground materials to annotate regions with different distributions and reduce hand-crafted cost in the field of remote sensing. However, existing approaches focus on single-source domain generalization to unseen target domains, and are easily confused by large real-world domain shifts due to the limited training information and insufficient diversity modeling capacity. To address this gap, we propose a novel multi-source collaborative domain generalization framework (MS-CDG) based on homogeneity and heterogeneity characteristics of multi-source remote sensing data, which considers data-aware adversarial augmentation and model-aware multi-level diversification simultaneously to enhance cross-scene generalization performance. The data-aware adversarial augmentation adopts an adversary neural network with semantic guide to generate MS samples by adaptively learning realistic channel and distribution changes across domains. In views of cross-domain and intra-domain modeling, the model-aware diversification transforms the shared spatial-channel features of MS data into the class-wise prototype and kernel mixture module, to address domain discrepancies and cluster different classes effectively. Finally, the joint classification of original and augmented MS samples is employed by introducing a distribution consistency alignment to increase model diversity and ensure better domain-invariant representation learning. Extensive experiments on three public MS remote sensing datasets demonstrate the superior performance of the proposed method when benchmarked with the state-of-the-art methods.

Low-Rank Tensor Learning by Generalized Nonconvex Regularization

In this paper, we study the problem of low-rank tensor learning, where only a few of training samples are observed and the underlying tensor has a low-rank structure. The existing methods are based on the sum of nuclear norms of unfolding matrices of a tensor, which may be suboptimal. In order to explore the low-rankness of the underlying tensor effectively, we propose a nonconvex model based on transformed tensor nuclear norm for low-rank tensor learning. Specifically, a family of nonconvex functions are employed onto the singular values of all frontal slices of a tensor in the transformed domain to characterize the low-rankness of the underlying tensor. An error bound between the stationary point of the nonconvex model and the underlying tensor is established under restricted strong convexity on the loss function (such as least squares loss and logistic regression) and suitable regularity conditions on the nonconvex penalty function. By reformulating the nonconvex function into the difference of two convex functions, a proximal majorization-minimization (PMM) algorithm is designed to solve the resulting model. Then the global convergence and convergence rate of PMM are established under very mild conditions. Numerical experiments are conducted on tensor completion and binary classification to demonstrate the effectiveness of the proposed method over other state-of-the-art methods.

Robust Instance Optimal Phase-Only Compressed Sensing

Phase-only compressed sensing (PO-CS) is concerned with the recovery of structured signals from the phases of complex measurements. Recent results show that structured signals in the standard sphere $\mathbb{S}^{n-1}$ can be exactly recovered from complex Gaussian phases, by recasting PO-CS as linear compressed sensing and then applying existing solvers such as basis pursuit. Known guarantees are either non-uniform or do not tolerate model error. We show that this linearization approach is more powerful than the prior results indicate. First, it achieves uniform instance optimality: Under complex Gaussian matrix with a near-optimal number of rows, this approach uniformly recovers all signals in $\mathbb{S}^{n-1}$ with errors proportional to the model errors of the signals. Specifically, for sparse recovery there exists an efficient estimator $\mathbf{x}^\sharp$ and some universal constant $C$ such that $\|\mathbf{x}^\sharp-\mathbf{x}\|_2\le \frac{C\sigma_s(\mathbf{x})_1}{\sqrt{s}}~(\forall\mathbf{x}\in\mathbb{S}^{n-1})$, where $\sigma_s(\mathbf{x})_1=\min_{\mathbf{u}\in\Sigma^n_s}\|\mathbf{u}-\mathbf{x}\|_1$ is the model error under $\ell_1$-norm. Second, the instance optimality is robust to small dense disturbances and sparse corruptions that arise before or after capturing the phases. As an extension, we also propose to recast sparsely corrupted PO-CS as a linear corrupted sensing problem and show that this achieves perfect reconstruction of the signals. Our results resemble the instance optimal guarantees in linear compressed sensing and, to our knowledge, are the first results of this kind for a non-linear sensing scenario.

Non-Negative Reduced Biquaternion Matrix Factorization with Applications in Color Face Recognition

Reduced biquaternion (RB), as a four-dimensional algebra highly suitable for representing color pixels, has recently garnered significant attention from numerous scholars. In this paper, for color image processing problems, we introduce a concept of the non-negative RB matrix and then use the multiplication properties of RB to propose a non-negative RB matrix factorization (NRBMF) model. The NRBMF model is introduced to address the challenge of reasonably establishing a non-negative quaternion matrix factorization model, which is primarily hindered by the multiplication properties of traditional quaternions. Furthermore, this paper transforms the problem of solving the NRBMF model into an RB alternating non-negative least squares (RB-ANNLS) problem. Then, by introducing a method to compute the gradient of the real function with RB matrix variables, we solve the RB-ANNLS optimization problem using the RB projected gradient algorithm and conduct a convergence analysis of the algorithm. Finally, we validate the effectiveness and superiority of the proposed NRBMF model in color face recognition.

Data Valuation by Leveraging Global and Local Statistical Information

Data valuation has garnered increasing attention in recent years, given the critical role of high-quality data in various applications, particularly in machine learning tasks. There are diverse technical avenues to quantify the value of data within a corpus. While Shapley value-based methods are among the most widely used techniques in the literature due to their solid theoretical foundation, the accurate calculation of Shapley values is often intractable, leading to the proposal of numerous approximated calculation methods. Despite significant progress, nearly all existing methods overlook the utilization of distribution information of values within a data corpus. In this paper, we demonstrate that both global and local statistical information of value distributions hold significant potential for data valuation within the context of machine learning. Firstly, we explore the characteristics of both global and local value distributions across several simulated and real data corpora. Useful observations and clues are obtained. Secondly, we propose a new data valuation method that estimates Shapley values by incorporating the explored distribution characteristics into an existing method, AME. Thirdly, we present a new path to address the dynamic data valuation problem by formulating an optimization problem that integrates information of both global and local value distributions. Extensive experiments are conducted on Shapley value estimation, value-based data removal/adding, mislabeled data detection, and incremental/decremental data valuation. The results showcase the effectiveness and efficiency of our proposed methodologies, affirming the significant potential of global and local value distributions in data valuation.

A New Cross-Space Total Variation Regularization Model for Color Image Restoration with Quaternion Blur Operator

The cross-channel deblurring problem in color image processing is difficult to solve due to the complex coupling and structural blurring of color pixels. Until now, there are few efficient algorithms that can reduce color infection in deblurring process. To solve this challenging problem, we present a novel cross-space total variation (CSTV) regularization model for color image deblurring by introducing a quaternion blur operator and a cross-color space regularization functional. The existence and uniqueness of the solution is proved and a new L-curve method is proposed to find a sweet balance of regularization functionals on different color spaces. The Euler-Lagrange equation is derived to show that CSTV has taken into account the coupling of all color channels and the local smoothing within each color channel. A quaternion operator splitting method is firstly proposed to enhance the ability of color infection reduction of the CSTV regularization model. This strategy also applies to the well-known color deblurring models. Numerical experiments on color image databases illustrate the efficiency and manoeuvrability of the new model and algorithms. The color images restored by them successfully maintain the color and spatial information and are of higher quality in terms of PSNR, SSIM, MSE and CIEde2000 than the restorations of the-state-of-the-art methods.

Multispectral Image Restoration by Generalized Opponent Transformation Total Variation

Multispectral images (MSI) contain light information in different wavelengths of objects, which convey spectral-spatial information and help improve the performance of various image processing tasks. Numerous techniques have been created to extend the application of total variation regularization in restoring multispectral images, for example, based on channel coupling and adaptive total variation regularization. The primary contribution of this paper is to propose and develop a new multispectral total variation regularization in a generalized opponent transformation domain instead of the original multispectral image domain. Here opponent transformations for multispectral images are generalized from a well-known opponent transformation for color images. We will explore the properties of generalized opponent transformation total variation (GOTTV) regularization and the corresponding optimization formula for multispectral image restoration. To evaluate the effectiveness of the new GOTTV method, we provide numerical examples that showcase its superior performance compared to existing multispectral image total variation methods, using criteria such as MPSNR and MSSIM.

On the Expressive Power of a Variant of the Looped Transformer

Besides natural language processing, transformers exhibit extraordinary performance in solving broader applications, including scientific computing and computer vision. Previous works try to explain this from the expressive power and capability perspectives that standard transformers are capable of performing some algorithms. To empower transformers with algorithmic capabilities and motivated by the recently proposed looped transformer (Yang et al., 2024; Giannou et al., 2023), we design a novel transformer block, dubbed Algorithm Transformer (abbreviated as AlgoFormer). Compared with the standard transformer and vanilla looped transformer, the proposed AlgoFormer can achieve significantly higher expressiveness in algorithm representation when using the same number of parameters. In particular, inspired by the structure of human-designed learning algorithms, our transformer block consists of a pre-transformer that is responsible for task pre-processing, a looped transformer for iterative optimization algorithms, and a post-transformer for producing the desired results after post-processing. We provide theoretical evidence of the expressive power of the AlgoFormer in solving some challenging problems, mirroring human-designed algorithms. Furthermore, some theoretical and empirical results are presented to show that the designed transformer has the potential to be smarter than human-designed algorithms. Experimental results demonstrate the empirical superiority of the proposed transformer in that it outperforms the standard transformer and vanilla looped transformer in some challenging tasks.

Uniform Recovery Guarantees for Quantized Corrupted Sensing Using Structured or Generative Priors

This paper studies quantized corrupted sensing where the measurements are contaminated by unknown corruption and then quantized by a dithered uniform quantizer. We establish uniform guarantees for Lasso that ensure the accurate recovery of all signals and corruptions using a single draw of the sub-Gaussian sensing matrix and uniform dither. For signal and corruption with structured priors (e.g., sparsity, low-rankness), our uniform error rate for constrained Lasso typically coincides with the non-uniform one [Sun, Cui and Liu, 2022] up to logarithmic factors. By contrast, our uniform error rate for unconstrained Lasso exhibits worse dependence on the structured parameters due to regularization parameters larger than the ones for non-uniform recovery. For signal and corruption living in the ranges of some Lipschitz continuous generative models (referred to as generative priors), we achieve uniform recovery via constrained Lasso with a measurement number proportional to the latent dimensions of the generative models. Our treatments to the two kinds of priors are (nearly) unified and share the common key ingredients of (global) quantized product embedding (QPE) property, which states that the dithered uniform quantization (universally) preserves inner product. As a by-product, our QPE result refines the one in [Xu and Jacques, 2020] under sub-Gaussian random matrix, and in this specific instance we are able to sharpen the uniform error decaying rate (for the projected-back projection estimator with signals in some convex symmetric set) presented therein from $O(m^{-1/16})$ to $O(m^{-1/8})$.