Does generalization performance of lql^q regularization learning depend on qq? A negative example

$l^q$-regularization has been demonstrated to be an attractive technique in machine learning and statistical modeling. It attempts to improve the generalization (prediction) capability of a machine (model) through appropriately shrinking its coefficients. The shape of a $l^q$ estimator differs in varying choices of the regularization order $q$. In particular, $l^1$ leads to the LASSO estimate, while $l^{2}$ corresponds to the smooth ridge regression. This makes the order $q$ a potential tuning parameter in applications. To facilitate the use of $l^{q}$-regularization, we intend to seek for a modeling strategy where an elaborative selection on $q$ is avoidable. In this spirit, we place our investigation within a general framework of $l^{q}$-regularized kernel learning under a sample dependent hypothesis space (SDHS). For a designated class of kernel functions, we show that all $l^{q}$ estimators for $0< q < \infty$ attain similar generalization error bounds. These estimated bounds are almost optimal in the sense that up to a logarithmic factor, the upper and lower bounds are asymptotically identical. This finding tentatively reveals that, in some modeling contexts, the choice of $q$ might not have a strong impact in terms of the generalization capability. From this perspective, $q$ can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..Comment: 35 pages, 3 figure

SpectralDiff: A Generative Framework for Hyperspectral Image Classification with Diffusion Models

Hyperspectral Image (HSI) classification is an important issue in remote sensing field with extensive applications in earth science. In recent years, a large number of deep learning-based HSI classification methods have been proposed. However, existing methods have limited ability to handle high-dimensional, highly redundant, and complex data, making it challenging to capture the spectral-spatial distributions of data and relationships between samples. To address this issue, we propose a generative framework for HSI classification with diffusion models (SpectralDiff) that effectively mines the distribution information of high-dimensional and highly redundant data by iteratively denoising and explicitly constructing the data generation process, thus better reflecting the relationships between samples. The framework consists of a spectral-spatial diffusion module, and an attention-based classification module. The spectral-spatial diffusion module adopts forward and reverse spectral-spatial diffusion processes to achieve adaptive construction of sample relationships without requiring prior knowledge of graphical structure or neighborhood information. It captures spectral-spatial distribution and contextual information of objects in HSI and mines unsupervised spectral-spatial diffusion features within the reverse diffusion process. Finally, these features are fed into the attention-based classification module for per-pixel classification. The diffusion features can facilitate cross-sample perception via reconstruction distribution, leading to improved classification performance. Experiments on three public HSI datasets demonstrate that the proposed method can achieve better performance than state-of-the-art methods. For the sake of reproducibility, the source code of SpectralDiff will be publicly available at https://github.com/chenning0115/SpectralDiff