Cross-domain classification of hyperspectral remote sensing images is one of the hotspots of research in recent years, and its main problem is insufficient training samples. To address this issue, few-shot learning (FSL) has emerged as a promising paradigm in cross-domain classification tasks. However, a notable limitation of most existing FSL methods is that they focus only on local information and less on the critical role of global information. Based on this, this paper proposes a new feature processing method with adaptive band selection, which takes into account the global nature of image features. Firstly, adaptive band analysis is performed in the target domain, and threshold analysis is used to determine the number of selected bands. Secondly, a band selection method is employed to select representative bands from the spectral bands of the high-dimensional data according to the determined band count. Finally, the weights of the selected bands are analyzed, fully considering the importance of pixel weight, and then the results are used as inputs for the classification model. The experimental results on various datasets show that this method can effectively improve the classification accuracy and generalization ability. Meanwhile, the results of the objective accuracy index of the proposed method in different databases improved by 3.9%, 4.7% and 5.4%.

This paper studies a general framework for high-order tensor SVD. We propose a new computationally efficient algorithm, tensor-train orthogonal iteration (TTOI), that aims to estimate the low tensor-train rank structure from the noisy high-order tensor observation. The proposed TTOI consists of initialization via TT-SVD [1] and new iterative backward/forward updates. We develop the general upper bound on estimation error for TTOI with the support of several new representation lemmas on tensor matricizations. By developing a matching information-theoretic lower bound, we also prove that TTOI achieves the minimax optimality under the spiked tensor model. The merits of the proposed TTOI are illustrated through applications to estimation and dimension reduction of high-order Markov processes, numerical studies, and a real data example on New York City taxi travel records. The software of the proposed algorithm is available online (https://github.com/Lili-Zheng-stat/TTOI).

This paper investigates the asymptotic behaviors of gradient descent algorithms (particularly accelerated gradient descent and stochastic gradient descent) in the context of stochastic optimization arising in statistics and machine learning, where objective functions are estimated from available data. We show that these algorithms can be computationally modeled by continuous-time ordinary or stochastic differential equations. We establish gradient flow central limit theorems to describe the limiting dynamic behaviors of these computational algorithms and the large-sample performances of the related statistical procedures, as the number of algorithm iterations and data size both go to infinity, where the gradient flow central limit theorems are governed by some linear ordinary or stochastic differential equations, like time-dependent Ornstein-Uhlenbeck processes. We illustrate that our study can provide a novel unified framework for a joint computational and statistical asymptotic analysis, where the computational asymptotic analysis studies the dynamic behaviors of these algorithms with time (or the number of iterations in the algorithms), the statistical asymptotic analysis investigates the large-sample behaviors of the statistical procedures (like estimators and classifiers) that are computed by applying the algorithms; in fact, the statistical procedures are equal to the limits of the random sequences generated from these iterative algorithms, as the number of iterations goes to infinity. The joint analysis results based on the obtained gradient flow central limit theorems lead to the identification of four factors---learning rate, batch size, gradient covariance, and Hessian---to derive new theories regarding the local minima found by stochastic gradient descent for solving non-convex optimization problems.