支持张量机的稀疏优化理论与算法研究

Support vector machine (SVM), which is recognized as an effective tool for classification and regression, has been widely used in pattern recognition, image processing, information retrieval and so on. Support tensor machine (STM), as an extension of SVM, has become a hot topic in the field of data mining. This project aims to establish sparse optimization theory and algorithms for STM. The main objective includes establishing some sparse STM models corresponding to 0-norm, 1-norm and adaptive p-norm (0<p<1) and some other models corresponding to tensor sparse decomposition and tensor sparse decomposition with orthogonal constraints, discussing their theoretical properties and designing some fast and efficient solution algorithms, finally making complexity analysis and convergence analysis. In the field of application, we will construct some sparse STM models and their optimization algorithms for the prediction of cancer cells to drug sensitivity. This research can enrich the theory and methods in data mining and promote the development of optimization and STM, and also has great scientific significance and practical value to solve practical problems.

支持向量机是求解分类问题和回归问题的有力工具，已被广泛应用于模式识别、图片处理、信息检索等领域。支持张量机是支持向量机的推广，因其很强的应用性而成为当今数据挖掘领域的一个研究热点。本项目旨在开展支持张量机的稀疏优化理论与算法研究。具体地，首先通过引入符合张量稀疏结构的0-范数、1-范数和可调节p-范数(0<p<1)正则项，构建对应的支持张量机稀疏模型，然后利用张量稀疏分解和稀疏正交分解技术，分别构建相应的稀疏优化模型，同时讨论这些新模型的理论性质，并设计快速高效的求解算法及对算法进行复杂性和收敛性分析。在应用领域，将针对癌细胞系对药物敏感度预测问题构建特定的支持张量机稀疏模型，并设计求解算法。本项目的研究成果能够丰富数据挖掘理论和方法，推动最优化和支持张量机方法的发展，对实际问题的解决具有重要的科学意义和实用价值。

支持张量机因其很强的应用性而成为当今数据挖掘领域的一个研究热点。本项目通过引入符合张量稀疏结构的0-范数、1-范数和可调节p-范数(0<p<1)，建立了相应的支持张量机稀疏模型，同时讨论这些新模型的理论性质，并设计快速高效的求解算法及对算法进行复杂性和收敛性分析。然后针对张量特征值互补问题，压电型张量和对称矩形张量，讨论了部分性质及其判定性条件。相关成果发表在《IEEE Geoscience and Remote Sensing Letters》《Journal of Inequalities and Applications》《Axioms》《Journal of Industrial and Management Optimization》等国内外重要学术期刊之上。本项目的研究成果能够丰富数据挖掘的理论和方法，推动最优化和支持张量机方法的发展，对实际问题的解决具有重要的科学意义和实用价值。