基于双层正交投影与稀疏字典学习的图像低维判别表示算法研究

How to effectively extract low-dimensional discriminative information from high-dimensional image data is a long-standing challenging problem in image processing and machine learning. The most classical low-dimensional representation methods often suffer from the noise corruption and the low efficiency for extracting discrimination between images, as well as the high computational complexity and the high sample complexity. In this project, we propose to jointly learn discriminative low-dimensional image representations based on two-layer orthogonal projections and dictionary learning. The former is a well-known efficient tool to project data on to low dimensional subspace, while the latter is known for disentangling nonlinear underlying discriminative factors in data. It has been empirically observed that the structure in sparse domain could make the hidden patterns more prominent and easier to be captured, and sparse coefficients are often interpreted as the extracted features to promote the discriminative prediction task. This project aims to take advantage of both orthogonal projection and dictionary learning. The procedure can be concluded as follows. To avoid the high computational cost of direct sparse coding on large scale input image, we first present to learn sparse representation (SR) in an orthogonal projected space over a task-driven sparsifying dictionary. We then exploit the discriminative projection on SR. The whole learning process is treated as an optimization problem of trace quotient maximization, which involves the orthogonal projection on original data space, dictionary and the discriminative projection on sparse codes. The related cost function is well defined on a product manifold of the Stiefel manifold and the Oblique manifold, and could be resolved via the geometrical gradient descent algorithm. Finally, the performance of the proposed algorithmic framework is investigated on image clustering and classification.

随着图像大数据的不断发展，如何从高维度的图像大数据中高效地提取有用的低维判别信息是当前研究的热点问题之一。针对高维图像大数据提取低维判别信息效率较低、抗噪鲁棒性较差等问题，以及低维表示学习的高计算复杂度问题和高样本复杂度问题，申请人提出构建基于双层正交投影映射和单层稀疏字典学习的三层低维判别信息表示算法。该算法利用双层正交映射提高计算效率，同时利用一层稀疏分解提取数据的非线性判别信息并提高噪声鲁棒性，以期为高维图像的高效聚类和分类应用提供有力工具。在理论上，我们通过研究稀疏字典学习的可微性，保证所提模型具有全局优化解。在关键性技术上，我们研究赋予字典参数和正交投影参数合适的光滑流形约束，开发光滑流形上的高效几何梯度下降算法。最后，我们将所提模型应用于高维图像的分类和聚类，并通过一系列的对比实验验证其效果。

如何从高维度图像大数据中高效提取判别信息是当今研究的热门问题。本项目针对目前所面临关键问题展开分析，利用稀疏编码、几何优化、经典降维映射算法等，构建高维数据的低维判别表示学习模型，高效地挖掘数据内部丰富的几何结构信息和潜在因素，并探索其多样化的视觉应用。主要完成工作如下。第一，研究稀疏表示关于字典参数一阶微分可导的条件，分析其一阶导数的闭合形式和唯一性问题，解决基于微分流形乘积的联合学习优化问题，最终提出正交投影映射结合稀疏字典学习的低维判别信息表示算法，充分利用经典正交投影降维的高效性、稀疏字典学习的非线性特征提取能力和噪声鲁棒性；第二，设计参数的几何优化策略，解决稀疏字典学习和深度神经网络模型间的梯度传递问题，实现深度学习和稀疏编码的有机结合，充分利用了深度学习在特征学习上的优势和字典学习方法提取判别性特征的能力；第三，在图像分类、高光谱图像分类、无人机目标追踪等应用问题进行实验，取得了较好的结果。基于本项目，已在国内外核心刊物和国际会议上发表25篇高质量的学术论文；已获得3项授权专利，包括1项发明专利，2项实用新型专利。本项研究的完成为在高维数据提取低维判别信息提供新理论依据和技术支撑，也为相关学科的发展和应用上提供方法和思路的借鉴。