连续吸引子的学习算法研究

Continuous attractors have been widely used in the study of many diverse brain functions, such as local cortical information processing, saccadic motor programming, head orientation, working memory and etc. Scientists have realized that continuous attractors can be used to explain the intelligent behavior of organisms. At present, the research of continuous attractors based machine intelligence is just beginning, it is urgently necessary to establish a fundamental theory in mathematics. In preliminary studies, we have known that symmetrical networks can exhibit a dynamical behavior of linear continuous attractors. However, how can we represent and store mass similar patterns as continuous attractors，the learing mechnism of continuous attractors is not yet clear. This project aims to study learning algorithms of continuous attractors, which are helpful to construct the methods of learning algorithm of continuous attractors in detail. It is well known that the learning algorithm is very important in the area of machine intelligence. However, currently, there are very few studies on the learning algorithms of continuous attractors. In this project, the main research contents include: (1) learning algorithms of linear continuous attractors; (2) learning algorithms of nonlinear continuous attractors; (3) learning algorithms of multiple continuous attractors coexisting in the same network. In summary, the research of continuous attractors learning algorithms will help to build the theoretical foundation of continuous attractor networks and provide theoretical and experimental evidence for similar pattern recognition.

连续吸引子已被广泛应用于各种脑部功能的研究，如局部脑皮层信息处理，眼球跳变性运动，头部方向定位，工作记忆等。科学家们已经认识到，连续吸引子可以用来解释生物体的许多智能行为。目前，基于连续吸引子的机器智能方法研究尚处于起步阶段，非常有必要建立完整的基础理论。申请者在前期研究中发现对称网络具有线性连续吸引子的动力学性质。但是，利用连续吸引子表达和存储大量相似模式的学习机制尚未明确。本项目旨在研究连续吸引子的学习算法，其目标是发展出构建连续吸引子的具体学习算法。学习算法在机器智能研究中有着十分重要的地位。然而，连续吸引子的学习算法目前仅有极少的成果。本项目的主要研究内容包括：（1）线性连续吸引子的学习算法；（2）非线性连续吸引子的学习算法；（3）多个连续吸引子并存于同一网络的学习算法。连续吸引子学习算法的建立，必将有助于进一步奠定连续吸引子的理论基础，为相似模式的识别提供理论与实验依据。

本项目针对连续吸引子的学习算法开展理论研究，重点研究了数据的稀疏表达以及在低维流形上的结构。从动态网络的角度看，传统的模式识别假定模式由单个稳定状态构成，而现实中的模式往往表现为连续的运动状态，原有的吸引子不足以描述认知中呈现的连续动力学性质。.将连续吸引子网络用于相似模式的表达及存储等工程应用，首先需要解决的便是连续吸引子的学习算法问题。目前，该方面的研究尚处于起步阶段。基于此，本项目的主要研究内容包括：（1）利用高维数据的稀疏性及其存在低维流形结构的特性，发展新的模式表达方法。（2）将数据的稀疏表达方法，用于发展连续吸引子的学习算法。.在过去三年的研究工作中，项目组取得了一系列的研究成果，其主要贡献与科学意义可以概括如下：（1）理论上首次证明了，当数据不包含噪声时，基于Frobenius范数的自表达与Nuclear范数的低秩表达是等价的。而且，数据不包含噪声条件下基于Frobenius范数的自表达相似矩阵是一个块对角矩阵。根据稀疏表达的研究成果，网络的连接矩阵分块后可进行非线性连续吸引子的表达。（2）提出了新的高维数据表达方法，将联合表达和低秩矩阵恢复技术结合起来，通过封闭解的形式学习一个对称的低秩表达。这种方法保存了高维数据的子空间结构信息，揭示了子空间之间的关系，也为线性与非线性连续吸引子的表达提供了方法。（3）利用深度神经网络连续吸引子的结构性质，开展了一系列图像识别问题的研究。为连续吸引子学习算法的工程应用提供了思路。