Theoretical Study Of Manifold Learning And Related Improvement

This article starts from the basic theory of the manifold learning firstly, and describes several kinds of well-known manifold learning algorithms. By the he research on these algorithms, this paper gives two ways to improve the algorithms creatively.Firstly, in the course of the research, we found that the tangent space of the data point plays an very important role in the manifold learning. In theory, many well-known manifold learning algorithms(such as HLLE, LTSA, etc)are based on the tangent space of the data point, but in practice, these algorithms always use the space constructed by the principal component of sample covariance matrix that created by the data point' neighborhood. So far, the industry of the manifold learning does not seem to have a clear understanding and specific comments between theory and practice for these algorithms. This paper firstly proves that the space constructed by the principal comment of sample covariance matrix is not the real tangent space of the data point, but just the tangent space of the neighborhood's center. So this paper proposes a new calculation of the data point's tangent space. By the mathematical deduction, We prove that the space calculated by this method is the tangent space of the data point itself in the approximation of the first-order Taylor expansion, more importantly, the proposed method dose not increase any computational complexity.Secondly, through the theoretical research of the local linear algorithm(LLE),we have found that there are many defects of the dimension reduction by the LLE algorithm, and under the assumptions of LLE algorithm: in the local neighborhood, there is one linear relationship between the points, so this paper gives out one algorithm whose selection of the local neighborhood by the larger absolute value of the correlation coefficient between the vectors, thereby reducing the effect on the dimension reduction caused by the improper selection of the local neighborhood, so performance will be significantly improved compared with LLE algorithm, while the concept of the correlation coefficient presented here can serve as a basis for data reduction and data mining, thus breaking the traditional that solely using the distance for the routine selection of the neighbor, so providing a new way of thinking for the exploration of the manifold learning algorithm.