computing similarity between two sequences can be performed by computing the distance between the model parameters, 统计流形上给出了两个点之间的距离，也是两个点之间的信息量，这个距离叫做Fisher信息度量，它等于流形上两个点之间的最小距离（也叫测地距离），而不是两个点之间的欧氏距离。

最近研究工作中涉及到一个新的数学方向，微分几何-》信息几何-》统计流形，而数学界华人的骄傲陈省身和丘成桐在此领域都很有建树，网上能够查到微分几何与广义相对论的关系。但现在我们应用的领域却是计算机视觉和数据库。

香农的信息论基础就是根据概率可以计算出信息量，给了信息一个定量的表示，也是所有通信类学科的基础。而信息几何就是从几何的角度研究概率统计分布。这里面流形和统计流形是个很重要的概念，流形其实就是一个光滑的曲线或者曲面，也可以看成数据点的集合；统计流形上的点则是另外一个含义，即参数化的概率密度函数，这些参数是统计流形的坐标。

在非线性数据分析、数据分类领域，流形学习得到了广泛的应用，主要是从2000年Science上连续发表了3篇流形学习的论文开始。现在的数据分析，数据一般是高维数据，如果看成一个向量的话，那么它就是高维欧氏空间中的一个点。一个高维数据集中，数据之间往往有许多隐含的联系，这些关联可能是以某种流形形式存在，找到了这种流形，就可以进行非线性降维或者聚类这样的操作了。

统计流形上的点主要是参数化的概率密度函数，一般来讲，直方图是概率密度分布的最大似然估计，那么统计流形上的点就可以看成是直方图向量。最重要的一点，统计流形上给出了两个点之间的距离，也是两个点之间的信息量，这个距离叫做Fisher信息度量，它等于流形上两个点之间的最小距离（也叫测地距离），而不是两个点之间的欧氏距离。

前几天见了北大的徐进老师，他的关于四色定理的证明据说快要在美国的顶级数学杂志上发表了，这也是他几十年研究的成果。静下心来做研究，做的多了发现数学真是大多数工程应用学科的基础，要解决新的问题，没有坚实的数学基础是

http://stat.fsu.edu/~anuj/pdf/papers/Y2009/TuragaChapterVideoManifolds.pdf

5.1 Feature Space Manifold: Kendall’s Shape Sphere for Human Gait Analysis Shape analysis plays a very important role in object recognition, matching and registration. There has been substantial work in shape representation and on defining a feature vector which captures the essential attributes of the shape. A description of shape must be invariant to translation, scale and rotation. The Kendall’s shape space is a natural feature to use in such cases. Given a binary image consisting of the silhouette of a person, we extract the shape from this binary image. The procedure for obtaining shapes from the video sequence is graphically illustrated in Figure 3(a). Note that each frame of the video sequence maps to a point on the spherical shape manifold. Consider a situation where there are two shape sequences and we wish to compare how similar these two shape sequences are. One may want to use non-parametric sequence matching such as Dynamic-Time warping or a parametric approach such as state-space modeling. In either case, we need to take into account the geometry of the shape-manifold for matching. Consider dynamic time warping, which has been successfully used by the speech recognition [34] community for performing non-linear time normalization. Pre-shape, as we have already discussed lies on a spherical manifold. In our experiments, we use the Procrustes shape distance described in section 3.2 during the DTW distance computations. For state-space modeling such as autoregressive (AR) or ARMA, we use the tangent structure of the manifold. We project a given sequence to the tangent plane constructed at the mean-point. The AR and ARMA model parameters are then estimated on the tangent-planes. The tangent structure for Kendall’s shape manifold was 20 Chellappa et al. discussed in 3.2.1. Once the model parameters are estimated, computing similarity between two sequences can be performed by computing the distance between the model parameters. We refer the reader to [48] for details of model fitting and computing similarity between the model-parameters. Next, we present some experiments that demonstrate the utility of these methods.