Cynosure对《Modern Approaches to Discrete Curvature》的笔记(1)

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one finds a new take on localdata (known to fail at pointwise convergence, as mentioned above), averaged as ameasure rather than a vertex/edge-based quantity.
local averaging是引入measures的原因：For a long time, research was concerned withsectional curvature. The reason is that, as explained above, sectional curvatureinequalities can be characterized in terms of relations involving only distancefunctions. Thus, no further structure beyond the metric is needed in principle. Incontrast, Ricci curvature in Riemannian geometry involves some averaging andcontains information about volumes and eigenvalues of a Laplace operator whichagain we have introduced and discussed in terms of some local averaging. Suchan operation, however, needs a measure. Therefore, in order to define some kindof generalized Ricci curvature, we need some measure on our space in addition tothe metric, that is, the distance function. 
The reason we study Metric Geometry is due to the fact that it provides us with aminimalistic framework that requires no additional smoothness, nor does it imposeany supplementary, ad hoc structure upon a given geometric object. It is thereforeour firm belief that this property renders the metric method as an approach ideallysuited for the study of the various structures and problems encountered in computerScience in general, and in Graphics, Imaging and Vision in particular and that,moreover, this Newtonian stance of “hypotheses non fingo” represents not only thephilosophically correct attitude, it also is ideally suited for the truthful intelligenceof Digital Spaces.
Unfortunately, the “linear”, or “first order Geometry” considered in the articlesmentioned above does not fulfill all its expectations, and particularly so in Manifold。Learning (as well as in Imaging and Graphics). To explain this, we could emphasizethe essential importance of “second order Geometry”, or succinctly, the Geometry ofcurvature, in Graphics, Imaging and Manifold Learning.