On the Shape of Data

One of the active areas in machine learning deals with the problem of determining the "shape" of data. In its simplest setting, one may be concerned with the geometric structure of spatial coordinates of data in 3-dimensional space. More abstractly, the variables in any multivariate data can be viewed as coordinates of some multi-dimensional space, in which case one may be interested in some geometric or topological feature of the point cloud data in that space. Methods for determining the "shape" of data are most useful in high-dimensional data, where visual methods cannot be employed. Also, given that identifying qualitative features of data is often the first step of a model-building endeavor, such methods are useful in narrowing down the types of models that are appropriate. The focus of this talk is on one particular method from algebraic topology, called Morse theory, which can reveal some important aspects of the underlying structure of data.