Davi Geiger, Ricardo Pereira
Learning a map from an input set to an output set is similar to the prob(cid:173) lem of reconstructing hypersurfaces from sparse data (Poggio and Girosi, 1990). In this framework, we discuss the problem of automatically select(cid:173) ing "minimal" surface data. The objective is to be able to approximately reconstruct the surface from the selected sparse data. We show that this problem is equivalent to the one of compressing information by data re(cid:173) moval and the one oflearning how to teach. Our key step is to introduce a process that statistically selects the data according to the model. During the process of data selection (learning how to teach) our system (teacher) is capable of predicting the new surface, the approximated one provided by the selected data. We concentrate on piecewise smooth surfaces, e.g. images, and use mean field techniques to obtain a deterministic network that is shown to compress image data.
1 Learning and surface reconstruction
Given a dense input data that represents a hypersurface, how could we automatically select very few data points such as to be able to use these fewer data points (sparse data) to approximately reconstruct the hypersurface ? We will be using the term surface to refer to hypersurface (surface in multidimen-