In the last few years, several interesting variants of kernel-based spectral methods have arisen in the applied mathematics literature. These ideas appeared in connection with new types of data, where pairs of objects or measurements of interest have a relationship that is "blurred" by the action of a nuisance parameter. More specifically, we can find this type of data in a wide range of problems, for instance in the class averaging algorithm for the cryo-electron microscope (cryo-EM) problem [62, 71], in a modern light source imaging technique known as ptychography [45], in graph realization problems [24, 25], in vectored PageRank [20], in multi-channels image processing [5], etc... Before we give further details about the cryo-EM problem, let us present the main building blocks of the methods we will study. They depend on the following three components: 1. an undirected graph G (V, E) which describes all observations.

Many relations of scientific interest are nonlinear, and even in linear systems distributions are often non-Gaussian, for example in fMRI BOLD data. A class of search procedures for causal relations in high dimensional data relies on sample derived conditional independence decisions. The most common applications rely on Gaussian tests that can be systematically erroneous in nonlinear non-Gaussian cases. Recent work (Gretton et al. (2009), Tillman et al. (2009), Zhang et al. (2011)) has proposed conditional independence tests using Reproducing Kernel Hilbert Spaces (RKHS). Among these, perhaps the most efficient has been KCI (Kernel Conditional Independence, Zhang et al. (2011)), with computational requirements that grow effectively at least as O(N3), placing it out of range of large sample size analysis, and restricting its applicability to high dimensional data sets. We propose a class of O(N2) tests using conditional correlation independence (CCI) that require a few seconds on a standard workstation for tests that require tens of minutes to hours for the KCI method, depending on degree of parallelization, with similar accuracy. For accuracy on difficult nonlinear, non-Gaussian data sets, we also compare a recent test due to Harris & Drton (2012), applicable to nonlinear, non-Gaussian distributions in the Gaussian copula, as well as to partial correlation, a linear Gaussian test.

Multi-layer perceptrons are often slow to learn nonlinear functions with complex local structure due to the global nature of their function approximations. It is shown that standard multi-layer perceptrons are actually a special case of a more general network formulation that incorporates B-splines into the node computations. This allows novel spline network architectures to be developed that can combine the generalization capabilities and scaling properties of global multi-layer feedforward networks with the computational efficiency and learning speed of local computational paradigms. Simulation results are presented for the well known spiral problem of Weiland and of Lang and Witbrock to show the effectiveness of the Spline Net approach.

Multi-layer perceptrons are often slow to learn nonlinear functions with complex local structure due to the global nature of their function approximations. It is shown that standard multi-layer perceptrons are actually a special case of a more general network formulation that incorporates B-splines into the node computations. This allows novel spline network architectures to be developed that can combine the generalization capabilities and scaling properties of global multi-layer feedforward networks with the computational efficiency and learning speed of local computational paradigms. Simulation results are presented for the well known spiral problem of Weiland and of Lang and Witbrock to show the effectiveness of the Spline Net approach.

Multi-layer perceptrons are often slow to learn nonlinear functions with complex local structure due to the global nature of their function approximations. It is shown that standard multi-layer perceptrons are actually a special case of a more general network formulation that incorporates B-splines into the node computations. This allows novel spline network architectures to be developed that can combine the generalization capabilities and scaling properties of global multi-layer feedforward networks with the computational efficiency and learning speed of local computational paradigms. Simulation results are presented for the well known spiral problem of Weiland and of Lang and Witbrock to show the effectiveness of the Spline Net approach.