Generative adversarial networks (GANs) are an exciting alternative to algorithms for solving density estimation problems---using data to assess how likely samples are to be drawn from the same distribution. Instead of explicitly computing these probabilities, GANs learn a generator that can match the given probabilistic source. This paper looks particularly at this matching capability in the context of problems with one-dimensional outputs. We identify a class of function decompositions with properties that make them well suited to the critic role in a leading approach to GANs known as Wasserstein GANs. We show that Taylor and Fourier series decompositions belong to our class, provide examples of these critics outperforming standard GAN approaches, and suggest how they can be scaled to higher dimensional problems in the future.

Diffusion distance has been shown to be significantlymore effective than Euclidean distance in multi-scalerecognition of similar experiences in Recognition-Primed Decision making In this paper, we first examine the experience data set used inthe previous study. The visualization of the data set(using the first three dominant eigenvectors of the diffusion space) suggests the applicability of the diffusion approach. Second, we investigate two approaches to the computation of diffusion distance: Spectrum based and Probability-Matching based. Specifically, by ‘Spectrumbased’ approach we refer to the one derived in terms of the eigenvalues/eigenvectors of the normalized diffusion matrix. We use the term ‘Probability-Matching’ to refer to the use of various probability distances, where the original L2 diffusion distance is treated as a special case. Our preliminary result indicates that the performance of using L2 diffusion distance at least is tied with the use of Spectrum based distance. Furthermore, when spectrum based approach is applied, we have to use the embedding and extending techniques for labeling new experience data, while such recomputation is not necessary when the L2 diffusion distance is used. We do not need to recompute the diffusion matrix, hence the diffusion map each time when adding a new data. It is more natural and robust especially for labeling new single experience data. The numerical examples also show the improvement on the performance. We are currently working on several other Probability-Matching approaches (e.g. the Earth-Mover’s Distance).