A Consistent Histogram Estimator for Exchangeable Graph Models

Exchangeable graph models (ExGM) subsume a number of popular network models. The mathematical object that characterizes an ExGM is termed a graphon. Finding scalable estimators of graphons, provably consistent, remains an open issue. In this paper, we propose a histogram estimator of a graphon that is provably consistent and numerically efficient. The proposed estimator is based on a sorting-and-smoothing (SAS) algorithm, which first sorts the empirical degree of a graph, then smooths the sorted graph using total variation minimization. The consistency of the SAS algorithm is proved by leveraging sparsity concepts from compressed sensing.Comment: 28 pages, 5 figure

Computational Image Formation

At the pinnacle of computational imaging is the co-optimization of camera and algorithm. This, however, is not the only form of computational imaging. In problems such as imaging through adverse weather, the bigger challenge is how to accurately simulate the forward degradation process so that we can synthesize data to train reconstruction models and/or integrating the forward model as part of the reconstruction algorithm. This article introduces the concept of computational image formation (CIF). Compared to the standard inverse problems where the goal is to recover the latent image $\mathbf{x}$ from the observation $\mathbf{y} = \mathcal{G}(\mathbf{x})$, CIF shifts the focus to designing an approximate mapping $\mathcal{H}_{\theta}$ such that $\mathcal{H}_{\theta} \approx \mathcal{G}$ while giving a better image reconstruction result. The word ``computational'' highlights the fact that the image formation is now replaced by a numerical simulator. While matching nature remains an important goal, CIF pays even greater attention on strategically choosing an $\mathcal{H}_{\theta}$ so that the reconstruction performance is maximized. The goal of this article is to conceptualize the idea of CIF by elaborating on its meaning and implications. The first part of the article is a discussion on the four attributes of a CIF simulator: accurate enough to mimic $\mathcal{G}$, fast enough to be integrated as part of the reconstruction, providing a well-posed inverse problem when plugged into the reconstruction, and differentiable in the backpropagation sense. The second part of the article is a detailed case study based on imaging through atmospheric turbulence. The third part of the article is a collection of other examples that fall into the category of CIF. Finally, thoughts about the future direction and recommendations to the community are shared

Stochastic blockmodel approximation of a graphon: Theory and consistent estimation

Non-parametric approaches for analyzing network data based on exchangeable graph models (ExGM) have recently gained interest. The key object that defines an ExGM is often referred to as a graphon. This non-parametric perspective on network modeling poses challenging questions on how to make inference on the graphon underlying observed network data. In this paper, we propose a computationally efficient procedure to estimate a graphon from a set of observed networks generated from it. This procedure is based on a stochastic blockmodel approximation (SBA) of the graphon. We show that, by approximating the graphon with a stochastic block model, the graphon can be consistently estimated, that is, the estimation error vanishes as the size of the graph approaches infinity.Comment: 20 pages, 4 figures, 2 algorithms. Neural Information Processing Systems (NIPS), 201