Geometry in Physics

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. T...

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Hodge Laplacians on graphs

The problem of predicting image or video interestingness from their low-level feature representations has received increasing interest. As a highly subjective visual attribute, annotating the interestingness value of training data for learning a prediction model is challenging. To make the annotation less subjective and more reliable, recent studies employ crowdsourcing tools to collect pairwise comparisons – relying on majority voting to prune the annotation outliers/errors. In this paper, we propose a more principled way to identify annotation outliers by formulating the interestingness prediction task as a unified robust learning to rank problem, tackling both the outlier detection and interestingness prediction tasks jointly. Extensive experiments on both image and video interestingness benchmark datasets demonstrate that our new approach significantly outperforms state-of-the-art alternatives.

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Interestingness Prediction by Robust Learning to Rank

In this paper, we provide a short review of Retinex and then present a unifying framework. The fundamental assumption of all Retinex models is that the observed image is a multiplication between the illumination and the true underlying reflectance of the object. Starting from Morel's 2010 PDE model, where illumination is supposed to vary smoothly and where the reflectance is thus recovered from a hard-thresholded Laplacian of the observed image in a Poisson equation, we define our unifying Retinex model in two similar, but more general, steps. We reinterpret the gradient thresholding model as variational models with sparsity constraints. First, we look for a filtered gradient that is the solution of an optimization problem consisting of two terms: a sparsity prior of the reflectance and a fidelity prior of the reflectance gradient to the observed image gradient. Second, since this filtered gradient almost certainly is not a consistent image gradient, we then fit an actual reflectance gradient to it, subje...

Non-Local Retinex---A Unifying Framework and Beyond

Can one reduce the size of a graph without significantly altering its basic properties? The graph reduction problem is hereby approached from the perspective of restricted spectral approximation, a modification of the spectral similarity measure used for graph sparsification. This choice is motivated by the observation that restricted approximation carries strong spectral and cut guarantees, and that it implies approximation results for unsupervised learning problems relying on spectral embeddings. 
The paper then focuses on coarsening---the most common type of graph reduction. Sufficient conditions are derived for a small graph to approximate a larger one in the sense of restricted similarity. These findings give rise to nearly-linear algorithms that, compared to both standard and advanced graph reduction methods, find coarse graphs of improved quality, often by a large margin, without sacrificing speed.

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Graph Reduction with Spectral and Cut Guarantees

Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph The vertices are the alternatives, and the edge values comprise the comparison data The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data Since an exact match will usually be impossible, one settles for matching in a least squares sense This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas ofresearch These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus); other mathematics (Hodge decomposition, and random clique complexes); and applications (arbitrage, and ranking of sports teams) Not all of these connections are explored in this paper, but many are The underlying ideas are easy to explain, requiring only the four fundamental subspaces from elementary linear algebra One of our aims is to explain these basic ideas and connections, to get researchers in many fields interested in this topic Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development

Least Squares Ranking on Graphs

We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.

Technical note: Delaunay Hodge star

We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other things, this allows localization near topological features of interest. We derive a weighted least squares method by proving a discrete Hodge-deRham theorem on the isomorphism between the space of harmonic cochains and cohomology. The solution obtained either satisfies the Whitney form finite element exterior calculus equations or the discrete exterior calculus equations for harmonic cochains, depending on the discrete Hodge star used.

Cohomologous Harmonic Cochains

A discrete diagonal Hodge star operator for pairwise Delaunay triangulations was described in Hirani et al. (2013). This note fixes errors in two figures and some indexing mistakes in the original paper. One consequence of this fix is that discrete exterior calculus may be applicable to a wider class of meshes.

Corrigendum to “Delaunay Hodge star” [Comput. Aided Des. 45 (2013) 540–544]

There are several classes of operators on graphs to consider in deciding on a collection of building blocks for graph algorithms. One class involves traditional graph operations such as breadth first or depth first search, finding connected components, spanning trees, cliques and other sub graphs, operations for editing graphs and so on. Another class consists of linear algebra operators where the matrices somehow depend on a graph. It is the latter class of operators that this paper addresses. We describe a least squares formulation on graphs that arises naturally in problems of ranking, distributed clock synchronization, social choice, arbitrage detection, and many other applications. The resulting linear systems are analogous to Poisson's equations. We show experimental evidence that some iterative methods that work very well for continuous domains do not perform well on graphs whereas some such methods continue to work well. By studying graph problems that are analogous to discretizations of partial differential equations (PDEs) one can hope to isolate the specific computational obstacles that graph algorithms present due to absence of spatial locality. In contrast, such locality is inherent in PDE problems on continuous domains. There is also evidence that PDE based methods may suggest improvements suitable for implementation on graphs.

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Kaushik Kalyanaraman

Papers

Least Squares Ranking on Graphs

Technical note: Delaunay Hodge star

Cohomologous Harmonic Cochains

Corrigendum to “Delaunay Hodge star” [Comput. Aided Des. 45 (2013) 540–544]

Graph Laplacians and Least Squares on Graphs