We give explicit polynomial-sized (in @math n and @math k) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree @math k in @math n variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we construct explicit semidefinite representations (polynomial-sized in @math k,m, and @math n) of the hyperbolicity cones associated with @math kth directional derivatives of polynomials of the form @math p(x)=det(?i=1nAixi) where the @math Ai are @math m×m symmetric matrices. These convex cones form an analogous family of outer approximations to any spectrahedral cone. Our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their (less well understood) dual cones.
Recently Br "a nd 'e n @cite_4 settled this conjecture in the affirmative giving spectrahedral representations of @math for @math of size @math . For each @math Br "a nd 'e n constructs a graph @math together with edge weights @math that are linear forms in @math so that where @math is the @math edge-weighted Laplacian of @math . Since @math is linear in the edge weights, and the edge weights are linear forms in @math , is a spectrahedral representation of size @math . With the exception of two distinguished vertices, the vertices of @math are indexed by all @math -tuples (for @math ) consisting of distinct elements of @math . Hence @math showing that Br "a nd 'e n's spectrahedral representation of @math has size @math . While Br "a nd 'e n's construction is of considerable theoretical interest, these representations (unlike ours) are not practical for optimization due to their prohibitive size.
@cite_4
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Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix @math , one may look at the spectrum of @math for a properly chosen @math . The issue is that the spectrum of @math might be contaminated by non-informational top eigenvalues, e.g., due to scale variations in the data, and the application of @math aims to remove these. Designing a good functional @math (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, @math where @math denotes the adjacency matrix and @math is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance @math , and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd o s-R 'enyi ensemble, which has no spectral gap, it is shown that graph powering produces a maximal' spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to massoulie-STOC , settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts massoulie-STOC,Mossel_SBM2,bordenave,colin3 . This is illustrated on a geometric block model that is dense in cliques. We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any @math , given cost and demand graphs with edge weights @math respectively, we can find a set @math with @math at most @math times the optimal non-uniform sparsest cut value, in time @math provided @math . Here @math is the @math 'th smallest generalized eigenvalue of the Laplacian matrices of cost and demand graphs; @math (resp. @math ) is the weight of edges crossing the @math cut in cost (resp. demand) graph and @math is the sparsity of the optimal cut. In words, we show that the non-uniform sparsest cut problem is easy when the generalized spectrum grows moderately fast. To the best of our knowledge, there were no results based on higher order spectra for non-uniform sparsest cut prior to this work. Even for uniform sparsest cut, the quantitative aspects of our result are somewhat stronger than previous methods. Similar results hold for other expansion measures like edge expansion, normalized cut, and conductance, with the @math 'th smallest eigenvalue of the normalized Laplacian playing the role of @math in the latter two cases. Our proof is based on an l1-embedding of vectors from a semi-definite program from the Lasserre hierarchy. The embedded vectors are then rounded to a cut using standard threshold rounding. We hope that the ideas connecting @math -embeddings to Lasserre SDPs will find other applications. Another aspect of the analysis is the adaptation of the column selection paradigm from our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather than vertices. This feature is important in order to extend the algorithms to non-uniform sparsest cut. We undertake a systematic study of sketching a quadratic form: given an @math matrix @math , create a succinct sketch @math which can produce (without further access to @math ) a multiplicative @math -approximation to @math for any desired query @math . While a general matrix does not admit non-trivial sketches, positive semi-definite (PSD) matrices admit sketches of size @math , via the Johnson-Lindenstrauss lemma, achieving the "for each" guarantee, namely, for each query @math , with a constant probability the sketch succeeds. (For the stronger "for all" guarantee, where the sketch succeeds for all @math 's simultaneously, again there are no non-trivial sketches.) We design significantly better sketches for the important subclass of graph Laplacian matrices, which we also extend to symmetric diagonally dominant matrices. A sequence of work culminating in that of Batson, Spielman, and Srivastava (SIAM Review, 2014), shows that by choosing and reweighting @math edges in a graph, one achieves the "for all" guarantee. Our main results advance this front. @math For the "for all" guarantee, we prove that 's bound is optimal even when we restrict to "cut queries" @math . In contrast, previous lower bounds showed the bound only for spectral-sparsifiers . @math For the "for each" guarantee, we design a sketch of size @math bits for "cut queries" @math . We prove a nearly-matching lower bound of @math bits. For general queries @math , we construct sketches of size @math bits. The spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure. It has also been employed as a graph representation for pattern matching since it is invariant to the labelling of the graph. There are, however, a number of potential drawbacks in using the spectrum as a representation of a graph. Firstly, more than one graph may share the same spectrum. It is well known, for example, that very few trees can be uniquely specified by their spectrum. Secondly, the spectrum may change dramatically with a small change structure. There are a wide variety of graph matrix representations from which the spectrum can be extracted. Among these are the adjacency matrix, combinatorial Laplacian, normalised Laplacian and unsigned Laplacian. Spectra can also be derived from the heat kernel matrix and path length distribution matrix. The choice of matrix representation clearly has a large effect on the suitability of spectrum in a number of pattern recognition tasks. In this paper we investigate the performance of the spectra as a graph representation in a variety of situations. Firstly, we investigate the cospectrality of the various matrix representations over large graph and tree sets, extending the work of previous authors. We then show that the Euclidean distance between spectra tracks the edit distance between graphs over a wide range of edit costs, and we analyse the accuracy of this relationship. We then use the spectra to both cluster and classify the graphs and demonstrate the effect of the graph matrix formulation on error rates. These results are produced using both synthetic graphs and trees and graphs derived from shape and image data.