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Register a new userGramian Tensor Decomposition via Semidefinite Programming
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In this paper we examine a symmetric tensor decomposition problem, the Gramian decomposition, posed as a rank minimization problem. We study the relaxation of the problem and consider cases when the relaxed solution is a solution to the original problem. In some instances of tensor rank and order, we prove generically that the solution to the relaxation will be optimal in the original. In other cases, we present interesting examples and approaches that demonstrate the intricacy of this problem.
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Given all (finite) moments of two measures $mu$ and $lambda$ on $R^n$, we provide a numerical scheme to obtain the Lebesgue decomposition $mu= u+psi$ with $ ulllambda$ and $psiperplambda$. When$ u$ has a density in $L_infty(lambda)$ then we obtain two sequences of finite moments vectorsof increasing size (the number of moments) which converge to the moments of $ u$ and $psi$ respectively, as the number of moments increases. Importantly, {it no} `a priori knowledge on the supports of $mu, u$ and $psi$ is required.
The tendency of semidefinite programs to compose perfectly under product has been exploited many times in complexity theory: for example, by Lovasz to determine the Shannon capacity of the pentagon; to show a direct sum theorem for non-deterministic communication complexity and direct product theorems for discrepancy; and in interactive proof systems to show parallel repetition theorems for restricted classes of games. Despite all these examples of product theorems--some going back nearly thirty years--it was only recently that Mittal and Szegedy began to develop a general theory to explain when and why semidefinite programs behave perfectly under product. This theory captured many examples in the literature, but there were also some notable exceptions which it could not explain--namely, an early parallel repetition result of Feige and Lovasz, and a direct product theorem for the discrepancy method of communication complexity by Lee, Shraibman, and Spalek. We extend the theory of Mittal and Szegedy to explain these cases as well. Indeed, to the best of our knowledge, our theory captures all examples of semidefinite product theorems in the literature.
In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $pi/3$ on the hemisphere in $mathbb R^8$, and we prove that the three-point bound for the $(3, 8, vartheta)$-spherical code, where $vartheta$ is such that $cos vartheta = (2sqrt{2}-1)/7$, is sharp by rounding to $mathbb Q[sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.
In this paper, we introduce a set of block factor-width-two matrices, which is a generalisation of factor-width-two matrices and is a subset of positive semidefinite matrices. The set of block factor-width-two matrices is a proper cone and we compute a closed-form expression for its dual cone. We use these cones to build hierarchies of inner and outer approximations of the cone of positive semidefinite matrices. The main feature of these cones is that they enable a decomposition of a large semidefinite constraint into a number of smaller semidefinite constraints. As the main application of these classes of matrices, we envision large-scale semidefinite feasibility optimisation programs including sum-of-squares (SOS) programs. We present numerical examples from SOS optimisation showcasing the properties of this decomposition.
In this paper, we show that the bundle method can be applied to solve semidefinite programming problems with a low rank solution without ever constructing a full matrix. To accomplish this, we use recent results from randomly sketching matrix optimization problems and from the analysis of bundle methods. Under strong duality and strict complementarity of SDP, our algorithm produces primal and the dual sequences converging in feasibility at a rate of $tilde{O}(1/epsilon)$ and in optimality at a rate of $tilde{O}(1/epsilon^2)$. Moreover, our algorithm outputs a low rank representation of its approximate solution with distance to the optimal solution at most $O(sqrt{epsilon})$ within $tilde{O}(1/epsilon^2)$ iterations.
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