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Register a new userBundle Method Sketching for Low Rank Semidefinite Programming
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Added by
Lijun Ding
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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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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We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It is thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem.
Semidefinite programming (SDP) is a central topic in mathematical optimization with extensive studies on its efficient solvers. In this paper, we present a proof-of-principle sublinear-time algorithm for solving SDPs with low-rank constraints; specifically, given an SDP with $m$ constraint matrices, each of dimension $n$ and rank $r$, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time $O(mcdotmathrm{poly}(log n,r,1/varepsilon))$ given access to a sampling-based low-overhead data structure for the constraint matrices, where $varepsilon$ is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC 12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC 19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest: $bullet$ Weighted sampling: assuming sampling access to each individual constraint matrix $A_{1},ldots,A_{tau}$, we propose a procedure that gives a good approximation of $A=A_{1}+cdots+A_{tau}$. $bullet$ Symmetric approximation: we propose a sampling procedure that gives the emph{spectral decomposition} of a low-rank Hermitian matrix $A$. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.
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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.
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