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Register a new userGeneralized low rank approximation to the symmetric positive semidefinite matrix
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In this paper, we investigate the generalized low rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $$underset{ rank(X)leq k}{min} sum^m_{i=1}left Vert A_i - B_i XB_i^T right Vert^2_F,$$ where $X$ is an unknown symmetric positive semidefinite matrix and $k$ is a positive integer. We firstly use the property of a symmetric positive semidefinite matrix $X=YY^T$, $Y$ with order $ntimes k$, to convert the generalized low rank approximation into unconstraint generalized optimization problem. Then we apply the nonlinear conjugate gradient method to solve the generalized optimization problem. We give a numerical example to illustrate the numerical algorithm is feasible.
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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.
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.
Low rank matrix recovery problems appear widely in statistics, combinatorics, and imaging. One celebrated method for solving these problems is to formulate and solve a semidefinite program (SDP). It is often known that the exact solution to the SDP with perfect data recovers the solution to the original low rank matrix recovery problem. It is more challenging to show that an approximate solution to the SDP formulated with noisy problem data acceptably solves the original problem; arguments are usually ad hoc for each problem setting, and can be complex. In this note, we identify a set of conditions that we call simplicity that limit the error due to noisy problem data or incomplete convergence. In this sense, simple SDPs are robust: simple SDPs can be (approximately) solved efficiently at scale; and the resulting approximate solutions, even with noisy data, can be trusted. Moreover, we show that simplicity holds generically, and also for many structured low rank matrix recovery problems, including the stochastic block model, $mathbb{Z}_2$ synchronization, and matrix completion. Formally, we call an SDP simple if it has a surjective constraint map, admits a unique primal and dual solution pair, and satisfies strong duality and strict complementarity. However, simplicity is not a panacea: we show the Burer-Monteiro formulation of the SDP may have spurious second-order critical points, even for a simple SDP with a rank 1 solution.
We provide a number of algorithmic results for the following family of problems: For a given binary mtimes n matrix A and integer k, decide whether there is a simple binary matrix B which differs from A in at most k entries. For an integer r, the simplicity of B is characterized as follows. - Binary r-Means: Matrix B has at most r different columns. This problem is known to be NP-complete already for r=2. We show that the problem is solvable in time 2^{O(klog k)}cdot(nm)^{O(1)} and thus is fixed-parameter tractable parameterized by k. We prove that the problem admits a polynomial kernel when parameterized by r and k but it has no polynomial kernel when parameterized by k only unless NPsubseteq coNP/poly. We also complement these result by showing that when being parameterized by r and k, the problem admits an algorithm of running time 2^{O(rcdot sqrt{klog{(k+r)}})}(nm)^{O(1)}, which is subexponential in k for rin O(k^{1/2 -epsilon}) for any epsilon>0. - Low GF(2)-Rank Approximation: Matrix B is of GF(2)-rank at most r. This problem is known to be NP-complete already for r=1. It also known to be W[1]-hard when parameterized by k. Interestingly, when parameterized by r and k, the problem is not only fixed-parameter tractable, but it is solvable in time 2^{O(r^{ 3/2}cdot sqrt{klog{k}})}(nm)^{O(1)}, which is subexponential in k. - Low Boolean-Rank Approximation: Matrix B is of Boolean rank at most r. The problem is known to be NP-complete for k=0 as well as for r=1. We show that it is solvable in subexponential in k time 2^{O(r2^rcdot sqrt{klog k})}(nm)^{O(1)}.
The goal of this work is to fill a gap in [Yang, SIAM J. Matrix Anal. Appl, 41 (2020), 1797--1825]. In that work, an approximation procedure was proposed for orthogonal low-rank tensor approximation; however, the approximation lower bound was only established when the number of orthonormal factors is one. To this end, by further exploring the multilinearity and orthogonality of the problem, we introduce a modified approximation algorithm. Approximation lower bound is established, either in deterministic or expected sense, no matter how many orthonormal factors there are. In addition, a major feature of the new algorithm is its flexibility to allow either deterministic or randomized procedures to solve a key step of each latent orthonormal factor involved in the algorithm. This feature can reduce the computation of large SVDs, making the algorithm more efficient. Some numerical studies are provided to validate the usefulness of the proposed algorithm.
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