Abstract
We propose an algorithm for approximately maximizing a concave function over the bounded semi-definite cone, which produces sparse solutions. Sparsity for SDP corresponds to low rank matrices, and is a important property for both computational as well as learning theoretic reasons. As an application, building on Aaronson’s recent work, we derive a linear time algorithm for Quantum State Tomography.
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References
Aaronson, S.: The learnability of quantum states. arXiv:quant-ph/0608142v3
Arora, S., Hazan, E., Kale, S.: Fast algorithms for approximate semide.nite programming using the multiplicative weights update method. In: 46th IEEE FOCS, pp. 339–348 (2005)
Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5(1), 13–51 (1995)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York, NY, USA (2004)
Clarkson, K.L.: Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm. In: SODA 2008: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms,
Frank, M., Wolfe, P.: An algorithm for quadratic programming. Naval Research Logistics Quarterly 3, 149–154 (1956)
Grigoriadis, M.D., Khachiyan, L.G.: Fast approximation schemes for convex programs with many block and coupling constraints. SIAM Journal on Optimization 4, 86–107 (1994)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1988)
Kuczyński, J., Woźniakowski, H.: Estimating the largest eigenvalue by the power and lanczos algorithms with a random start. SIAM Journal on Matrix Analysis and Applications 13(4), 1094–1122 (1992)
Nesterov, Y., Nemirovskii, A.: Interior Point Polynomial Methods in Convex Programming: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia (1994)
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Hazan, E. (2008). Sparse Approximate Solutions to Semidefinite Programs. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_27
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DOI: https://doi.org/10.1007/978-3-540-78773-0_27
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