We show how to sketch semidefinite programs (SDPs) using positive maps in order to reduce their dimension. More precisely, we use Johnson-Lindenstrauss transforms to produce a smaller SDP whose solution preserves feasibility or approximates the value of the original problem with high probability. These techniques allow us to improve both complexity and storage space requirements necessary to solve SDPs. They apply to problems in which the Schatten 1-norm of the matrices specifying the SDP and of a solution to the problem is constant in the problem size. Furthermore, we provide some no-go results which clarify the limitations of positive, linear sketches in this setting. Finally, we discuss an application to uncertainty relations.
De Anthony Leverrier
De Zbigniew Puchała
De Benoit Collins
De Satya Majumdar
De Motohisa Fukuda
De Nilanjana Datta
De Daniel Stilck Franca
De Radosław Adamczak
De Madalin Guta
De Antonio Acín
De Ashley Montanaro
De Ping Zhong
De James Mingo
De Camille Male
De Clément Delcamp
De Clément Delcamp
De Claudio Dappiaggi
