# Recovering the Optimal Solution by Dual Random Projection

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Lijun Zhang,
Mehrdad Mahdavi,
Rong Jin,
Tianbao Yang,
Shenghuo Zhu
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Proceedings of the 26th Annual Conference on Learning Theory, PMLR 30:135-157, 2013.

#### Abstract

Random projection has been widely used in data classification. It maps high-dimensional data into a low-dimensional subspace in order to reduce the computational cost in solving the related optimization problem. While previous studies are focused on analyzing the classification performance of using random projection, in this work, we consider the recovery problem, i.e., how to accurately recover the optimal solution to the original optimization problem in the high-dimensional space based on the solution learned from the subspace spanned by random projections. We present a simple algorithm, termed Dual Random Projection, that uses the dual solution of the low-dimensional optimization problem to recover the optimal solution to the original problem. Our theoretical analysis shows that with a high probability, the proposed algorithm is able to accurately recover the optimal solution to the original problem, provided that the data matrix is of low rank or can be well approximated by a low rank matrix.

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