# A Well-Tempered Landscape

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Abstract: We present a mathematical analysis of a non-convex energy landscape for Robust Subspace Recovery. We prove that an underlying subspace is the only stationary point and local minimizer in a large neighborhood if a generic condition holds for a dataset. We further show that if the generic condition is satisfied, a geodesic gradient descent method over the Grassmannian manifold can exactly recover the underlying subspace with proper initialization. The condition is shown to hold with high probability for a certain model of data.

## Acknowledgments

This work was supported by NSF award DMS-14-18386 and a UMII MnDRIVE graduate assistantship. The authors would like to thank Chao Gao and Nati Srebro for helpful discussion on a preliminary version of the work.