  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 specified neighborhood under a
deterministic condition on a dataset. If the deterministic condition is
satisfied, we further show that a geodesic gradient descent method over the
Grassmannian manifold can exactly recover the underlying subspace when the
method is properly initialized. Proper initialization by principal component
analysis is guaranteed with a simple deterministic condition. Under slightly
stronger assumptions, the gradient descent method with a piecewise constant
step-size scheme achieves linear convergence. The practicality of the
deterministic condition is demonstrated on some statistical models of data, and
the method achieves almost state-of-the-art recovery guarantees on the Haystack
Model for different regimes of sample size and ambient dimension. In
particular, when the ambient dimension is fixed and the sample size is large
enough, we show that our gradient method can exactly recover the underlying
subspace for any fixed fraction of outliers (less than 1).
