High-dimensional Matched Subspace Detection when data are missing

We consider the problem of deciding whether a highly incomplete signal lies within a given subspace. This problem, Matched Subspace Detection, is a classical, well-studied problem when the signal is completely observed. High-dimensional testing problems in which it may be prohibitive or impossible to obtain a complete observation motivate this work. The signal is represented as a vector in ℝⁿ, but we only observe m ≪ n of its elements.We show that reliable detection is possible, under mild incoherence conditions, as long as m is slightly greater than the dimension of the subspace in question.