Subspace detection of high-dimensional vectors using compressive sampling

We consider the problem of detecting whether a high dimensional vector ∈ ℝⁿ lies in a r-dimensional subspace S, where r ≪ n, given few compressive measurements of the vector. This problem arises in several applications such as detecting anomalies, targets, interference and brain activations. In these applications, the object of interest is described by a large number of features and the ability to detect them using only linear combination of the features (without the need to measure, store or compute the entire feature vector) is desirable. We present a test statistic for subspace detection using compressive samples and demonstrate that the probability of error of the proposed detector decreases exponentially in the number of compressive samples, provided that the energy off the subspace scales as n. Using information-theoretic lower bounds, we demonstrate that no other detector can achieve the same probability of error for weaker signals. Simulation results also indicate that this scaling is near-optimal.