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Abstract:
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A number of statistical problems today require the estimation of singular vectors from noisy observations. For matrices, results in this direction are based on the Davis-Kahan theorem, which shows that estimation of individual singular vectors is possible only if there is a gap between the singular values. We show that for orthogonally decomposable tensors, a small perturbation affects each singular vector in isolation, and hence their recovery does not depend on the multiplicity of the singular values or the gap between consecutive singular values. This in turn provides sufficient conditions on signal to noise ratio (SNR) for consistent estimation in different problems, including independent component analysis, tensor PCA and tensor completion. We show that the requirements on SNR are rate-optimal. We also provide polynomial time computable algorithms for the recovery task under suitable assumptions on the SNR.
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