Confidence region of singular vectors for high-dimensional and low-rank matrix regression
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Let M∈R^m_1× m_2 be an unknown matrix with r= rank( M)≪(m_1,m_2) whose thin singular value decomposition is denoted by M= UΛ V^ where Λ= diag(λ_1,...,λ_r) contains its non-increasing singular values. Low rank matrix regression refers to instances of estimating M from n i.i.d. copies of random pair {( X, y)} where X∈R^m_1× m_2 is a random measurement matrix and y∈R is a noisy output satisfying y= tr( M^ X)+ξ with ξ being stochastic error independent of X. The goal of this paper is to construct efficient estimator (denoted by Û and V̂) and confidence region of U and V. In particular, we characterize the distribution of dist^2[(Û,V̂), ( U, V)]=ÛÛ^- U U^_ F^2+V̂V̂^- V V^_ F^2. We prove the asymptotic normality of properly centered and normalized dist^2[(Û,V̂), ( U, V)] with data-dependent centering and normalization when r^5/2(m_1+m_2)^3/2=o(n/ n), based on which confidence region of U and V is constructed achieving any pre-determined confidence level asymptotically.
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