Stable recovery of low-rank matrix via nonconvex Schatten p-minimization

AbstractIn this paper, a sufficient condition is obtained to ensure the stable recovery (ɛ ≠ 0) or exact recovery (ɛ = 0) of all r-rank matrices X ∈ ℝm×n from $$b = \mathcal{A}(X) + z$$ via nonconvex Schatten p-minimization for any $$\delta _{4r} \in \left[ {\frac{{\sqrt 3 }} {2},1} \right)$$. Moreover, we determine the range of parameter p with any given δ$$\delta _{4r} \in \left[ {\frac{{\sqrt 3 }} {2},1} \right)$$. In fact, for any given $$\delta _{4r} \in \left[ {\frac{{\sqrt 3 }} {2},1} \right)$$, p ∈ (0, 2(1 − δ4r)] suffices for the stable recovery or exact recovery of all r-rank matrices.

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