Sparse and Low Rank Recovery

Compressive sensing (sparse recovery) is a new area in mathematical image and signal processing that predicts that sparse signals can be recovered from what was previously believed to be highly incomplete measurement [3, 5, 7, 12]. Recently, the ideas of this field have been extended to the recovery of low rank matrices from undersampled information [6, 8]; most notably to the matrix completion problem [4, 11]. A vector x ∈ C is called s-sparse if ‖x‖0 := #{`, x` 6= 0} ≤ s. In practice, vectors will usually not be exactly s-sparse, but can be well-approximated by a sparse vector. In order to quantify this notion one introduces the best s-term approximation error in `p by σs(x)p := infz∈C,‖z‖0≤s ‖x− z‖p. Informally, a vector x is called compressible if σs(x)p decays quickly in s. The basic task of compressive sensing is to recover a sparse (or compressible) vector from undersampled linear information, that is, from y = Ax ∈ C, A ∈ Cm×N