Compressive Sensing - Incoherence Property Compressive Sensing is built on 2 properties: 1) the sparsity of the representation basis relative to the sampling basis and 2) the incoherence between the singular vectors from each of the 2 bases in a). On the surface this seems fine to me but he "incoherence" relationship is confusing me a little. 

Some texts refer to the coherence between the bases (representation, sampling) and other refer to the coherence between each basis and the standard basis (e1, e2, e3 ...).

Is there a difference between these two statements?


  "We are in the position to state our main result: if a matrix has row
  and column spaces that are incoherent with the standard basis, then
  nuclear norm minimization can recover this matrix from a random
  sampling of a small number of entries."


Page 6: http://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf

and

"Incoherent sampling ... The coherence between the sensing basis and and the representation basis".
Page 3: http://authors.library.caltech.edu/10092/1/CANieeespm08.pdf

My question is related to this question:
https://dsp.stackexchange.com/a/13017/4038
I am trying to answer your question about incoherence here rather than update my previous answer on another question of yours.

Compressive sensing requires low coherent pairs. So the lower $\mu(\Phi,\Psi)$, the better. Actually is $\Phi$ is spike basis (identity matrix) with $\phi_k(t) = \delta(t-k)$, and $\Psi$ is Fourier basis with $\psi_j(t) = 1/\sqrt n e^{-i \cdot 2\pi \cdot jt/n}$, which is just the example I showed you in another question, $\mu(\Phi,\Psi) = 1$, and the maximal incoherence is achieved. It is similar that if you apply the spike basis and the orthogonal basis obtained from SVD. 

Other coherence includes but not limited to: coherence between noiselets and Haar wavelets is $\sqrt2$; coherence between noiselets and Baubechies D4 and D8 are $2.2$ and $2.9$, respectively; random matrices are largely incoherent with any fixed basis $\Psi$.

You may refer to Candes-Romberg theorem for more details on incoherent sampling.