Blind Deconvolution Using Convex Programming

  • Ahmed A
  • Recht B
  • Romberg J
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We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in both w and x, it is linear in the rank-1 matrix formed by their outer product wx*. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that, for “generic” signals, the program can deconvolve w and x exactly when the maximum of N and K is almost on the order of L. That is, we show that if x is drawn from a random subspace of dimension N, and w is a vector in a subspace of dimension K whose basis vectors are spread out in the frequency domain, then nuclear norm minimization recovers wx* without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length N, which we code using a random L x N coding matrix, and the encoded message travels through an unknown linear time-invariant channel of maximum length K, then the receiver can recover both the channel response and the message when L ≳ N + K, to within constant and log factors.

Author-supplied keywords

  • Convolution
  • Equations
  • Frequency-domain analysis
  • Government
  • Minimization
  • Vectors
  • blind channel estimation
  • blind deconvolution
  • blind source separation
  • channel coding
  • channel estimation
  • channel response
  • circular convolution
  • coding matrix
  • compressed sensing
  • convex programming
  • deconvolution
  • image deblurring
  • linear time-invariant channel
  • log factor
  • low-rank matrix
  • low-rank matrix recovery problem
  • matrix algebra
  • matrix factorizations
  • natural convex relaxation
  • nuclear norm minimization
  • nuclear norm minimization program
  • rank-1 matrix

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  • A Ahmed

  • B Recht

  • J Romberg

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