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I would like to minimize the following objective function

$$ \| H \ast A - (H \cdot I) \ast B \|_F^2 $$

w.r.t. $H$, where

  • $H$, $I$, $A$, and $B$ are all square matrices of the same size ($I$ is a flipped identity matrix i.e., off-diagonal elements are 1 and others 0),
  • $\ast$ denotes convolution and
  • $\cdot$ is the matrix multiplication.

I would like to ask: are there any common routines to solve a problem involving convolution? Also, is this problem convex?

Edit:

I first considered to apply FFT on both $H \ast A$ and $ (H \cdot I) \ast B$. This however still cannot eliminate all convolution operations from the objective.

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  • $\begingroup$ Are $H \ast A$ and $ (H \cdot I) \ast B$ linear (affine) functions of the elements of $H$? If so, this is just a linear least squares problem (even if transformation to standard form is "convoluted") , and is therefore convex. $\endgroup$
    – Mark L. Stone
    Commented Oct 11, 2021 at 17:04
  • $\begingroup$ @MarkL.Stone Why convolutions are affine functions? $\endgroup$
    – lisi
    Commented Oct 12, 2021 at 0:04
  • $\begingroup$ Only need them to be affine w.r.t. H, (optimization variables), not with respect to the other matrices, which are input to the optimization problem. $\endgroup$
    – Mark L. Stone
    Commented Oct 12, 2021 at 1:15

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