A similarity/metric learning method that takes in the form of x T W y = z, where x and y are real valued vectors. For example, two images.Breaking it into a more familiar form: x T W y = ∑ i j w i j x i y j = zThis essentially looks very similar to polynomial regression with only interactions between features (without the polynomials). i.e. z = f w ( x ) = ∑ i w i x i + ∑ i ∑ j = i + 1 w i j x i x j I was curious to see if the optimization for the matrix W is the same as doing optimization for multivariate linear/polynomial regression, since x and y are fixed, and the only variate is the weight matrix W?

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A similarity/metric learning method that takes in the form of       x    T    W  y  =  z, where   x and   y are real valued vectors. For example, two images.Breaking it into a more familiar form:      x    T    W  y  =      ∑          i      j            w          i      j            x      i        y      j    =  zThis essentially looks very similar to polynomial regression with only interactions between features (without the polynomials). i.e.  z  =      f    w    (  x  )  =      ∑      i        w    i        x    i    +      ∑    i        ∑          j      =      i      +      1            w          i      j            x    i        x    j  I was curious to see if the optimization for the matrix   W is the same as doing optimization for multivariate linear/polynomial regression, since   x and   y are fixed, and the only variate is the weight matrix   W?

concludirgt · Accepted Answer

If you have only one vector   (  z  ,      x    1    ,  .  .  .  ,      x    p    ,      y    1    ,  .  .  .  ,      y    p    ) and you want to find matrix   W such that   z  =      x    ′    W  y, then this is simple linear equation with infinite set of solutions, i.e., infinitely many   Ws can solve it. If you have   p  (  p  +  1  )  /  2 distinct vectors of that kind, i.e.,  {  (      z    i    ,      x          1      i        ,  .  .  .  ,      x          p      i        ,      y          1      i        ,  .  .  .  ,      y          p      i        )      }          i      =      1              p      (      p      +      1      )          /      2        ,then you&#039;ll have unique solution (no optimization is required, just inverting matrices in whatever method you like). And if you have more than   p  (  p  +  1  )  /  2 distinct such vectors, then this is a linear regression problem, i.e.,      z    i    =      ∑          i      ≥      j            β          i      j            x    i        y    j    +      ϵ    i    ,that you can solve using ordinary least squares, which is indeed convex optimization problem.

A similarity/metric learning method that takes in the form of x T </msup> W y =

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