Obtaining a thin/compact SVD

Question

I'm using SingularValueDecomposition for a least-squares regression, the instruction that works fine for what I need is

SingularValueDecomposition[X,Length[SingularValueList[X]]]

Is there a way to do the same thing but without calling SingularValueList as I suppose similar computations are done two times in SingularValueDecomposition and SingularValueList ?

Answer 1

You can find the full svd, then use the number of nonzero singular values to recover the thin svd.

thinSVD[mat_] := Module[
  {u, w, v, wprime, len},
  {u, w, v} = SingularValueDecomposition[mat];
  wprime = DeleteCases[w, {_?(# == 0 &) ..}];
  len = Length[wprime];
  wprime = wprime[[All, 1 ;; len]];
  {u[[All, 1 ;; len]], wprime, v[[All, 1 ;; len]]}
  ]

Here is a fairly standard example.

In[66]:= m = N[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}];
{ut, wt, vt} = thinSVD[m]
Chop[ut.wt.Transpose[vt] - m](*want zeros*)

Out[67]= {{{-0.214837, -0.887231}, {-0.520587, -0.249644}, {-0.826338,
    0.387943}}, {{16.8481, 0.}, {0., 1.06837}}, {{-0.479671, 
   0.776691}, {-0.572368, 0.0756865}, {-0.665064, -0.625318}}}

Out[68]= {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}