After some searching I couldn't find an existing solution that lets you compute orthogonal (not orthonormal basis) without computing huge performance penalty relative to Orthogonalize. Any suggestions how to speed it up?
Two versions below compute desired quantity, but 30x slower than built-in Orthogonalize. It doesn't seem to be possible to use Orthogonalize with custom dot product function to achieve this (suggested in earlier version)
d = 200;
numSamples = d;
sigma = DiagonalMatrix[Table[1., {i, 1, d}]];
dist = MultinormalDistribution[sigma];
data = RandomVariate[dist, numSamples];
data = Normalize /@ data;
(* Version from https://mathematica.stackexchange.com/a/37357/217 )
GramSchmidt[w_?MatrixQ] := Module[{v = ConstantArray[0, Length[w]]},
Table[
v[[n]] =
w[[n]] -
Sum[(v[[i]] . w[[n]]/v[[i]] . v[[i]])v[[i]], {i, 1,
n - 1}], {n, 1, Length[w]}];
v];
(* modified version from
https://mathematica.stackexchange.com/a/101463/217 *)
proj[u_, v_] := v . u/u . u u;
gsOrthogonal[vectorSet_?MatrixQ] :=
Map[With[{norm = Norm[#]}, If[norm > 0, #, #]] &,
Fold[Function[{spanVec, v},
Join[spanVec, {v -
Total@Map[proj[#, v] &,
spanVec]}]][#1, #2] &, {vectorSet[[1]]},
Drop[vectorSet, 1]]];
Print["Built-in timing: ", First@Timing@Orthogonalize[data]]
Print["version1 timing: ", First@Timing@GramSchmidt@data]
Print["version2 timing: ", First@Timing@gsOrthogonal@data]
Produces
Built-in timing: 0.021098
version1 timing: 0.829932
version2 timing: 0.629197
