Counter - example

(*KhatriRaoproduct*)​​khatriRao[A_,B_]:=MapThread[Flatten[{#1}{#2}]&,Transpose/@{A,B}]//Transpose​​​​(*Symmetricsquareroot*)​​symsqrt[m_]:=Module[{U,S,W},​​{U,S,W}=SingularValueDecomposition[m];​​U.Sqrt[S].Transpose[W]​​];​​​​dsize=10;(*datasize*)​​fsize=3;(*numberofactivations*)​​A=RandomReal[{0,1},{fsize,dsize}];​​B=RandomReal[{0,1},{fsize,dsize}];​​mean[A_]:=Table[Mean[A],Dimensions[A][[2]]];​​cov[A_]:=(A-mean[A]).(A-mean[A]);​​whiten[A_]:=symsqrt@PseudoInverse@cov@A;
cov[A]//MatrixForm
0.74137
0.131775
0.539301
0.131775
0.56786
0.0647092
0.539301
0.0647092
0.871038
A2=whiten[A].A;​​cov[A2]//MatrixForm
1.
0
0
0
1.
0
0
0
1.
B2=whiten[B].B;​​cov[B2]//MatrixForm
1.
0
0
0
1.
0
0
0
1.
cov[khatriRao[A,B]]//MatrixForm
0.571116
0.0361183
0.213684
0.334681
-0.0908269
0.012452
0.430007
-0.0469041
0.088791
0.0361183
0.133577
0.132738
-0.0255711
0.0837464
-0.0113528
0.0215404
0.0992301
0.097021
0.213684
0.132738
0.377636
0.111855
-0.0416776
0.186022
0.0841341
0.00133955
0.281309
0.334681
-0.0255711
0.111855
0.372078
-0.00344318
0.154363
0.215251
-0.0222328
-0.000172265
-0.0908269
0.0837464
-0.0416776
-0.00344318
0.200056
-0.0118707
-0.0289404
0.160138
-0.0289253
0.012452
-0.0113528
0.186022
0.154363
-0.0118707
0.442733
-0.107718
-0.0961924
0.0629019
0.430007
0.0215404
0.0841341
0.215251
-0.0289404
-0.107718
0.422975
0.0357447
0.0828251
-0.0469041
0.0992301
0.00133955
-0.0222328
0.160138
-0.0961924
0.0357447
0.19823
0.0506098
0.088791
0.097021
0.281309
-0.000172265
-0.0289253
0.0629019
0.0828251
0.0506098
0.337163