Wolfram Cloud Document

In[]:=

CompoundExpression[

]

deploy

Mon 27 Nov 2023 11:24:51

- “Frobenius norm of a product of Gaussian matrices” math.SE post
- random-matrix-moments colab
- top level: Free Probability section under Notes: Optimization

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Second moment

In[]:=

d=1000;getNorm[n_]:=x:=RandomVariate[NormalDistribution[0,

-1/2

d

],{d,d}];Cn=Nest[x.#&,x,n-1];

2

Norm[Cn.Cn,"Frobenius"]

;TableForm[Table[{n,getNorm[n],getNorm[n],getNorm[n],getNorm[n]},{n,1,5}],TableHeadings->{{},{"n","sample1","sample2","sample3","sample4"}}]

Out[]//TableForm=

n	sample1	sample2	sample3	sample4
1	1994.13	2004.98	2013.08	2001.93
2	3017.27	3005.96	2996.5	2995.37
3	4042.05	4010.99	3968.99	4053.55
4	5036.93	5029.61	5067.01	5092.75
5	5873.58	6115.03	6127.35	6032.45

Mixed Moments

Comparing XX’XX’ vs XXX’X’

In[]:=

n=40;numTrials=10;maxMoment=5;sq[mat_]:=mat.mat;randMat:=RandomVariate[NormalDistribution[],{n,n}]

n

;sq[mat_]:=mat.mat;method1:=With[{X=randMat},X.X.X.X];method2:=With[{X=randMat},X.X.X.X];method1f:=With[{X=randMat,Y=randMat},X.X.Y.X];method2f:=With[{X=randMat,Y=randMat},X.X.Y.X];(*momentofamatrix*)moment[mat_,s_]:=Tr[MatrixPower[mat,s]]/n;(*Estimatessmoment*)momentEstimate[s_,method_]:=If[s==0,0,Mean@Table[moment[method,s],{numTrials}]];momentsXY=Table[momentEstimate[s,sq@randMat],{s,1,maxMoment+1}];momentsXYXY=Table[momentEstimate[s,method1],{s,0,maxMoment}];momentsXXYY=Table[momentEstimate[s,method2],{s,0,maxMoment}];momentsXYXYf=Table[momentEstimate[s,method1f],{s,0,maxMoment}];momentsXXYYf=Table[momentEstimate[s,method2f],{s,0,maxMoment}];TableForm[{momentsXY,momentsXYXY,momentsXXYY,momentsXYXYf,momentsXXYYf},TableHeadings->{{"s=1","s=2","s=3"},{"XY","XYXY","XXYY","fXYXY","fXXYY"}}]

Out[]//TableForm=

	XY	XYXY	XXYY	fXYXY	fXXYY
s=1	1.03831	0	0	0	0
s=2	1.89654	2.31175	0.90341	-0.0592956	0.0135517
s=3	6.78403	14.168	3.48215	-0.185881	-0.0379916
	13.0656	163.782	14.8713	-0.882164	0.0905633
	40.7758	910.889	42.7098	-0.589415	0.0387337
	153.486	19358.	248.225	1.2882	0.817834

In[]:=

momentsXXYYf

Out[]=

{0,0.0135517,-0.0379916,0.0905633,0.0387337,0.817834}