In[]:=
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Wed 27 Mar 2024 16:20:16
Free probability
parent doc: Free Probability section​
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Log:
- Mar 27: diagrams for AAAA’A’A’ question​
- Mar 25: wicks visualization: question​
- Feb 19: disentangling question on math.SE with Robert Israel answer, cleaned-up forum-disentangling.nb
- Feb 18: Mathematica formulas question of Ben’s code mathematica.SE, getting formulas like
4
n
+10
2
n
+4n
​
- Dec 4: stats.SE Sangchul update, also math.SE Ben update on Mathematica code to compute arbitrary formulas
- Dec 3: mathoverflow
3
n
expression for trace of product,
- Dec 2: forum-gaussian-second-moment.nb, update Disentanging results
- Dec 1:
3
n
+n
formula from Sangchul Lee about counting cycles, math.SE “Showing
ETr(AA
T
A
T
A
)
3
n
+2n
” post​
- Nov 25: math.SE q on product of matrices X1 X2 X3 with inductive proof
- Nov 24: simulation results colab​
- Nov 24: forum-random-matrix-moments.nb​
- Nov 24: stats.SE q on product of matrices X1 X2 X3 with inductive proof on E[XCX’] from amoeba
- Oct: forked from NN<>LeastSquares-2.nb
- July 19: mathematica.SE “Obtaining asymptotics of parametrically defined function” post​
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Mathematica:
- purity-of-matrix-products.nb (Mathoverflow post, my own answer)
- Frobenius norm of product of random matrices
- sent to Thomas thomas-convergence-of-rank.nb​
- forum-burda-evals.nb (checking for commutativity)
- forum-product-of-matrices.nb​
- formula-check.nb​
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Notability:
- Formulas-contents notability (”dot product formulas”)
- “NN<>Least Squares” notability (Frobenius/Trace/Rank)
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Posts:
- Nov 24: math.SE and stats.SE “Frobenius norm of a product of Gaussian matrices” math.SE post, stats.SE post
- Sep 11 “Estimating the sum of 4th powers of singular values?” scicomp post (notebook)
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Burda, Eigenvalues of Powers of Gaussian
(
n
Marchenko
law)

3.59-3.60 of https://arxiv.org/pdf/1510.06128.pdf
Also https://iopscience.iop.org/article/10.1088/1742-6596/473/1/012002/pdf
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forum-burda-evals.nb
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​
In[]:=
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1/3
2
1/2
3
12Pi
1/3
2
2/3
27+
27(27-4x)

-6
1/3
x
2/3
x
1/3
27+
27(27-4x)

;​​​​ClearAll["Global`*"];​​Clear[x];​​​​x[t_]:=
n+1
Sin[(n+1)t]
Sin[t]
n
Sin[nt]
;​​y[t_]:=
2
Sin[t]
n-1
Sin[nt]
π
n
Sin[(n+1)t]
;​​clip[a_]:=Clipa,0,
Pi
n+1
;​​nvals=Range[3];​​​​sf={"Log","Log"};​​SF=StringForm;​​legends=SF["n=``",#]&/@nvals;​​funcs=Table[{x@clip@t,y@clip@t},{n,nvals}];​​parametric=ParametricPlot@@{funcs,{t,0,Pi},PlotRange->{{0,4},{0,1}},AspectRatio->1,PlotLabel->"Sine vs explicit formula",PlotLegends->legends};​​​​​​pdf2=
1/3
2
1/2
3
12Pi
1/3
2
2/3
27+
27(27-4x)

-6
1/3
x
2/3
x
1/3
27+
27(27-4x)

;​​marchenko=PDF[MarchenkoPasturDistribution[1],x];​​(*cdf2=Assuming[{0<x<1/100},Integrate[pdf2,{x,0,y}]]/.y->x*)​​explicit=Plot[{marchenko,pdf2},{x,0,4},PlotStyle->{{Black,Bold,Dashed},{Black,Bold,Dashed}},PlotLegends->{"Marchenko Pastur","Burda"}];​​Show[parametric,explicit]​​​​explicit=Plot[{marchenko,pdf2},{x,0,4},PlotStyle->{{Bold,Dashed},{Bold,Dashed}},PlotLegends->{"Marchenko Pastur","Burda"},ScalingFunctions->sf];​​​​approx=Plot@@Table
-1+
1
1+n
x
Sin
π
1+n

π
,{n,{1,2}},{x,0,4},ScalingFunctions->sf,PlotLabel->"Asymptotic vs explicit";​​Show[approx,explicit]​​​​n=2;​​Clear[x];​​makeFoxH[n_]:=(​​top={{0,1},{2-n,n}};​​topleft={};​​topright={{0,1},{1,1}};​​bottom={{-1,2}};​​bottomleft={{-n,n+1}};​​bottomright={};​​specs={{topleft,topright},{bottomleft,bottomright}};​​FoxH[{topleft,topright},{bottomleft,bottomright},x]​​);​​​​explicit=Plot[{marchenko,pdf2},{x,0,4},PlotStyle->{{Bold,Dashed},{Bold,Dashed}},PlotLegends->{"Marchenko Pastur","Burda"},ScalingFunctions->sf];​​​​fox=Plot@@{makeFoxH/@{1,2},{x,0,1},ScalingFunctions->sf};​​Show[explicit,fox,PlotLabel->"FoxH vs explicit"]​​​​​​
Out[]=
n=1
n=2
n=3
Marchenko Pastur
Burda
Out[]=
Marchenko Pastur
Burda
Out[]=
Marchenko Pastur
Burda
Construct MeijerG:
​
1/n
In[]:=
Solve
1/3
2
1/2
3
12Pi
1/3
2
2/3
27+
27(27-4x)

-6
1/3
x
2/3
x
1/3
27+
27(27-4x)

==0,x
Out[]=
x
27
4


Moments of Fuchs-Catalan density

Ipsen, J. R. 2015. “Products of Independent Gaussian Random Matrices.” arXiv [math-Ph]. arXiv. http://arxiv.org/abs/1510.06128.
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Formula 3.59

Gaussian matrix moments

Solve using S-transform

​
Now for squared Wishart

Simplified solving using S-transform

https://mathematica.stackexchange.com/questions/287953/obtaining-moments-from-the-inverse-of-the-moment-generating-function

Least squares on Gaussian random data

Density after many matrix products

Main doc: https://www.wolframcloud.com/obj/yaroslavvb/nn-linear/forum-product-of-matrices.nb
Mo question: https://mathoverflow.net/questions/451732/what-does-a-product-of-many-gaussian-matrices-converge-to
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Convergence of effective rank

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Isometry/Ahle

TLDR; information is kept on isometries, but isometry + small noise will lose it