Random matrix theory and rank-1 signal + noise

This example compares results from random matrix theory with empirical results for rank-1 matrices with additive white Gaussian noise using the Julia language. This demo illustrates the phase transition that occurs when the singular value is sufficiently large relative to the matrix aspect ratio $c = M/N$.

This page comes from a single Julia file: gauss1.jl.

You can access the source code for such Julia documentation using the 'Edit on GitHub' link in the top right. You can view the corresponding notebook in nbviewer here: gauss1.ipynb, or open it in binder here: gauss1.ipynb.

Add the Julia packages that are need for this demo. Change false to true in the following code block if you are using any of the following packages for the first time.

if false
    import Pkg
    Pkg.add([
        "InteractiveUtils"
        "LaTeXStrings"
        "LinearAlgebra"
        "MIRTjim"
        "Plots"
        "Random"
        "StatsBase"
    ])
end

Tell Julia to use the following packages for this example. Run Pkg.add() in the preceding code block first, if needed.

using InteractiveUtils: versioninfo
using LaTeXStrings
using LinearAlgebra: dot, rank, svd, svdvals
using MIRTjim: prompt, jim
using Plots: default, gui, plot, plot!, scatter!, savefig, histogram
using Plots.PlotMeasures: px
using Random: seed!
using StatsBase: mean, var
default(markerstrokecolor=:auto, label="", widen=true, markersize = 6,
 labelfontsize = 24, legendfontsize = 18, tickfontsize = 14, linewidth = 3,
)
seed!(0)

Random.TaskLocalRNG()

The following line is helpful when running this file as a script; this way it will prompt user to hit a key after each image is displayed.

isinteractive() && prompt(:prompt);

Helper functions

Generate random data for one trial:

function gen1(
    θ::Real = 3,
    M::Int = 100,
    N::Int = 2M,
    T::Type{<:Real} = Float32,
)
    u = randn(T, M) / T(sqrt(M))
    v = randn(T, N) / T(sqrt(N))
    X = θ * u * v' # theoretically rank-1 matrix
    Z = randn(T, M, N) / T(sqrt(N)) # gaussian noise
    Y = X + Z
    return Y, u, v, θ
end;

SVD results for 1 trial:

function trial1(args...)
    Y, u, v, θ = gen1(args...)
    fac = svd(Y)
    σ1 = fac.S[1]
    u1 = fac.U[:,1]
    v1 = fac.Vt[1,:]
    return [σ1, abs2(dot(u1, u)), abs2(dot(v1, v))]
end;

Average nrep trials:

trial2(nrep::Int, args...) = mean((_) -> trial1(args...), 1:nrep);

SVD for each of multiple trials, for different SNRs and matrix sizes:

if !@isdefined(vgrid)

    # Simulation parameters
    T = Float32
    Mlist = [30, 300]
    θmax = 4
    nθ = θmax * 4 + 1
    nrep = 100
    θlist = T.(range(0, θmax, nθ));
    labels = map(n -> latexstring("\$M = $n\$"), Mlist)

    c = 1 # square matrices for simplicity
    c4 = c^0.25
    tmp = ((θ, M) -> trial2(nrep, θ, M, ceil(Int, M/c) #= N =#)).(θlist, Mlist')
    σgrid = map(x -> x[1], tmp)
    ugrid = map(x -> x[2], tmp)
    vgrid = map(x -> x[3], tmp)
end;

Results

Compare theory predictions and empirical results. There is again notable agreement between theory and empirical results here.

σ1 plot

colors = [:orange, :red]
θfine = range(0, θmax, 40θmax + 1)
sbg(θ) = θ > c4 ? sqrt((1 + θ^2) * (c + θ^2)) / θ : 1 + √(c)
stheory = sbg.(θfine)
bm = s -> "\\mathbf{\\mathit{$s}}"
ylabel = latexstring("\$σ_1($(bm(:Y)))\$ (Avg)")
ps = plot(θfine, θfine, color=:black,
    aspect_ratio = 1, linewidth = 2,
    xaxis = (L"θ", (0,θmax), 0:θmax),
    yaxis = (ylabel, (1,θmax), 1:θmax),
    annotate = (2.1, 3.6, latexstring("c = $c"), :left),
)
plot!(θfine, stheory, color=:blue, label="theory")
scatter!(θlist, σgrid[:,1], marker=:square, color=colors[1], label = labels[1])
scatter!(θlist, σgrid[:,2], marker=:circle, color=colors[2], label = labels[2])

prompt()

u1 plot

ubg(θ) = (θ > c4) ? 1 - c * (1 + θ^2) / (θ^2 * (θ^2 + c)) : 0
utheory = ubg.(θfine)
ylabel = latexstring("\$|⟨\\hat{$(bm(:u))}, $(bm(:u))⟩|^2\$ (Avg)")
pu = plot(θfine, utheory, color=:blue, label="theory",
    left_margin = 10px, legend = :bottomright,
    xaxis = (L"θ", (0,θmax), 0:θmax),
    yaxis = (ylabel, (0,1), 0:0.5:1),
)
scatter!(θlist, ugrid[:,1], marker=:square, color=colors[1], label = labels[1])
scatter!(θlist, ugrid[:,2], marker=:circle, color=colors[2], label = labels[2])

prompt()

v1 plot

vbg(θ) = (θ > c^0.25) ? 1 - (c + θ^2) / (θ^2 * (θ^2 + 1)) : 0
vtheory = vbg.(θfine)
ylabel = latexstring("\$|⟨\\hat{$(bm(:v))}, $(bm(:v))⟩|^2\$ (Avg)")
pv = plot(θfine, vtheory, color=:blue, label="theory",
    left_margin = 10px, legend = :bottomright,
    xaxis = (L"θ", (0,θmax), 0:θmax),
    yaxis = (ylabel, (0,1), 0:0.5:1),
)
scatter!(θlist, vgrid[:,1], marker=:square, color=colors[1], label = labels[1])
scatter!(θlist, vgrid[:,2], marker=:circle, color=colors[2], label = labels[2])

prompt()


if false
    savefig(ps, "gauss1-s.pdf")
    savefig(pu, "gauss1-u.pdf")
    savefig(pv, "gauss1-v.pdf")
end

Marčenko–Pastur distribution

Examine the singular values of the noise-only matrix $Z$. having elements $z_{ij} ∼ N(0, 1/N)$. and compare to the asymptotic prediction by the Marčenko–Pastur distribution. The agreement is remarkably good, even for a modest matrix size of 100 × 100.

Marčenko–Pastur pdf

function mp_predict(x::Real, c::Real)
    σm = 1 - sqrt(c)
    σp = 1 + sqrt(c)
    return (σm < x < σp) ?
        sqrt(4c - (x^2 - 1 - c)^2) / (π * c * x) : 0.
end;


function mp_plot(M::Int, N::Int, rando::Function, name::String;
    ntrial = 150,
    bins = range(0, 2, 101),
)
    c = M//N
    pred = mp_predict.(bins, c)
    pmax = ceil(maximum(pred), digits=1)
    data = [svdvals(rando()) for _ in 1:ntrial]
    data = reduce(hcat, data)
    σm = 1 - sqrt(c)
    σp = 1 + sqrt(c)
    xticks = (c == 1) ? (0:2) : round.([0, σm, 1, σp, 2]; digits=2)
    cstr = c == 1 ? L"c = 1" : latexstring("c = $(c.num)/$(c.den) $name")
    histogram(vec(data); bins, linewidth=0,
     xaxis = (L"σ", (0, 2), xticks),
     yaxis = ("", (0, 2.0), [-1, 0, pmax]),
     label = "Empirical", normalize = :pdf,
     left_margin = 10px,
     annotate = (0.1, 1.5, cstr, :left),
    )
    return plot!(bins, pred, label="Predicted")
end;

M = 100
Nlist = [1, 4, 9] * M
fun1 = N -> mp_plot(M, N, () -> randn(M, N) / sqrt(N), "")
pp = fun1.(Nlist)
p3 = plot(pp...; layout=(3,1), size=(600,800))

# savefig(p3, "gauss-mp.pdf")
# savefig(pp[2], "gauss-mp-c4.pdf")

prompt()

Universality

Repeat the previous experiment with a (zero-mean) Bernoulli distribution.

M = 100
N = 4 * M
randb = () -> rand((-1,1), M, N) / sqrt(N) # Bernoulli, variance 1/N
if false
    tmp = randb()
    @show mean(tmp) # check mean is 0
    @show mean(abs2, tmp), 1/N # check variance is 1/N (exact!)
end
pb = mp_plot(M, N, randb, ", \\mathrm{Bernoulli}")

prompt()

Show a typical Bernoulli matrix realization

pb0 = jim(randb()', "'Bernoulli' matrix"; clim = (-1,1) .* 0.05,
 size=(600,200), right_margin = 30px, cticks=(-1:1)*0.05)

Sparsity

Universality can break down if the data is too sparse. Here we modify Bernoulli to be a categorical distribution with values $(-a, 0, a)$ and probabilities $((1-p)/2, p, (1-p)/2)$, with $a$ set so that the variance is $1/N$.

Here we set $p$ so that most of the random matrix elements are zero. In this extremely sparse case, the Marčenko–Pastur distribution no longer applies.

M = 100
N = 4 * M
p = (1 - 8/N) # just a few non-zero per row

rands = () -> rand((-1,1), M, N) / sqrt(N * (1-p)) .* (rand(M,N) .> p)
if false
    tmp = rands()
    @show count(==(0), tmp) / (M*N), p
    @show mean(tmp) # check mean is 0
    @show mean(abs2, tmp), 1/N # check variance is 1/N (exact!)
end

Show a typical matrix realization to illustrate the sparsity

pb1 = jim(rands()', "Very sparse 'Bernoulli' matrix";
 size=(600,200), right_margin = 20px)

Now make the plot

pss = mp_plot(M, N, rands, ", \\mathrm{Sparse},  p = $p")

prompt()

Reproducibility

This page was generated with the following version of Julia:

using InteractiveUtils: versioninfo
io = IOBuffer(); versioninfo(io); split(String(take!(io)), '\n')

11-element Vector{SubString{String}}:
 "Julia Version 1.11.2"
 "Commit 5e9a32e7af2 (2024-12-01 20:02 UTC)"
 "Build Info:"
 "  Official https://julialang.org/ release"
 "Platform Info:"
 "  OS: Linux (x86_64-linux-gnu)"
 "  CPU: 4 × AMD EPYC 7763 64-Core Processor"
 "  WORD_SIZE: 64"
 "  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)"
 "Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)"
 ""

And with the following package versions

import Pkg; Pkg.status()

Status `~/work/book-la-demo/book-la-demo/docs/Project.toml`
  [6e4b80f9] BenchmarkTools v1.5.0
  [aaaa29a8] Clustering v0.15.7
  [35d6a980] ColorSchemes v3.27.1
⌅ [3da002f7] ColorTypes v0.11.5
⌃ [c3611d14] ColorVectorSpace v0.10.0
⌅ [717857b8] DSP v0.7.10
  [72c85766] Demos v0.1.0 `~/work/book-la-demo/book-la-demo`
  [e30172f5] Documenter v1.8.0
  [4f61f5a4] FFTViews v0.3.2
  [7a1cc6ca] FFTW v1.8.0
  [587475ba] Flux v0.15.2
  [a09fc81d] ImageCore v0.10.5
  [71a99df6] ImagePhantoms v0.8.1
  [b964fa9f] LaTeXStrings v1.4.0
  [7031d0ef] LazyGrids v1.0.0
  [599c1a8e] LinearMapsAA v0.12.0
  [98b081ad] Literate v2.20.1
  [7035ae7a] MIRT v0.18.2
  [170b2178] MIRTjim v0.25.0
  [eb30cadb] MLDatasets v0.7.18
  [efe261a4] NFFT v0.13.5
  [6ef6ca0d] NMF v1.0.3
  [15e1cf62] NPZ v0.4.3
  [0b1bfda6] OneHotArrays v0.2.6
  [429524aa] Optim v1.10.0
  [91a5bcdd] Plots v1.40.9
  [f27b6e38] Polynomials v4.0.12
  [2913bbd2] StatsBase v0.34.4
  [d6d074c3] VideoIO v1.1.1
  [b77e0a4c] InteractiveUtils v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [44cfe95a] Pkg v1.11.0
  [9a3f8284] Random v1.11.0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

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