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fix: some small problem #1

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fix: some small problem #1

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32 changes: 17 additions & 15 deletions experiments/density_estimation.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -31,8 +31,10 @@ Random.seed!(0)
# generate independent samples from Gaussian mixture
function rand_gmm(n)
indicator = rand(n)
out = (indicator .> p).*(sqrt(σ2_1)*randn(n) .+ μ1)
out += (indicator .≤ p).*(sqrt(σ2_2)*randn(n) .+ μ2)
# FIX: make sampling weights consistent with pdf(x) below:
# P(component 1) = p -> (μ1, σ2_1), P(component 2) = 1-p -> (μ2, σ2_2)
out = (indicator .≤ p).*(sqrt(σ2_1)*randn(n) .+ μ1)
out += (indicator .> p).*(sqrt(σ2_2)*randn(n) .+ μ2)
end
# this is our "calibration" data (iid samples from distribution)
x_cal = rand_gmm(n)
Expand All @@ -41,21 +43,26 @@ x_cal = rand_gmm(n)
# the goal is to estimate P(x | x_cal)
# we'll use a semidefinite model
B = Semidefinite(n)
v(x) = k.([x], x_cal)
# FIX: v(x) must be an n-vector v(x) = (k(x, x_i))_i as in the paper.
# Using k.(x, x_cal) ensures we get a length-n vector (not a 1×n matrix).
v(x) = k.(x, x_cal)
# Model: P(x | x_cal) = v(x)'*B*v(x)

# maximum likelihood estimate
# max P(x | x_cal)
# take the -log likelihood:
objective = -1/n * sum([log(v(xi_cal)'*B*v(xi_cal)) for xi_cal in x_cal])
# regularization term
objective += λ*(nuclearnorm(B) + 0.005*tr(B))
z_cal = [v(xi_cal)'*B*v(xi_cal) for xi_cal in x_cal]
# FIX: avoid log(0) by enforcing strict positivity on training evaluations.
ϵ = 1e-9
objective = -1/n * sum(log.(z_cal))
# FIX: elastic-net style regularizer (paper): nuclear norm + Frobenius^2
objective += λ*(nuclearnorm(B) + 0.01/2 * sumsquares(B))

# integral constraint
# enforces integral of P(x | x_cal) over x = 1
# M is just the integral of the product of kernels
M = s^2*exp.(-1/(4σ^2)*((x_cal .- x_cal').^2))*sqrt(σ^2*π) # closed form from kernel
constraints = [tr(B*M) == 1]
constraints = [tr(B*M) == 1, z_cal .>= ϵ]

# solve!
problem = minimize(objective, constraints)
Expand All @@ -64,18 +71,13 @@ solve!(problem, Mosek.Optimizer)
# we have our model!
B_val = evaluate(B)

# ## Mode estimation
# x = Variable()
# obj_mode = sum([-(1/(2σ^2))*((x-x_cal[i])^2 + (x-x_cal[j])^2) for i = 1:n, j = 1:n])
# prob_mode = maximize(obj_mode)
# solve!(prob_mode, Mosek.Optimizer)


## plot
pdf_n(x, μ, σ2) = 1/(sqrt(2π*σ2))*exp(-(x-μ)^2 / (2σ2))
pdf(x) = p*pdf_n(x, μ1, σ2_1) + (1-p)*pdf_n(x, μ2, σ2_2)

x_dense = collect(-4:0.01:4)
Plots.plot(x_dense, transpose.(v.(x_dense)) .* [B_val] .* v.(x_dense), label="learned",lw=2)
# FIX: plot the scalar quadratic form f(x)=v(x)'*B*v(x) (no elementwise broadcasting).
learned = [v(x)'*B_val*v(x) for x in x_dense]
Plots.plot(x_dense, learned, label="learned",lw=2)
Plots.plot!(x_dense, pdf.(x_dense), label="ground truth", ylabel="p(x | cal)",lw=2)
Plots.scatter!(x_cal, 0 .*x_cal, label="training points", markershape=:x, msw=2)