The minimum distortion problem

This example arises from computational geometry, in particular the problem of embedding a general finite metric space into a euclidean space.

It is known that the 4-point metric space defined by the star graph:

x
 \\
  x — x
 /
x

where distances are computed by length of the shortest path between vertices, cannot be exactly embedded into a euclidean space of any dimension.

Here we will formulate and solve an SDP to compute the best possible embedding, that is, the embedding f() that minimizes the distortion c such that

(1 / c) * D(a, b) ≤ ||f(a) - f(b)|| ≤ D(a, b)

for all points (a, b), where D(a, b) is the distance in the metric space.

Any embedding can be characterized by its Gram matrix Q, which is PSD, and

||f(a) - f(b)||^2 = Q[a, a] + Q[b, b] - 2 * Q[a, b]

We can therefore constrain

D[i, j]^2 ≤ Q[i, i] + Q[j, j] - 2 * Q[i, j] ≤ c^2 * D[i, j]^2

and minimize c^2, which gives us the SDP formulation below.

For more detail, see "Lectures on discrete geometry" by J. Matoušek, Springer, 2002, pp. 378-379.

using JuMP
import SCS
import Test

function example_min_distortion()
    model = Model(SCS.Optimizer)
    set_silent(model)
    D = [
        0.0 1.0 1.0 1.0;
        1.0 0.0 2.0 2.0;
        1.0 2.0 0.0 2.0;
        1.0 2.0 2.0 0.0
    ]
    @variable(model, c² >= 1.0)
    @variable(model, Q[1:4, 1:4], PSD)
    for i in 1:4
        for j in (i + 1):4
            @constraint(model, D[i, j]^2 <= Q[i, i] + Q[j, j] - 2 * Q[i, j])
            @constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] <= c² * D[i, j]^2)
        end
    end
    @objective(model, Min, c²)
    optimize!(model)
    Test.@test termination_status(model) == MOI.OPTIMAL
    Test.@test primal_status(model) == MOI.FEASIBLE_POINT
    Test.@test objective_value(model) ≈ 4/3 atol = 1e-4
    return
end

example_min_distortion()

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