New comparison exmaple.

examples/example3.jl

@@ -0,0 +1,129 @@
+using Plots
+using Convex
+using Mosek
+using SCS
+using Interpolations
+
+# This is a julia implementaiton of an example
+# from "ℓ1 Trend Filtering" by Kim et al (2009)
+
+I_sets = [
+[1, 2000],
+[1, 968, 2000],
+[1, 339, 897, 2000],
+[1, 337, 983, 1145, 2000],
+[1, 317, 791, 885, 1207, 2000],
+[1, 350, 636, 755, 884, 1208, 2000],
+[1, 349, 631, 788, 834, 991, 1150, 2000],
+[1, 349, 631, 787, 835, 988, 1191, 1853, 2000],
+[1, 350, 636, 755, 887, 922, 976, 1181, 1853, 2000],
+[1, 365, 515, 545, 628, 769, 837, 987, 1190, 1853, 2000]
+]
+
+cost_l0 = [dev.find_optimal_y_values(ℓ, I_sets[k])[2] for k=1:10]
+
+
+#---
+
+function is_local_max(y)
+    v = zeros(y, Bool)
+    for k=2:length(y)-1
+        v[k] = (y[k] >= y[k-1]) && (y[k] >= y[k+1])
+    end
+    return v
+end
+
+#---
+data = readdlm(joinpath(Pkg.dir("DynamicApproximations"),"examples","data","snp500.txt"))
+
+N = length(data)
+H =  spdiagm((ones(N-2), -2*ones(N-2), ones(N-2)), (0,1,2))
+
+x = Variable(N)
+
+
+#---
+r = 6
+I = I_sets[r]
+Y, _ = dev.find_optimal_y_values(ℓ, I)
+y_l0 = interpolate((I,), Y, Gridded(Linear()))[1.0:N]
+
+
+
+
+
+λ = 100
+problem = minimize(0.5*sumsquares(x - data) + λ*norm(H*x, 1))
+
+
+solve!(problem, MosekSolver([("QUIET", true)]))
+println("Opt. val: ",problem.optval, "   (problem status: ", problem.status, ")")
+
+y_l1 = evaluate(x)[:]
+
+
+
+v = sortperm(abs.(H*y_l1) + 1*is_local_max(abs.(H*y_l1)))
+
+plot(H*y_l1)
+plot!(v[end-r+2:end], zeros(r-1), m=:circle)
+
+I_l1 = [1; sort(v[end-r+2:end]); N]
+Y_l1, f_l1 = dev.find_optimal_y_values(ℓ, I_l1)
+
+y_l10 = interpolate((I_l1,), Y_l1, Gridded(Linear()))[1.0:N]
+
+println("ℓ1: ", sum((evaluate(x) - data).^2))
+println("ℓ0/ℓ1: ", sum((y_l10 - data).^2))
+println("ℓ0: ", sum((y_l0 - data).^2))
+
+
+#subplot(211)
+plot(data)
+#plot!(evaluate(x), color="green", linewidth=3)
+plot!(I, Y, color="red", linewidth=3)
+plot!(I_l1, Y_l1, l=(:cyan, :dash), linewidth=3)
+#---
+
+λ_vec = logspace(1,4,60)
+cost = zeros(length(λ_vec), 10)
+
+for (k, λ)=enumerate(λ_vec)
+    problem = minimize(0.5*sumsquares(x - data) + λ*norm(H*x, 1))
+
+    solve!(problem, MosekSolver([("QUIET", true)]))
+    println("Opt. val: ",problem.optval, "   (problem status: ", problem.status, ")")
+
+    y_l1 = evaluate(x)[:]
+    v = sortperm(abs.(H*y_l1) + 1*is_local_max(abs.(H*y_l1)))
+    for r=1:10
+        I_l1 = [1; sort(v[end-r+2:end]); N]
+        _, f_l1 = dev.find_optimal_y_values(ℓ, I_l1)
+        cost[k, r] = f_l1
+    end
+end
+
+#---
+
+#---
+plot(log10(λ_vec), cost[:,3:end], show=true, yaxis=((0,10), 0:0.5:10) )
+
+plot!([1.5, 2.5], [1,1]*cost_l0[3:end]')
+
+#---
+[minimum(cost,1)' cost_l0]
+
+
+#---
+ costs = [42.8484   42.8484
+  9.25796   9.25796
+  4.46404   4.26137
+  3.13439   3.02902
+  3.09011   2.71251
+  2.68868   2.27944
+  2.16277   2.03848
+  1.94739   1.8517
+  1.89786   1.76981
+  1.75176   1.59654]
+
+plot(costs)