Visualize neighborhoods of global minima

engine-proto/gram-test/basin-shapes.jl

@@ -0,0 +1,99 @@
+include("Engine.jl")
+
+using LinearAlgebra
+using SparseArrays
+
+function sphere_in_tetrahedron_shape()
+  # initialize the partial gram matrix for a sphere inscribed in a regular
+  # tetrahedron
+  J = Int64[]
+  K = Int64[]
+  values = BigFloat[]
+  for j in 1:5
+    for k in 1:5
+      push!(J, j)
+      push!(K, k)
+      if j == k
+        push!(values, 1)
+      elseif (j <= 4 && k <= 4)
+        push!(values, -1/BigFloat(3))
+      else
+        push!(values, -1)
+      end
+    end
+  end
+  gram = sparse(J, K, values)
+  
+  # plot loss along a slice
+  loss_lin = []
+  loss_sq = []
+  mesh = range(0.9, 1.1, 101)
+  for t in mesh
+    L = hcat(
+      Engine.plane(normalize(BigFloat[ 1,  1,  1]), BigFloat(1)),
+      Engine.plane(normalize(BigFloat[ 1, -1, -1]), BigFloat(1)),
+      Engine.plane(normalize(BigFloat[-1,  1, -1]), BigFloat(1)),
+      Engine.plane(normalize(BigFloat[-1, -1,  1]), BigFloat(1)),
+      Engine.sphere(BigFloat[0, 0, 0], BigFloat(t))
+    )
+    Δ_proj = Engine.proj_diff(gram, L'*Engine.Q*L)
+    push!(loss_lin, norm(Δ_proj))
+    push!(loss_sq, dot(Δ_proj, Δ_proj))
+  end
+  mesh, loss_lin, loss_sq
+end
+
+function circles_in_triangle_shape()
+  # initialize the partial gram matrix for a sphere inscribed in a regular
+  # tetrahedron
+  J = Int64[]
+  K = Int64[]
+  values = BigFloat[]
+  for j in 1:8
+    for k in 1:8
+      filled = false
+      if j == k
+        push!(values, 1)
+        filled = true
+      elseif (j == 1 || k == 1)
+        push!(values, 0)
+        filled = true
+      elseif (j == 2 || k == 2)
+        push!(values, -1)
+        filled = true
+      end
+      #=elseif (j <= 5 && j != 2 && k == 9 || k == 9 && k <= 5 && k != 2)
+        push!(values, 0)
+        filled = true
+      end=#
+      if filled
+        push!(J, j)
+        push!(K, k)
+      end
+    end
+  end
+  append!(J, [6, 4, 6, 5, 7, 5, 7, 3, 8, 3, 8, 4])
+  append!(K, [4, 6, 5, 6, 5, 7, 3, 7, 3, 8, 4, 8])
+  append!(values, fill(-1, 12))
+  
+  # plot loss along a slice
+  loss_lin = []
+  loss_sq = []
+  mesh = range(0.99, 1.01, 101)
+  for t in mesh
+    L = hcat(
+      Engine.plane(BigFloat[0, 0, 1], BigFloat(0)),
+      Engine.sphere(BigFloat[0, 0, 0], BigFloat(t)),
+      Engine.plane(BigFloat[1, 0, 0], BigFloat(1)),
+      Engine.plane(BigFloat[cos(2pi/3), sin(2pi/3), 0], BigFloat(1)),
+      Engine.plane(BigFloat[cos(-2pi/3), sin(-2pi/3), 0], BigFloat(1)),
+      Engine.sphere(4//3*BigFloat[-1, 0, 0], BigFloat(1//3)),
+      Engine.sphere(4//3*BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//3)),
+      Engine.sphere(4//3*BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//3))
+    )
+    Δ_proj = Engine.proj_diff(gram, L'*Engine.Q*L)
+    push!(loss_lin, norm(Δ_proj))
+    push!(loss_sq, dot(Δ_proj, Δ_proj))
+  end
+  mesh, loss_lin, loss_sq
+end