splines_showcase.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b3bc5a79",
   "metadata": {},
   "outputs": [],
   "source": [
    "using NNlib\n",
    "using LaTeXStrings\n",
    "using Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "021d5d89",
   "metadata": {},
   "source": [
    "# Defining spline transformation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "bcc1d813",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "rls_inverse (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "    spline_trafo(widths::AbstractVector, heigths::AbstractVector, derivatives::AbstractVector, trafo::String)\n",
    "\n",
    "Create a spline transformation object. \n",
    "\n",
    "'trafo' ∈ [\"rqs\", \"rqs_inverse\", \"rls\", \"rls_inverse\"] specifies which type of spline is to be used, defaults to 'trafo = \"rqs\"'.\n",
    "\n",
    "'trafo' = \"rqs\" or \"rqs_inverse\" implements a rational quadratic spline or its inverse as defined in [1]\n",
    "'trafo' = \"rls\" or \"rls_inverse\" implements a rational linear spline or its inverse as defined in [2]\n",
    "\n",
    "'widths', 'heights' and 'derivatives' must satisfy the following constraints:\n",
    "\n",
    "    widths[i] <= widths[i + 1] \n",
    "    heigths[i] <= heights[i + 1]\n",
    "        \n",
    "    widths[1] = heigths[1] = -B with [-B, B] being the desired interval mask\n",
    "    widths[end] = heigths[end] = B\n",
    "\n",
    "    length(widths) = length(heigths) = (length(derivatives) + 1) = K (= desired number of bins into which to partition the input interval)\n",
    "\n",
    "To use a spline transformation on an univariate input do for example:\n",
    "\n",
    "    julia> rqs = spline_trafo(widths, heights, derivatives, \"rqs\")\n",
    "    julia> rqs(0.4)\n",
    "\n",
    "Where widths, heights and derivatives are the values parameterizing your desired interval mask, and 0.4 the value you wish to transform.\n",
    "\n",
    "Passing a multivariate input to such a transfromation brodcasts the univariate trafo over the multivariate input vector.\n",
    "\n",
    "[1] 'Neural Spline Flows' Papamakarios et al \n",
    "[2] 'Invertible Generative Modeling using Linear Rational Splines' Dolatabadi et al\n",
    "\"\"\"\n",
    "struct spline_trafo{P<:AbstractVector}\n",
    "    widths::P\n",
    "    heights::P\n",
    "    derivatives::P\n",
    "    trafo::String \n",
    "end\n",
    "\n",
    "function (g::spline_trafo{<:AbstractVector})(x::Real)\n",
    "    return spline_univariate(g.widths, g.heights, g.derivatives, x; g.trafo)\n",
    "end\n",
    "\n",
    "(g::spline_trafo{<:AbstractVector})(x::AbstractVector) = g.(x)\n",
    "\n",
    "function spline_univariate(\n",
    "    widths::AbstractVector,\n",
    "    heights::AbstractVector,\n",
    "    derivatives::AbstractVector,\n",
    "    x::Real;\n",
    "    trafo::String = \"rqs\"\n",
    ")\n",
    "    if !(trafo in [\"rqs\", \"rqs_inverse\", \"rls\", \"rls_inverse\"])\n",
    "        throw(ArgumentError(\"Please select a valid trafo keyword: trafo ∈ [\\\"rqs\\\", \\\"rqs_inverse\\\", \\\"rls\\\", \\\"rls_inverse\\\"]\"))\n",
    "    end\n",
    "\n",
    "    T = promote_type(eltype(widths), eltype(heights), eltype(derivatives), eltype(x))\n",
    "\n",
    "    # Number of bins K\n",
    "    K = length(widths)\n",
    "    \n",
    "    # Find in which bin x lies, -1 is so bin idxs start at k = 0\n",
    "    if trafo == \"rqs\" || trafo == \"rls\"\n",
    "        k = searchsortedfirst(widths, x) - 1 \n",
    "    else \n",
    "        k = searchsortedfirst(heights, x) - 1 \n",
    "    end\n",
    "\n",
    "    # If x outside interval mask apply identity transform\n",
    "    if (k >= K) || (k == 0)\n",
    "        return (one(T) * x, (trafo == \"rqs\" || trafo == \"rls\") ? zero(T) * x : nothing)\n",
    "    end\n",
    "\n",
    "    # Width\n",
    "    # If k == 0 put it in the bin `[-B, widths[1]]`\n",
    "    w_k = widths[k] #(k == 0) ? -widths[end] :\n",
    "    w = widths[k + 1] - w_k\n",
    "\n",
    "    # Height\n",
    "    h_k = heights[k] # (k == 0) ? -heights[end] : \n",
    "    h_kplus1 = (k == K) ? heights[end] : heights[k + 1]\n",
    "    \n",
    "    # Slope \n",
    "    Δy = heights[k + 1] - h_k\n",
    "    s = Δy / w\n",
    "\n",
    "    # Derivatives\n",
    "    d_k = derivatives[k] #(k == 0) ? one(T) :\n",
    "    d_kplus1 = (k == K - 1) ? one(T) : derivatives[k + 1]\n",
    "    \n",
    "    params = [x, w_k, w, h_k, h_kplus1, Δy, s, d_k, d_kplus1]\n",
    "\n",
    "    if trafo == \"rqs\"\n",
    "        return rqs_forward(params...)\n",
    "\n",
    "    elseif trafo == \"rqs_inverse\"\n",
    "        return rqs_inverse(params...)\n",
    "\n",
    "    elseif trafo == \"rls\"\n",
    "        return rls_forward(params...)\n",
    "\n",
    "    elseif trafo == \"rls_inverse\"\n",
    "        return rls_inverse(params...)\n",
    "\n",
    "    end\n",
    "end\n",
    "\n",
    "\n",
    "function rqs_forward(x::Real, w_k::Real, w::Real, h_k::Real, h_kplus1::Real, Δy::Real, s::Real, d_k::Real, d_kplus1::Real)\n",
    "\n",
    "    ξ = (x - w_k) / w\n",
    "    # Eq. (14) from [1]\n",
    "    numerator = Δy * (s * ξ^2 + d_k * ξ * (1 - ξ))\n",
    "    numerator_jl = s^2 * (d_kplus1 * ξ^2 + 2 * s * ξ * (1 - ξ) + d_k * (1 - ξ)^2)\n",
    "\n",
    "    denominator = s + (d_kplus1 + d_k - 2s) * ξ * (1 - ξ)\n",
    "\n",
    "    g = h_k + numerator / denominator\n",
    "    logjac = log(numerator_jl) - 2 * log(denominator)\n",
    "\n",
    "    return (g, logjac)\n",
    "end\n",
    "\n",
    "function rqs_inverse(x::Real, w_k::Real, w::Real, h_k::Real, h_kplus1::Real, Δy::Real, s::Real, d_k::Real, d_kplus1::Real)\n",
    "\n",
    "    ds = d_kplus1 + d_k - 2 * s\n",
    "\n",
    "    # Eq.s (25) through (27) from [1]\n",
    "    a1 = Δy * (s - d_k) + (x - h_k) * ds\n",
    "    a2 = Δy * d_k - (x - h_k) * ds\n",
    "    a3 = - s * (x - h_k)\n",
    "\n",
    "    # Eq. (24) from [1]. There's a mistake in the paper; says `x` but should be `ξ`\n",
    "    numerator = - 2 * a3\n",
    "    denominator = (a2 + sqrt(a2^2 - 4 * a1 * a3))\n",
    "    ξ = numerator / denominator\n",
    "\n",
    "    g = ξ * w + w_k\n",
    "    return (g, nothing)\n",
    "end\n",
    "\n",
    "function rls_forward(x::Real, w_k::Real, w::Real, h_k::Real, h_kplus1::Real, Δy::Real, s::Real, d_k::Real, d_kplus1::Real)\n",
    "    \n",
    "    lam = 0.95 * sigmoid(0.5) + 0.025 # arbitrary value, sigmoid should take unnormalized_lambda as another free param.\n",
    "    wa  = 1\n",
    "    wb  = sqrt(d_k / d_kplus1) * wa\n",
    "    wc  = (lam * wa * d_k + (1 - lam) * wb *  d_kplus1) / s\n",
    "    yc  = ((1 - lam) * wa * h_k + lam * wb * h_kplus1)/ ((1 - lam) * wa + lam * wb)\n",
    "\n",
    "    # the boolean factors determine in which 'virtual bin' the input x lies (see [2]], sec. 3.2)\n",
    "    numerator = (lam * wa * (h_k - x)) * float(x <= yc) + ((wc - lam * wb) * x + lam * wb * h_kplus1 - wc * yc) * float(x > yc)\n",
    "\n",
    "    denominator = ((wc - wa) * x + wa * h_k - wc * yc) * float(x <= yc) + ((wc - wb) * x + wb * h_kplus1 - wc * yc) * float(x > yc)\n",
    "\n",
    "    theta = numerator / denominator\n",
    "\n",
    "    g = w_k + theta * w\n",
    "\n",
    "    derivative_numerator = (wa * wc * lam * (yc - h_k) * float(x <= yc) + wb * wc * (1 - lam) * (h_kplus1 - yc) * float(x > yc)) * w_k\n",
    "\n",
    "    logjac = log(derivative_numerator) - 2 * log(abs(denominator))\n",
    "\n",
    "    return (g, logjac)\n",
    "end\n",
    "\n",
    "function rls_inverse(x::Real, w_k::Real, w::Real, h_k::Real, h_kplus1::Real, Δy::Real, s::Real, d_k::Real, d_kplus1::Real)\n",
    "\n",
    "    lam = 0.95 * sigmoid(0.5) + 0.025 \n",
    "    wa  = 1\n",
    "    wb  = sqrt(d_k / d_kplus1) * wa\n",
    "    wc  = (lam * wa * d_k + (1 - lam) * wb *  d_kplus1) / s\n",
    "    yc  = ((1 - lam) * wa * h_k + lam * wb * h_kplus1)/ ((1 - lam) * wa + lam * wb)\n",
    "\n",
    "    theta = (x - w_k) / w\n",
    "\n",
    "    numerator = (wa * h_k * (lam - theta) + wc * yc * theta) * float(theta <= lam) + (wc * yc * (1 - theta) + wb * h_kplus1 * (theta - lam)) * float(theta > lam)\n",
    "\n",
    "    denominator = (wa * (lam - theta) + wc * theta) * float(theta <= lam) + (wc * (1 - theta) + wb * (theta - lam)) * float(theta > lam)\n",
    "\n",
    "    g = numerator / denominator\n",
    "\n",
    "    derivative_numerator = (wa * wc * lam * (yc - h_k) * float(theta <= lam) + wb * wc * (1 - lam) * (h_kplus1 - yc) * float(theta > lam)) / w_k\n",
    "\n",
    "    logjac = log(derivative_numerator) - 2 * log(abs(denominator))\n",
    "\n",
    "    return g, logjac\n",
    "end "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84cb7d0b",
   "metadata": {},
   "source": [
    "# Visualization:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b3c26e0d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7-element Vector{Float64}:\n",
       " 0.0\n",
       " 0.1\n",
       " 0.4\n",
       " 0.6\n",
       " 0.65\n",
       " 0.8\n",
       " 1.0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create mock inverval mask (needs to satisfy constraints as specified in spline_trafo() docstring):\n",
    "\n",
    "widths = [-1, -0.9, -0.6, -0.1, 0.4, 0.9, 1.0]\n",
    "heigths = [-1, -0.9, -0.6, -0.1, 0.4, 0.9, 1.0]\n",
    "derivatives = [0.4, 0.5, 1.2, 0.5, 0.1, 1.9]\n",
    "\n",
    "# For linear rational spline:\n",
    "# (The linear rational spline currently only supports positive input values. I'm working on fixing that)\n",
    "widths_pos = [0.0, 0.1, 0.4, 0.6, 0.65, 0.8, 1.0]\n",
    "heigths_pos = [0.0, 0.1, 0.4, 0.6, 0.65, 0.8, 1.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "94ad9cf5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "spline_trafo{Vector{Float64}}([0.0, 0.1, 0.4, 0.6, 0.65, 0.8, 1.0], [0.0, 0.1, 0.4, 0.6, 0.65, 0.8, 1.0], [0.4, 0.5, 1.2, 0.5, 0.1, 1.9], \"rls_inverse\")"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Create different spline transformation objects, using above intervals:\n",
    "rqs = spline_trafo(widths, heigths, derivatives, \"rqs\")\n",
    "\n",
    "rqs_inv = spline_trafo(widths, heigths, derivatives, \"rqs_inverse\")\n",
    "\n",
    "rls = spline_trafo(widths_pos, heigths_pos, derivatives, \"rls\")\n",
    "\n",
    "rls_inv = spline_trafo(widths_pos, heigths_pos, derivatives, \"rls_inverse\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96b33542",
   "metadata": {},
   "source": [
    "### Plot "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "17a209b6",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Forward and inverse rational quadratic splines\n",
    "\n",
    "rectangle(w, h, x, y) = Shape(x .+ [0,w,w,0], y .+ [0,0,h,h]) # for interval mask\n",
    "\n",
    "plot(widths, heigths, seriestype = :scatter, markercolor = :black, label = \"Knots\", legend = :topleft)\n",
    "plot!(rectangle(2,2,-1,-1), opacity=.2, label = \"Interval mask\")\n",
    "plot!(title = \"Forward and inverse rational quadratic Splines\", xlabel = L\"$x$\", ylabel = L\"$g_{\\theta}(x)$\")\n",
    "xlims!(-1.3, 1.3)\n",
    "ylims!(-1.3, 1.3)\n",
    "\n",
    "plot!(x -> rqs(x)[1], -1.2, 1.2, linewidth = 2.0, linecolor = :red, label = \"RQ Spline\")\n",
    "plot!(x -> rqs_inv(x)[1], -1.2, 1.2, linewidth = 2.0, linecolor = :blue, label = \"Inverse RQ Spline\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "38aebf6a",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Forward and inverse rational linear splines (currently only support positive inputs)\n",
    "\n",
    "plot(widths_pos, heigths_pos, seriestype = :scatter, markercolor = :black, label = \"Knots\", legend = :topleft)\n",
    "plot!(rectangle(1,1,0,0), opacity=.2, label = \"Interval mask\")\n",
    "plot!(title = \"Forward and inverse rational linear Splines\", xlabel = L\"$x$\", ylabel = L\"$g_{\\theta}(x)$\")\n",
    "xlims!(-0.3, 1.3)\n",
    "ylims!(-0.3, 1.3)\n",
    "\n",
    "plot!(x -> rls(x)[1], -0.0, 1.2, linewidth = 2.0, linecolor = :red, label = \"RQ Spline\")\n",
    "plot!(x -> rls_inv(x)[1], -0.0, 1.2, linewidth = 2.0, linecolor = :blue, label = \"Inverse RQ Spline\")"
   ]
  }
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