✨ add DDIM scheduler

src/Diffusers.jl

@@ -29,6 +29,7 @@ include("Schedulers/Schedulers.jl")
 import .Schedulers:
   # Scheduler
   DDPM,
+  DDIM,
 
   # Scheduler methods
   forward,
@@ -52,6 +53,7 @@ import .Schedulers:
 export
   # Scheduler
   DDPM,
+  DDIM,
 
   # Scheduler methods
   forward,

src/Schedulers/DDIM.jl

@@ -0,0 +1,177 @@
+include("Abstract.jl")
+
+using ShiftedArrays
+
+function _extract(
+  target::AbstractArray,
+  reference::AbstractArray,
+)
+  new_size = tuple(
+    fill(1, ndims(reference) - 1)...,
+    size(target, 1)
+  )
+  return reshape(target, new_size)
+end
+
+"""
+Denoising Diffusion Implicit Models (DDIM) scheduler.
+
+## References
+  * [[2010.02502] Denoising Diffusion Implicit Models](https://arxiv.org/abs/2010.02502)
+"""
+struct DDIM{V<:AbstractVector} <: Scheduler
+  T::Integer # length of markov chain
+
+  α::V # 1 - beta
+  β::V # beta variance schedule
+
+  ⎷α::V # square root of α
+  ⎷β::V # square root of β
+
+  α̅::V # cumulative product of α
+  β̅::V # 1 - α̅ (≠ cumprod(β))
+
+  # α̅₋₁::V # right-shifted α̅
+  # β̅₋₁::V # right-shifted β̅
+
+  ⎷α̅::V # square root of α̅
+  ⎷β̅::V # square root of β̅
+
+  # ⎷α̅₋₁::V # right-shifted ⎷α̅
+  # ⎷β̅₋₁::V # right-shifted ⎷β̅
+end
+
+function DDIM(β::AbstractVector)
+  T = length(β)
+
+  α = 1 .- β
+
+  ⎷α = sqrt.(α)
+  ⎷β = sqrt.(β)
+
+  α̅ = cumprod(α)
+  β̅ = 1 .- α̅
+
+  # α̅₋₁ = ShiftedArray(α̅, 1, default=1.0)
+  # β̅₋₁ = ShiftedArray(β̅, 1, default=0.0)
+
+  ⎷α̅ = sqrt.(α̅)
+  ⎷β̅ = sqrt.(β̅)
+
+  # ⎷α̅₋₁ = ShiftedArray(⎷α̅, 1, default=1.0)
+  # ⎷β̅₋₁ = ShiftedArray(⎷β̅, 1, default=0.0)
+
+  DDIM{typeof(β)}(
+    T,
+    α, β,
+    ⎷α, ⎷β,
+    α̅, β̅,
+    # α̅₋₁, β̅₋₁,
+    ⎷α̅, ⎷β̅,
+    # ⎷α̅₋₁, ⎷β̅₋₁,
+  )
+end
+
+function forward(
+  scheduler::DDIM,
+  x₀::AbstractArray,
+  ϵ::AbstractArray,
+  t::AbstractArray,
+)
+  # retreive scheduler variables at timesteps t
+  ⎷α̅ₜ = _extract(scheduler.⎷α̅[t], x₀)
+  ⎷β̅ₜ = _extract(scheduler.⎷β̅[t], x₀)
+
+  # noisify clean data
+  # arxiv:2006.11239 Eq. 4
+  xₜ = ⎷α̅ₜ .* x₀ + ⎷β̅ₜ .* ϵ
+
+  return xₜ
+end
+
+function reverse(
+  scheduler::DDIM,
+  xₜ::AbstractArray,
+  ϵᵧ::AbstractArray,
+  t::AbstractArray,
+  η::Real=0.0f0,
+  prediction_type::PredictionType=EPSILON,
+)
+  Δₜ = 1
+  α̅ = _extract(scheduler.α̅, xₜ)
+  α̅₋ₚ = ShiftedArray(scheduler.α̅, Δₜ, default=1.0f0)
+  α̅ₜ = _extract(α̅[t], xₜ)
+  α̅ₜ₋ₚ = _extract(α̅₋ₚ[t], xₜ)
+  β̅ = _extract(scheduler.β̅, xₜ)
+  β̅₋ₚ = ShiftedArray(scheduler.β̅, Δₜ, default=0.0f0)
+  β̅ₜ = _extract(β̅[t], xₜ)
+  β̅ₜ₋ₚ = _extract(β̅₋ₚ[t], xₜ)
+
+  println("α̅ₜ = ", α̅ₜ)
+  println("α̅ₜ₋ₚ = ", α̅ₜ₋ₚ)
+
+  # compute x₀ (approximation)
+  x̂₀ = get_prediction(scheduler, prediction_type, xₜ, ϵᵧ, t)
+
+  # clip between -1 and 1
+  x̂₀ = clamp.(x̂₀, -3, 3)
+
+  # compute σ (exact)
+  σ²ₜ = (β̅ₜ₋ₚ ./ β̅ₜ) .* (1 .- α̅ₜ ./ α̅ₜ₋ₚ)
+  println("β̅ₜ₋ₚ = ", β̅ₜ₋ₚ)
+  println("β̅ₜ = ", β̅ₜ)
+  println("α̅ₜ = ", α̅ₜ)
+  println("α̅ₜ₋ₚ = ", α̅ₜ₋ₚ)
+  println("σ²ₜ = ", σ²ₜ)
+  σₜ = η * sqrt.(σ²ₜ)
+
+  # compute direction
+  Δₓ = sqrt.(1 .- α̅ₜ₋ₚ .- σₜ^2) .* ϵᵧ
+
+  # sample xₜ₋ₚ
+  ϵ = randn(Float32, size(xₜ))
+  xₜ₋ₚ = sqrt.(α̅ₜ₋ₚ) .* x̂₀ .+ Δₓ .+ σₜ .* ϵ
+
+  return xₜ₋ₚ, x̂₀
+end
+
+function get_velocity(
+  scheduler::DDIM,
+  x₀::AbstractArray,
+  ϵ::AbstractArray,
+  t::AbstractArray,
+)
+  ⎷α̅ₜ = _extract(scheduler.⎷α̅[t], x₀)
+  ⎷β̅ₜ = _extract(scheduler.⎷β̅[t], x₀)
+
+  vₜ = ⎷α̅ₜ .* ϵ - ⎷β̅ₜ .* x₀
+
+  return vₜ
+end
+
+function get_prediction(
+  scheduler::DDIM,
+  prediction_type::PredictionType,
+  xₜ::AbstractArray,
+  ϵᵧ::AbstractArray,
+  t::AbstractArray,
+)
+  ⎷α̅ₜ = _extract(scheduler.⎷α̅[t], xₜ)
+  ⎷β̅ₜ = _extract(scheduler.⎷β̅[t], xₜ)
+
+  if prediction_type == EPSILON
+    # arxiv:2006.11239 Eq. 15
+    # arxiv:2208.11970 Eq. 115
+    x̂₀ = (xₜ - ⎷β̅ₜ .* ϵᵧ) ./ ⎷α̅ₜ
+    # elseif prediction_type == SAMPLE
+    #   # arxiv:2208.11970 Eq. 99
+    #   x̂₀ = ϵᵧ
+    # elseif prediction_type == VELOCITY
+    #   # arxiv:2202.00512 Eq. 31
+    #   x̂₀ = ⎷α̅ₜ .* xₜ - ⎷β̅ₜ .* ϵᵧ
+  else
+    throw("unimplemented prediction type")
+  end
+
+  return x̂₀
+end

src/Schedulers/Schedulers.jl

@@ -1,10 +1,12 @@
 module Schedulers
 
 include("DDPM.jl")
+include("DDIM.jl")
 
 export
   # Schedulers
   DDPM,
+  DDIM,
 
   # Scheduler methods
   forward,