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📝 (BetaSchedules) rework references in docstrings + add equations
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@ -1,6 +1,10 @@
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"""
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Cosine beta schedule.
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```math
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\\overline{\\alpha}_t = \\cos \\left( \\frac{t / T + \\epsilon}{1 + \\epsilon} \\frac{\\pi}{2} \\right)
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```
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## Input
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* `T::Int`: number of timesteps
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* `βₘₐₓ::Real=0.999f0`: maximum value of β
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@ -10,8 +14,8 @@ Cosine beta schedule.
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* `β::Vector{Real}`: βₜ values at each timestep t
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## References
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* [[2102.09672] Improved Denoising Diffusion Probabilistic Models](https://arxiv.org/abs/2102.09672)
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* [github:openai/improved-diffusion](https://github.com/openai/improved-diffusion/blob/783b6740edb79fdb7d063250db2c51cc9545dcd1/improved_diffusion/gaussian_diffusion.py#L36)
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* [nichol2021improved; Improved Denoising Diffusion Probabilistic Models](@cite)
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* [github:openai/improved-diffusion/improved_diffusion/gaussian_diffusion.py](https://github.com/openai/improved-diffusion/blob/783b6740edb79fdb7d063250db2c51cc9545dcd1/improved_diffusion/gaussian_diffusion.py#L36)
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"""
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function cosine_beta_schedule(T::Integer, βₘₐₓ::Real=0.999f0, ϵ::Real=1.0f-3)
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α̅(t) = cos((t / T + ϵ) / (1 + ϵ) * π / 2)^2
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"""
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Exponential beta schedule.
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```math
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\\overline{\\alpha}_t = \\exp \\left( \\frac{-12 t}{T} \\right)
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```
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## Input
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* `T::Int`: number of timesteps
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* `βₘₐₓ::Real=0.999f0`: maximum value of β
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## Output
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* `β::Vector{Real}`: βₜ values at each timestep t
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## References
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"""
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function exponential_beta_schedule(T::Integer, βₘₐₓ::Real=0.999f0)
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α̅(t) = exp(-12 * t / T)
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"""
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Linear beta schedule.
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```math
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\\beta_t = \\beta_1 + \\frac{t - 1}{T - 1} (\\beta_{-1} - \\beta_1)
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```
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## Input
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* `T::Integer`: number of timesteps
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* `β₁::Real=1.0f-4`: initial (t=1) value of β
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@ -10,7 +14,7 @@ Linear beta schedule.
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* `β::Vector{Real}`: βₜ values at each timestep t
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## References
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* [[2006.11239] Denoising Diffusion Probabilistic Models](https://arxiv.org/abs/2006.11239)
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* [ho2020denoising; Denoising Diffusion Probabilistic Models](@cite)
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"""
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function linear_beta_schedule(T::Integer, β₁::Real=1.0f-4, β₋₁::Real=2.0f-2)
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return range(start=β₁, stop=β₋₁, length=T)
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"""
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Scaled linear beta schedule.
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```math
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\\beta_t = \\left( \\sqrt{\\beta_1} + \\frac{t - 1}{T - 1} \\left( \\sqrt{\\beta_{-1}} - \\sqrt{\\beta_1} \\right) \\right)^2
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```
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## Input
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* `T::Int`: number of timesteps
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* `β₁::Real=1.0f-4`: initial value of β
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* `β::Vector{Real}`: βₜ values at each timestep t
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## References
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* [[2006.11239] Denoising Diffusion Probabilistic Models](https://arxiv.org/abs/2006.11239)
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* [ho2020denoising; Denoising Diffusion Probabilistic Models](@cite)
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"""
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function scaled_linear_beta_schedule(T::Integer, β₁::Real=1.0f-4, β₋₁::Real=2.0f-2)
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return range(start=√β₁, stop=√β₋₁, length=T) .^ 2
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@ -3,6 +3,10 @@ import NNlib: sigmoid
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"""
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Sigmoid beta schedule.
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```math
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\\beta_t = \\sigma \\left( 12 \\frac{t - 1}{T - 1} - 6 \\right) ( \\beta_{-1} - \\beta_1 ) + \\beta_1
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```
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## Input
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* `T::Int`: number of timesteps
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* `β₁::Real=1.0f-4`: initial value of β
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* `β::Vector{Real}`: βₜ values at each timestep t
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## References
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* [[2203.02923] GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://arxiv.org/abs/2203.02923)
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* [github.com:MinkaiXu/GeoDiff](https://github.com/MinkaiXu/GeoDiff/blob/ea0ca48045a2f7abfccd7f0df449e45eb6eae638/models/epsnet/diffusion.py#L57)
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* [xu2022geodiff; GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](@cite)
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* [github.com:MinkaiXu/GeoDiff/models/epsnet/diffusion.py](https://github.com/MinkaiXu/GeoDiff/blob/ea0ca48045a2f7abfccd7f0df449e45eb6eae638/models/epsnet/diffusion.py#L57)
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"""
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function sigmoid_beta_schedule(T::Integer, β₁::Real=1.0f-4, β₋₁::Real=2.0f-2)
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x = range(start=-6, stop=6, length=T)
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"""
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Rescale betas to have zero terminal Signal to Noise Ratio (SNR).
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Rescale β to have zero terminal Signal to Noise Ratio (SNR).
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## Input
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* `β::AbstractArray`: βₜ values at each timestep t
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* `β::Vector{Real}`: rescaled βₜ values at each timestep t
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## References
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* [[2305.08891] Rescaling Diffusion Models](https://arxiv.org/abs/2305.08891) (Alg. 1)
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* [lin2023common; Rescaling Diffusion Models (Alg. 1)](@cite)
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"""
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function rescale_zero_terminal_snr(β::AbstractArray)
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# convert β to ⎷α̅
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