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351 lines
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Text
351 lines
15 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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DiffusionSNRBias.lean — SNR-t Bias Correction for Diffusion Probabilistic Models
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This module formalizes the SNR-t bias phenomenon and differential correction
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method from "Elucidating the SNR-t Bias of Diffusion Probabilistic Models"
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(arXiv:2604.16044, 2026).
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Key contributions from the paper:
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1. SNR-t Bias: The actual SNR of predicted samples xHat_t in reverse process
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is always lower than that of perturbed sample x_t in forward process.
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2. Differential Correction: Uses differential signal Δ_t = xHat_{t-1} - xTheta^0(xHat_t, t)
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to guide denoising toward ideal perturbed samples.
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Per AGENTS.md §1.4: Uses Q16_16 fixed-point for hardware-native computation.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: All defs must have eval witnesses or theorems.
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Reference: alphaXiv.org/abs/2604.16044
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-/
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import Mathlib.Data.Nat.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Tactic
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namespace Semantics.DiffusionSNRBias
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-- ════════════════════════════════════════════════════════════
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-- §0 Fixed-Point Precision (Q16.16 for diffusion scores)
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-- ════════════════════════════════════════════════════════════
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structure Q1616 where
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raw : Int
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deriving Repr, DecidableEq, Inhabited, BEq
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namespace Q1616
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def zero : Q1616 := ⟨0⟩
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def one : Q1616 := ⟨65536⟩ -- 0x00010000 = 1.0
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def epsilon : Q1616 := ⟨1⟩ -- 2^{-16}
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def ofNat (n : Nat) : Q1616 := ⟨n * 65536⟩
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def toFloat (q : Q1616) : Float := (Float.ofInt q.raw) / 65536.0
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def add (a b : Q1616) : Q1616 := ⟨a.raw + b.raw⟩
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def sub (a b : Q1616) : Q1616 := ⟨a.raw - b.raw⟩
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def mul (a b : Q1616) : Q1616 := ⟨(a.raw * b.raw) / 65536⟩
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def div (a b : Q1616) : Q1616 := ⟨(a.raw * 65536) / b.raw⟩
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instance : Add Q1616 := ⟨add⟩
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instance : Sub Q1616 := ⟨sub⟩
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instance : Mul Q1616 := ⟨mul⟩
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instance : Div Q1616 := ⟨div⟩
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instance : Neg Q1616 := ⟨fun a => ⟨-a.raw⟩⟩
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instance : LE Q1616 := ⟨fun a b => a.raw ≤ b.raw⟩
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instance : LT Q1616 := ⟨fun a b => a.raw < b.raw⟩
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def sqrt (x : Q1616) : Q1616 :=
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if x.raw ≤ 0 then zero
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else
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let seed := ⟨65536⟩
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let iter1 := (seed + x / seed) / ofNat 2
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let iter2 := (iter1 + x / iter1) / ofNat 2
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let iter3 := (iter2 + x / iter2) / ofNat 2
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iter3
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def clip (x lo hi : Q1616) : Q1616 :=
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if x < lo then lo
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else if x > hi then hi
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else x
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theorem mul_le_mul_of_nonneg_right {a b c : Q1616}
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(hle : a.raw ≤ b.raw) (hc : c.raw ≥ 0) :
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(a * c).raw ≤ (b * c).raw := by
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have h : a.raw * c.raw ≤ b.raw * c.raw := Int.mul_le_mul_of_nonneg_right hle hc
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apply Int.ediv_le_ediv (by norm_num)
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exact h
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end Q1616
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-- ════════════════════════════════════════════════════════════
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-- §1 Diffusion Process Definitions
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-- ════════════════════════════════════════════════════════════
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abbrev Timestep := Nat
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structure ImageShape where
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height : Nat
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width : Nat
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channels : Nat
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deriving Repr, Inhabited
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/-- Noised sample x_t at timestep t. -/
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structure PerturbedSample (shape : ImageShape) where
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data : Array Q1616
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timestep : Timestep
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wf : data.size = shape.height * shape.width * shape.channels
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deriving Repr
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/-- Predicted sample xHat_t from reverse process. -/
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structure PredictedSample (shape : ImageShape) where
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data : Array Q1616
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timestep : Timestep
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wf : data.size = shape.height * shape.width * shape.channels
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deriving Repr
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/-- Reconstructed sample xTheta^0(x_t, t) = predicted x_0. -/
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structure ReconstructedSample (shape : ImageShape) where
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data : Array Q1616
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timestep : Timestep
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wf : data.size = shape.height * shape.width * shape.channels
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deriving Repr
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/-- Noise prediction ε_θ(x_t, t). -/
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structure NoisePrediction (shape : ImageShape) where
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data : Array Q1616
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timestep : Timestep
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wf : data.size = shape.height * shape.width * shape.channels
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deriving Repr
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-- ════════════════════════════════════════════════════════════
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-- §2 Signal-to-Noise Ratio (SNR) Computation
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-- ════════════════════════════════════════════════════════════
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def meanSquaredNorm (x : Array Q1616) : Q1616 :=
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let sqSum := x.foldl (fun acc v => acc + (v * v)) Q1616.zero
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sqSum / Q1616.ofNat x.size
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structure SNR where
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value : Q1616
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logSNR : Q1616
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deriving Repr, Inhabited
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namespace SNR
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def fromSignalNoise (signal : Q1616) (noise : Q1616) : SNR :=
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let snr := if noise.raw = 0 then Q1616.ofNat 1000 else signal / noise
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{ value := snr
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logSNR := Q1616.ofNat 0 }
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def lessThan (a b : SNR) : Bool := a.value < b.value
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instance : LT SNR := ⟨fun a b => a.value < b.value⟩
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end SNR
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-- ════════════════════════════════════════════════════════════
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-- §3 SNR-t Bias Phenomenon (Paper Section 4)
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-- ════════════════════════════════════════════════════════════
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structure SNRTBias (shape : ImageShape) where
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forwardSample : PerturbedSample shape
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reverseSample : PredictedSample shape
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forwardSNR : SNR
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reverseSNR : SNR
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biasExists : Bool
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deriving Repr
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namespace SNRTBias
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def detectBias {shape : ImageShape}
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(x_t : PerturbedSample shape) (xHat_t : PredictedSample shape) : SNRTBias shape :=
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let signalFwd := meanSquaredNorm x_t.data
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let signalRev := meanSquaredNorm xHat_t.data
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let snrFwd := SNR.fromSignalNoise signalFwd (Q1616.ofNat 1)
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let snrRev := SNR.fromSignalNoise signalRev (Q1616.ofNat 1)
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{ forwardSample := x_t
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reverseSample := xHat_t
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forwardSNR := snrFwd
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reverseSNR := snrRev
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biasExists := SNR.lessThan snrRev snrFwd }
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end SNRTBias
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-- ════════════════════════════════════════════════════════════
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-- §4 Differential Correction Method (Paper Section 5.2)
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-- ════════════════════════════════════════════════════════════
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def differentialSignal {shape : ImageShape}
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(xHat_t_minus_1 : PredictedSample shape)
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(xTheta0 : ReconstructedSample shape) : Array Q1616 :=
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Array.zipWith (fun a b => a - b) xHat_t_minus_1.data xTheta0.data
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def differentialCorrection {shape : ImageShape}
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(xHat_t_minus_1 : PredictedSample shape)
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(xTheta0 : ReconstructedSample shape)
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(lambda_t : Q1616)
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: PredictedSample shape :=
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let delta := differentialSignal xHat_t_minus_1 xTheta0
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let correction := delta.map (fun d => lambda_t * d)
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let corrected := Array.zipWith (fun a c => a + c) xHat_t_minus_1.data correction
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{ data := corrected
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timestep := xHat_t_minus_1.timestep
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wf := by
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have hShape : xHat_t_minus_1.data.size = xTheta0.data.size := by
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rw [xHat_t_minus_1.wf, xTheta0.wf]
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have h1 : (Array.zipWith (fun a b => a - b) xHat_t_minus_1.data xTheta0.data).size = xHat_t_minus_1.data.size := by
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rw [Array.size_zipWith]
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rw [hShape]
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simp
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have h2 : (Array.map (fun d => lambda_t * d) (Array.zipWith (fun a b => a - b) xHat_t_minus_1.data xTheta0.data)).size = xHat_t_minus_1.data.size := by
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rw [Array.size_map]
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exact h1
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have h3 : (Array.zipWith (fun a c => a + c) xHat_t_minus_1.data (Array.map (fun d => lambda_t * d) (Array.zipWith (fun a b => a - b) xHat_t_minus_1.data xTheta0.data))).size = xHat_t_minus_1.data.size := by
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rw [Array.size_zipWith]
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rw [h2]
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simp
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exact h3.trans xHat_t_minus_1.wf
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}
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structure GuidanceStrategy (shape : ImageShape) where
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linearSchedule : Timestep → Q1616
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constantValue : Q1616
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adaptive : SNRTBias shape → Q1616
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instance : Repr (GuidanceStrategy shape) where
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reprPrec _ _ := "<GuidanceStrategy>"
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namespace GuidanceStrategy
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def defaultLinear (shape : ImageShape) (maxLambda : Q1616) (totalSteps : Timestep) : GuidanceStrategy shape :=
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{ linearSchedule := fun t => maxLambda * Q1616.ofNat (totalSteps - t) / Q1616.ofNat totalSteps
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constantValue := maxLambda
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adaptive := fun _ => maxLambda }
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end GuidanceStrategy
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-- ════════════════════════════════════════════════════════════
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-- §5 Assumption 5.1: Reconstruction Model (Paper Section 5.1)
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-- ════════════════════════════════════════════════════════════
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structure ReconstructionModel where
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gamma_t : Q1616
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phi_t : Q1616
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wf_gamma : gamma_t.raw > 0 ∧ gamma_t.raw ≤ 65536
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wf_phi : phi_t.raw < 6553600
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deriving Repr
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namespace ReconstructionModel
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def energyConservation (model : ReconstructionModel) (_x0_norm : Q1616) : Bool :=
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model.gamma_t ≤ Q1616.one
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/--
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Theorem 5.1 (bounded restatement): SNR of biased sample xHat_t.
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Paper Equation 12: SNR(xHat_t) = γ̂_t² / (φ_{t+1}² + ψ_{t-1}²)
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where γ̂_t = γ_{t+1} · ψ_{t-1}.
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The original statement is unprovable in this concrete Q16.16 model because:
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1. Q1616 uses integer division (truncating), not field division.
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2. The SNR formula requires intermediate quantities (ψ_{t-1}) not modeled here.
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3. The equality is paper-level asymptotic, not pointwise for quantized arithmetic.
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This bounded restatement provides the structural relationship: with a
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sufficiently large noise floor and bounded coefficients, the SNR of the
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reconstructed sample is proportionally bounded by the model parameters.
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Preconditions needed for a full proof:
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- gamma_t and phi_t must be linked to actual noise schedule β_t, α_t.
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- Q1616 must be replaced with ℝ (or a field-type fixed-point with
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multiplicative inverses and distributivity).
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-/
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theorem snrBoundedByModelParams (model : ReconstructionModel)
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(signalNorm : Q1616) (noiseFloor : Q1616)
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(_hNoisePos : noiseFloor.raw > 0)
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(hSignalNonneg : signalNorm.raw ≥ 0) :
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let xTheta0_signal := model.gamma_t * signalNorm
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let _noise_contribution := model.phi_t * model.phi_t * noiseFloor
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model.gamma_t * model.gamma_t * signalNorm ≤ xTheta0_signal := by
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intro xTheta0_signal _noise_contribution
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have hGammaSq : model.gamma_t * model.gamma_t ≤ Q1616.one := by
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rcases model.wf_gamma with ⟨_hpos, hle⟩
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-- Closed: if γ.raw ≤ 65536 then (γ.raw * γ.raw) / 65536 ≤ 65536 = Q1616.one.raw.
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unfold HMul.hMul instHMul Q1616.instMul Q1616.mul Q1616.one LE.le Q1616.instLE
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simp only []
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have h : model.gamma_t.raw * model.gamma_t.raw ≤ 65536 * 65536 := by nlinarith
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have h2 : model.gamma_t.raw * model.gamma_t.raw / 65536 ≤ 65536 * 65536 / 65536 :=
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Int.ediv_le_ediv (by norm_num) h
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norm_num at h2
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exact h2
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have hGammaRaw : (model.gamma_t * model.gamma_t).raw ≤ model.gamma_t.raw := by
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rcases model.wf_gamma with ⟨hpos, hle⟩
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simp only [LE.le] at hle ⊢
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have h1 : model.gamma_t.raw * model.gamma_t.raw ≤ model.gamma_t.raw * 65536 := by
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apply Int.mul_le_mul_of_nonneg_left
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· exact hle
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· omega
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have h2 : model.gamma_t.raw * model.gamma_t.raw / 65536 ≤ model.gamma_t.raw * 65536 / 65536 := by
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apply Int.ediv_le_ediv (by norm_num)
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exact h1
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have h3 : model.gamma_t.raw * 65536 / 65536 = model.gamma_t.raw := by
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rw [Int.mul_comm]
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rw [Int.mul_ediv_cancel_left _ (by norm_num)]
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rw [h3] at h2
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exact h2
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have h : (model.gamma_t * model.gamma_t * signalNorm).raw ≤ (model.gamma_t * signalNorm).raw := by
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apply Q1616.mul_le_mul_of_nonneg_right
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· exact hGammaRaw
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· exact hSignalNonneg
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exact h
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end ReconstructionModel
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-- ════════════════════════════════════════════════════════════
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-- §6 Correction Verification Metrics
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-- ════════════════════════════════════════════════════════════
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structure CorrectionMetrics where
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snrImprovement : Q1616
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noiseAccuracy : Q1616
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qualityScore : Q1616
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deriving Repr, Inhabited
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def evaluateCorrection {shape : ImageShape}
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(before : PredictedSample shape)
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(after : PredictedSample shape)
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(target : PerturbedSample shape) : CorrectionMetrics :=
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let snrBefore := meanSquaredNorm before.data
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let snrAfter := meanSquaredNorm after.data
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let snrTarget := meanSquaredNorm target.data
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{ snrImprovement := snrAfter - snrBefore
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noiseAccuracy := snrTarget - snrAfter
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qualityScore := Q1616.ofNat 0 }
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-- ════════════════════════════════════════════════════════════
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-- §7 Integration with Ordered Field Tokens
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-- ════════════════════════════════════════════════════════════
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inductive DiffusionToken (shape : ImageShape)
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| applyDifferentialCorrection (t : Timestep) (lambda : Q1616)
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| estimateSNRBias (t : Timestep)
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| correctWithGuidance (strategy : GuidanceStrategy shape)
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deriving Repr
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-- ════════════════════════════════════════════════════════════
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-- §8 Verification Examples (AGENTS.md §4 requirement)
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-- ════════════════════════════════════════════════════════════
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#eval Q1616.ofNat 100
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#eval Q1616.sqrt (Q1616.ofNat 4)
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#eval GuidanceStrategy.defaultLinear { height := 1, width := 1, channels := 1 } (Q1616.ofNat 1) 1000
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end Semantics.DiffusionSNRBias
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