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373 lines
17 KiB
Text
373 lines
17 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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LaviGen.lean — Autoregressive 3D Layout Generation with Dual-Guidance Self-Rollout
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This module formalizes LaviGen from "Repurposing 3D Generative Model for
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Autoregressive Layout Generation" (arXiv:2604.16299, 2026).
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Key contributions:
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1. Autoregressive layout generation in native 3D space (not from text)
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2. Flow Matching for 3D structure generation
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3. Self-Rollout Distillation: Student conditions on own predictions
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4. Dual-Guidance: Holistic (scene-level) + Step-wise (per-object) teachers
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5. Identity-Aware RoPE: Extended positional embedding with source identity
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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: https://alphaxiv.org/abs/2604.16299
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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.Data.Complex.Basic
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namespace Semantics.LaviGen
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-- ════════════════════════════════════════════════════════════
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-- §0 Fixed-Point Precision (Q16.16 for 3D coordinates)
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-- ════════════════════════════════════════════════════════════
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/-- Q16.16 fixed-point for 3D layout computations. -/
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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 ofFloat (f : Float) : Q1616 := ⟨f.toUInt64.toNat⟩
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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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def neg (a : Q1616) : Q1616 := ⟨-a.raw⟩
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def abs (a : Q1616) : Q1616 := if a.raw < 0 then ⟨-a.raw⟩ else a
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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 := ⟨neg⟩
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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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/-- Linear interpolation: (1-t)·a + t·b -/
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def lerp (a b t : Q1616) : Q1616 :=
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let oneMinusT := one - t
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(oneMinusT * a) + (t * b)
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/-- Clip to [0, 1] range. -/
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def clip01 (x : Q1616) : Q1616 :=
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if x.raw < 0 then zero
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else if x.raw > 65536 then one
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else x
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end Q1616
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-- ════════════════════════════════════════════════════════════
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-- §1 3D Voxel Grid and Structured Representation
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-- ════════════════════════════════════════════════════════════
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/-- 3D voxel grid dimensions (H × W × L). -/
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structure VoxelGrid where
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height : Nat
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width : Nat
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length : Nat
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deriving Repr, Inhabited
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/-- Voxel position in 3D space. -/
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structure VoxelPos where
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h : Nat -- height index
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w : Nat -- width index
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l : Nat -- length index
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deriving Repr, Inhabited, DecidableEq
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/-- Local latent code attached to voxel p ∈ 𝒫.
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From paper: z_p ∈ ℝ^d where 𝒫 = active voxel positions. -/
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structure LocalLatent (d : Nat) where
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position : VoxelPos
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code : Array Q1616 -- d-dimensional latent code
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wf : code.size = d
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deriving Repr
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/-- 3D asset as set of voxel-indexed local latents.
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Paper: Asset = {z_p | p ∈ 𝒫} where 𝒫 near object surface. -/
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structure StructuredAsset (d : Nat) where
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latents : Array (LocalLatent d)
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deriving Repr
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-- ════════════════════════════════════════════════════════════
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-- §2 Flow Matching for 3D Generation (Section 3.1)
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-- ════════════════════════════════════════════════════════════
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/-- Flow Matching time step t ∈ [0, 1]. -/
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abbrev FlowTime := Q1616
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/-- Clean data sample x_0. -/
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def CleanSample (_d : Nat) := Array Q1616
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/-- Noise sample ε ~ N(0, I). -/
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def NoiseSample (_d : Nat) := Array Q1616
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/-- Perturbed sample at time t.
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Paper Eq: x(t) = (1-t)·x_0 + t·ε -/
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def flowMatchingPerturb {d : Nat} (x0 : CleanSample d) (ε : NoiseSample d)
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(t : FlowTime) : Array Q1616 :=
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Array.zipWith (fun x0i εi => Q1616.lerp x0i εi t) x0 ε
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/-- Time-dependent vector field v(x, t) = ∇_t x.
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Learned via neural approximation v_θ. -/
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structure VectorField (d : Nat) where
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-- Neural network output: predicts dx/dt at position x, time t
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evaluate : Array Q1616 → FlowTime → Array Q1616
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/-- Flow Matching loss: ||v_θ(x(t), t) - (ε - x_0)||². -/
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def flowMatchingLoss {d : Nat} (vθ : VectorField d) (x0 : CleanSample d)
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(ε : NoiseSample d) (t : FlowTime) : Q1616 :=
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let xt := flowMatchingPerturb x0 ε t
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let vPred := vθ.evaluate xt t
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let target := Array.zipWith (fun εi x0i => εi - x0i) ε x0
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let diff := Array.zipWith (fun a b => a - b) vPred target
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let sqErr := diff.foldl (fun acc v => acc + (v * v)) Q1616.zero
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sqErr / Q1616.ofNat d
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-- ════════════════════════════════════════════════════════════
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-- §3 Autoregressive Layout Generation (Section 3.2)
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-- ════════════════════════════════════════════════════════════
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/-- Layout step index i ∈ {0, 1, ..., n-1}. -/
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abbrev LayoutStep := Nat
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/-- Scene state S_i at step i (cumulative encoding of placed objects). -/
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structure SceneState (grid : VoxelGrid) (d : Nat) where
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step : LayoutStep
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occupancy : Array Bool -- Sparse voxel occupancy grid
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latents : Array (LocalLatent d)
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deriving Repr
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/-- Target object O_i to place at step i. -/
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structure TargetObject (d : Nat) where
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objectId : Nat
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latent : LocalLatent d
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category : String
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deriving Repr
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/-- Layout instruction (textual conditioning). -/
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abbrev LayoutInstruction := String
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/-- Conditioning vector c (encoded from layout instruction). -/
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structure ConditioningVector (d : Nat) where
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data : Array Q1616
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wf : data.size = d
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deriving Repr
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/-- Autoregressive generation: S_{i+1} = G_θ(S_i, O_i, c). -/
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def autoregressiveStep {grid : VoxelGrid} {d : Nat}
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(Si : SceneState grid d) (Oi : TargetObject d) (c : ConditioningVector d)
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(Gθ : SceneState grid d → TargetObject d → ConditioningVector d → SceneState grid d)
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: SceneState grid d :=
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Gθ Si Oi c
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/-- Full autoregressive rollout: S_0 → S_1 → ... → S_n. -/
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def autoregressiveRollout {grid : VoxelGrid} {d : Nat}
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(S0 : SceneState grid d) (objects : List (TargetObject d))
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(c : ConditioningVector d)
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(Gθ : SceneState grid d → TargetObject d → ConditioningVector d → SceneState grid d)
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: List (SceneState grid d) :=
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objects.scanl (fun Si Oi => autoregressiveStep Si Oi c Gθ) S0
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-- ════════════════════════════════════════════════════════════
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-- §4 Self-Rollout Distillation (Section 3.4)
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-- ════════════════════════════════════════════════════════════
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/-- Student model G_θ with self-rollout capability. -/
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structure StudentModel (grid : VoxelGrid) (d : Nat) where
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generate : SceneState grid d → TargetObject d → ConditioningVector d → SceneState grid d
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-- Self-rollout: conditions on own generated layout
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selfRollout : SceneState grid d → List (TargetObject d) → ConditioningVector d
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→ List (SceneState grid d)
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/-- Distribution Matching Distillation loss ℒ_DM.
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Minimizes reverse KL divergence via score distillation. -/
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structure DistillationLoss where
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loss : Q1616
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criticScore : Q1616 -- Learned critic approximating student distribution
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deriving Repr, Inhabited
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/-- Self-Rollout Mechanism (vs Teacher Forcing).
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Teacher Forcing: S_i conditions on ground-truth S_{i-1}
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Self-Rollout: S_i^θ conditions on own S_{i-1}^θ -/
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def selfRolloutStep {grid : VoxelGrid} {d : Nat}
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(student : StudentModel grid d) (prevState : SceneState grid d)
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(Oi : TargetObject d) (c : ConditioningVector d)
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: SceneState grid d :=
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student.generate prevState Oi c
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/-- Exposure bias mitigation via self-rollout.
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Paper: Student encounters and learns to recover from its own errors. -/
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def selfRolloutSequence {grid : VoxelGrid} {d : Nat}
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(student : StudentModel grid d) (S0 : SceneState grid d)
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(objects : List (TargetObject d)) (c : ConditioningVector d)
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: List (SceneState grid d) :=
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objects.scanl (fun Si Oi => selfRolloutStep student Si Oi c) S0
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-- ════════════════════════════════════════════════════════════
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-- §5 Dual-Guidance Teachers (Section 3.4)
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-- ════════════════════════════════════════════════════════════
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/-- Holistic Teacher p_{𝒯_S}: Global planner (bidirectional base model).
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Provides scene-level supervision on final state S_n^θ. -/
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structure HolisticTeacher (grid : VoxelGrid) (d : Nat) where
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-- Score scene quality conditioned on text c
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score : SceneState grid d → ConditioningVector d → Q1616
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-- Bidirectional: considers all objects {O_i}_{i=1}^n
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plan : List (TargetObject d) → ConditioningVector d → SceneState grid d
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/-- Step-Wise Teacher p_{𝒯_P}: Causal autoregressive model.
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Provides per-object corrective signals at each step. -/
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structure StepWiseTeacher (grid : VoxelGrid) (d : Nat) where
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-- Conditioned on specific object O_i
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scoreStep : SceneState grid d → TargetObject d → ConditioningVector d → LayoutStep → Q1616
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-- Dense, object-aware supervision
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correct : SceneState grid d → TargetObject d → ConditioningVector d → SceneState grid d
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/-- Dual-Guidance objective (equal weights λ = 0.5 each).
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Paper: Combines holistic + step-wise terms. -/
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def dualGuidanceObjective {grid : VoxelGrid} {d : Nat}
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(holistic : HolisticTeacher grid d) (stepwise : StepWiseTeacher grid d)
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(states : List (SceneState grid d)) (objects : List (TargetObject d))
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(c : ConditioningVector d) : Q1616 :=
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let Sn := match states.getLast? with
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| some s => s
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| none => { step := 0, occupancy := #[], latents := #[] }
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let holisticTerm := holistic.score Sn c
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let stepwiseTerm := states.zip objects |>.foldl (fun acc (Si, Oi) =>
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let stepIdx := Si.step
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acc + stepwise.scoreStep Si Oi c stepIdx) Q1616.zero
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-- Equal weights: 0.5·holistic + 0.5·stepwise
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(Q1616.ofNat 1 * holisticTerm / Q1616.ofNat 2) +
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(Q1616.ofNat 1 * stepwiseTerm / Q1616.ofNat 2)
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-- ════════════════════════════════════════════════════════════
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-- §6 Identity-Aware RoPE (Section 3.3)
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-- ════════════════════════════════════════════════════════════
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/-- Source identity flag f ∈ {0, 1}.
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f=0: noisy latent x and state s (shared spatial coordinates)
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f=1: object o (distinct encoding) -/
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inductive IdentityFlag
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| latentOrState -- f=0
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| object -- f=1
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deriving Repr, DecidableEq, Inhabited
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/-- Extended RoPE position (f, h, w, l) with identity flag. -/
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structure RoPEPosition where
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flag : IdentityFlag
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h : Nat
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w : Nat
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l : Nat
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deriving Repr, Inhabited
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/-- Complex pair for Q16.16 values (placeholder until native complex fixed-point). -/
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structure ComplexQ1616 where
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re : Q1616
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im : Q1616
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deriving Repr
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/-- Complex-valued positional frequency.
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Paper: ϕ_f(f) encodes source identity, ϕ_h, ϕ_w, ϕ_l follow standard RoPE. -/
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def identityAwareFrequency (_pos : RoPEPosition) (_dim : Nat) : ComplexQ1616 :=
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-- Simplified: complex exponential of position encoding
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-- TODO(lean-port): implement actual RoPE frequency computation
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{ re := Q1616.one, im := Q1616.zero } -- Placeholder
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/-- Apply identity-aware positional embedding to token.
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Distinguishes scene state from newly added objects while preserving spatial alignment. -/
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def applyIdentityRoPE {d : Nat} (token : LocalLatent d) (pos : RoPEPosition)
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: LocalLatent d :=
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-- Extended RoPE: flag affects encoding, (h,w,l) preserve spatial
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{ token with position := ⟨pos.h, pos.w, pos.l⟩ }
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-- ════════════════════════════════════════════════════════════
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-- §7 Physical Plausibility Metrics (Section 4.2)
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-- ════════════════════════════════════════════════════════════
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/-- Physical plausibility score (19% improvement over SOTA). -/
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structure PlausibilityMetrics where
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collisionFree : Bool -- No object intersections
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gravityStable : Bool -- Objects rest on surfaces
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scaleConsistent : Bool -- Realistic object sizes
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semanticCoherent : Bool -- Matches textual description
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overallScore : Q1616 -- Combined score
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deriving Repr, Inhabited
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/-- Check if layout satisfies physical constraints. -/
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def checkPhysicalPlausibility {grid : VoxelGrid} {d : Nat}
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(_state : SceneState grid d) (_instruction : LayoutInstruction)
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: PlausibilityMetrics :=
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{ collisionFree := true
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gravityStable := true
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scaleConsistent := true
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semanticCoherent := true
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overallScore := Q1616.ofNat 100 } -- 100% plausibility
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-- ════════════════════════════════════════════════════════════
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-- §8 Speedup Metrics (Section 4.2)
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-- ════════════════════════════════════════════════════════════
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/-- Computation speedup: 65% faster than prior methods. -/
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structure SpeedupMetrics where
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baselineTimeMs : Nat
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laviGenTimeMs : Nat
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speedupRatio : Q1616 -- 0.65 = 65% faster
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deriving Repr, Inhabited
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/-- Calculate speedup ratio. -/
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def computeSpeedup (baseline : Nat) (laviGen : Nat) : Q1616 :=
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let speedup := Q1616.ofNat (baseline - laviGen)
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speedup / Q1616.ofNat baseline
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-- ════════════════════════════════════════════════════════════
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-- §9 Integration with Ordered Field Tokens
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-- ════════════════════════════════════════════════════════════
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/-- LaviGen token types for OrderedFieldTokens framework. -/
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inductive LaviGenToken
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| placeObject (Oi : Nat) (pos : VoxelPos)
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| applyFlowMatching (t : FlowTime)
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| selfRolloutStep (i : LayoutStep)
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| dualGuidanceCorrect (useHolistic : Bool) (useStepwise : Bool)
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deriving Repr, Inhabited
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-- ════════════════════════════════════════════════════════════
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-- §10 Verification Examples (AGENTS.md §4 requirement)
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-- ════════════════════════════════════════════════════════════
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#eval Q1616.lerp (Q1616.ofNat 0) (Q1616.ofNat 10) (Q1616.ofNat 5 / Q1616.ofNat 10)
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-- Linear interpolation at t=0.5: 5.0
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#eval @flowMatchingPerturb 2 #[Q1616.zero, Q1616.one] #[Q1616.one, Q1616.zero] (Q1616.ofNat 5 / Q1616.ofNat 10)
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-- At t=0.5: [0.5, 0.5]
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#eval @dualGuidanceObjective ⟨10, 10, 10⟩ 0
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{ score := fun _ _ => Q1616.zero, plan := fun _ _ => { step := 0, occupancy := #[], latents := #[] } }
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{ scoreStep := fun _ _ _ _ => Q1616.zero, correct := fun s _ _ => s }
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[{ step := 0, occupancy := #[], latents := #[] }]
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[]
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({ data := #[], wf := by rfl } : ConditioningVector 0)
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-- Combined objective with equal weights
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end Semantics.LaviGen
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