| import Mathlib |
| import OAI.Probability.LogConcave.TensorGraphs.Chain |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling.CycleTrace |
| open AdjointRemoval MeasureTheory TensorEnergy |
| open scoped Classical BigOperators NNReal ENNReal Matrix.Norms.L2Operator |
|
|
| variable {s a b d : ℕ} |
| local instance cutDecEqDT (β : Fin (2*s+1) → ℕ) (q : ℕ) : |
| DecidableEq (InternalEdge β ⊕ Fin q) := |
| @instDecidableEqSum _ _ (Classical.decEq _) (Classical.decEq _) |
|
|
| def fieldPosition (v : Fin d → Point d) |
| (F : (Fin a ⊕ Fin b → Fin d) → Fin d → Point d → ℝ) |
| (e : ℕ) (x : Point d) (c : (Fin a ⊕ Fin b) ⊕ (Unit ⊕ Fin e) → Fin d) : ℝ := |
| JetCalculus.jet (fun k => v (c (Sum.inr (Sum.inr k)))) (List.finRange e) |
| (F (fun t => c (Sum.inl t)) (c (Sum.inr (Sum.inl ())))) x |
|
|
| lemma lift_position_bound (v : Fin d → Point d) |
| (F : (Fin a ⊕ Fin b → Fin d) → Fin d → Point d → ℝ) |
| (i : Primary s) (e : ℕ) (x : Point d) (M : ℝ) |
| (hM : AllSplitBound (fieldPosition v F e x) M) : |
| AllSplitBound (chainPositionTensor (bundle a b) a v i (liftField F i) e x) M := by |
| exact hM.reindex ((slotEquivAt i).sumCongr (Equiv.refl (Unit ⊕ Fin e))) |
|
|
| def scorePosition (H : Point d → ℝ) (v : Fin d → Point d) |
| (e : ℕ) (x : Point d) (c : Fin (e+2) → Fin d) : ℝ := |
| JetCalculus.jet (fun k : Fin e => v (c ⟨k.val+2,by omega⟩)) (List.finRange e) |
| (directional (v (c ⟨0,by omega⟩)) (directional (v (c ⟨1,by omega⟩)) H)) x |
|
|
| lemma scorePosition_eq (H : Point d → ℝ) (v : Fin d → Point d) |
| (h : Hessian (Primary s)) (x : Point d) : |
| scorePosition H v h.extra.length x=hessianPositionTensor H v h x := rfl |
|
|
| lemma trace_polySmooth {H : Point d → ℝ} (hH : PolySmooth H) |
| (v : Fin d → Point d) (F : (Fin a ⊕ Fin b → Fin d) → Fin d → Point d → ℝ) |
| (hF : ∀c z,PolySmooth (F c z)) : |
| PolySmooth (fun x => ((adjointMatrix H v F x*(adjointMatrix H v F x).transpose)^(s+1)).trace) := by |
| simp_rw [trace_eq_chain] |
| exact PolySmooth.sum Finset.univ (fun c _ => PolySmooth.prod Finset.univ |
| (fun i _ => hH.tensorAdjoint (fun z => hF _ z) v)) |
|
|
| theorem adjoint_moment_of_branches {H : Point d → ℝ} (v : Fin d → Point d) |
| (F : (Fin a ⊕ Fin b → Fin d) → Fin d → Point d → ℝ) |
| {K : ℝ≥0} (hH : PolySmooth H) (ht : HasGaussianLowerTail H) |
| (hK : LipschitzWith K (gradient H)) (hF : ∀c z,PolySmooth (F c z)) |
| (B : ℝ≥0∞) |
| (hB : ∀S∈expand (Fintype.card (Primary s)) initial, |
| (∫⁻x,ENNReal.ofReal |chainBranch (bundle (s:=s) a b) a H v (liftField F) S x| |
| ∂gibbs H)≤B) : |
| (∫⁻x,ENNReal.ofReal ‖adjointMatrix H v F x‖^(2*(s+1)) ∂gibbs H) ≤ |
| ((3*(2*s+2)+1:ℕ):ℝ≥0∞)^(2*s+2)*B := by |
| let T := fun x => ((adjointMatrix H v F x*(adjointMatrix H v F x).transpose)^(s+1)).trace |
| have hi : Integrable T (gibbs H) := (trace_polySmooth hH v F hF).integrable hH.smooth.continuous ht |
| have hp (x) : ‖adjointMatrix H v F x‖^(2*(s+1))≤T x := |
| TraceMoment.rectangular_trace_moment _ (Nat.succ_pos s) |
| have h0 (x) : 0≤T x := (pow_nonneg (norm_nonneg _) _).trans (hp x) |
| have hb := integral_chain_bound (bundle (s:=s) a b) a v (liftField F) hH ht hK |
| (fun i c z => hF _ z) B hB |
| calc |
| _ ≤ ∫⁻x,ENNReal.ofReal (T x) ∂gibbs H := by |
| apply lintegral_mono |
| intro x |
| change ENNReal.ofReal ‖adjointMatrix H v F x‖ ^ (2*(s+1)) ≤ _ |
| rw [←ENNReal.ofReal_pow (norm_nonneg _) (2*(s+1))] |
| exact ENNReal.ofReal_le_ofReal (hp x) |
| _ = ENNReal.ofReal (∫x,T x ∂gibbs H) := |
| (ofReal_integral_eq_lintegral_ofReal hi (Filter.Eventually.of_forall h0)).symm |
| _ = ENNReal.ofReal |∫x,T x ∂gibbs H| := by |
| rw [abs_of_nonneg (integral_nonneg h0)] |
| _ ≤ _ := by |
| simpa only [T,trace_eq_chain,show 2*s+1+1=2*s+2 from by omega] using hb |
|
|
| theorem adjoint_trace_moment {H : Point d → ℝ} (v : Fin d → Point d) |
| (F : (Fin a ⊕ Fin b → Fin d) → Fin d → Point d → ℝ) |
| (ha : 0<a) (hb : 0<b) {K : ℝ≥0} (hH : PolySmooth H) |
| (ht : HasGaussianLowerTail H) (hK : LipschitzWith K (gradient H)) |
| (hF : ∀c z,PolySmooth (F c z)) |
| (M N : ℕ → Point d → ℝ) |
| (hM0 : ∀e x,0≤M e x) (hN0 : ∀e x,0≤N e x) |
| (hM : ∀e x,AllSplitBound (fieldPosition v F e x) (M e x)) |
| (hN : ∀e x,AllSplitBound (scorePosition H v e x) (N e x)) |
| (j k : ℕ) (A R : ℝ≥0∞) (hA : 1≤A) (hR : 1≤R) (hAf : A≠⊤) (hRf : R≠⊤) |
| (hMm : ∀e,AEMeasurable (fun x => ENNReal.ofReal (M e x)) (gibbs H)) |
| (hp : ∀e≤2*(s+1),(∫⁻x,ENNReal.ofReal (M e x)^(2*(s+1)) ∂gibbs H) ≤ |
| (A^(j+e+1)*R^(k+j+e))^(2*(s+1))) |
| (hh : ∀e≤2*(s+1),∀x,ENNReal.ofReal (N e x)≤A^(e+1)*R^e) : |
| (∫⁻x,ENNReal.ofReal ‖adjointMatrix H v F x‖^(2*(s+1)) ∂gibbs H)≤ |
| ((3*(2*s+2)+1:ℕ):ℝ≥0∞)^(2*s+2) * (d:ℝ≥0∞)^a * |
| (A^(2*(2*s+2)*(j+1))*R^((2*s+2)*(k+j+1))) := by |
| have he : 2*s+1+1=2*(s+1) := by omega |
| have hβ : ∀t : Fin (2*s+1),0<bundle a b t := by |
| intro t |
| unfold bundle |
| split <;> assumption |
| rw [mul_assoc] |
| apply adjoint_moment_of_branches v F hH ht hK hF |
| intro S hS |
| have hpder (i : Primary s) : (S.derivatives i).length≤2*(s+1) := by |
| simpa only [Fintype.card_fin,he] using (terminal_moment_budgets hS j k).1 i |
| have hhder (h : S.hessians) : (h : Hessian (Primary s)).extra.length≤2*(s+1) := by |
| simpa only [Fintype.card_fin,he] using (terminal_moment_budgets hS j k).2.1 h Multiset.coe_mem |
| have hbound := chain_branch_bound (gibbs H) hS (bundle a b) hβ a ha H v (liftField F) |
| (fun i => M (S.derivatives i).length) (fun h => N (h : Hessian (Primary s)).extra.length) |
| (fun i x => hM0 _ x) (fun h x => hN0 _ x) |
| (fun i x => lift_position_bound v F i _ x _ (hM _ x)) |
| (fun h x => hN _ x) j k A R hA hR hAf hRf |
| (fun i => hMm _) (fun i => by simpa only [he] using hp _ (hpder i)) |
| (fun h x => hh _ (hhder h) x) |
| simpa only [show 2*s+1+1=2*s+2 from by omega] using hbound |
|
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| end LogConcaveSampling.CycleTrace |
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| end |
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| end |
|
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| end |
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| end OAI |
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