| import Mathlib |
| import OAI.Probability.BinarySweep.FiniteLaws.LinearEvenMoment |
| import OAI.Probability.BinarySweep.SparseBounds.ConditionalCentered |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| open Irrep Representation |
|
|
| variable {b h k : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) |
| (x : Placement H k 0) (g₀ : ConditionalChoices H) |
| {A W : Type*} [Fintype A] [NormedAddCommGroup W] [InnerProductSpace ℂ W] |
| [FiniteDimensional ℂ W] (e : A ⊕ Fin k ≃ FreeSlot H 0) |
| (σ : Representation ℂ (Equiv.Perm A) W) |
|
|
| def augmentedFrame : Equiv.Perm A := |
| ((augmentedInputBasis H x g₀ e).trans (freeIdentification (augmentByChoice H x g₀))).trans |
| (augmentedOutputBasis H x g₀ e).symm |
|
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| def augmentedRepresentation : Representation ℂ (Equiv.Perm (FreeSlot (augmentByChoice H x g₀) 0)) W := |
| σ.comp (augmentedInputBasis H x g₀ e).symm.permCongrHom.toMonoidHom |
|
|
| lemma augmented_subgroup_perm (g : ConditionalChoices (augmentByChoice H x g₀)) : |
| ((cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).symm |
| ((placementSection e (augmentedOutputIndex H x g₀))⁻¹* |
| remainingPerm H g.val ((augmentByChoice_extension H x g₀).pathEvent_mono g.property))).1 = |
| augmentedFrame H x g₀ e * (augmentedInputBasis H x g₀ e).symm.permCongr |
| (remainingPerm (augmentByChoice H x g₀) g.val g.property) := by |
| rw [augmented_subgroup_bijection] |
| ext a |
| simp only [augmentedFrame,Equiv.Perm.mul_apply,Equiv.permCongr_apply,Equiv.trans_apply, |
| Equiv.symm_symm,Equiv.apply_symm_apply,remainingPerm] |
|
|
| omit [FiniteDimensional ℂ W] in |
| lemma placementBlock_expectation (z : ℝ) (y : Placement H k 0) : |
| hilbertBlock (groupAverage (coindHilbertRep (blockPerm e) σ |
| (fun x => (placementSection e x)⁻¹) (placement_cosets_bijective e)) |
| (fun a => (conditionalGroupLaw H z a:ℂ))) y x = |
| ∑g : ConditionalChoices H, conditionalChoiceWeight H z g • |
| (if x.trans (remainingPerm H g.val g.property).toEmbedding=y then |
| σ ((cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).symm ((placementSection e y)⁻¹* |
| remainingPerm H g.val g.property)).1 else 0) := by |
| have hs (r : ℝ) (T : W →ₗ[ℂ] W) : (r:ℂ) • T = r • T := |
| algebraMap_smul ℂ r T |
| calc |
| _ = ∑a, conditionalGroupLaw H z a • |
| hilbertBlock (coindHilbertRep (blockPerm e) σ (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e) a) y x := by |
| ext v |
| simp only [groupAverage,hilbertBlock,LinearMap.comp_apply,LinearMap.sum_apply,LinearMap.smul_apply,map_sum,map_smul] |
| apply Finset.sum_congr rfl |
| intro a _ |
| exact algebraMap_smul ℂ (conditionalGroupLaw H z a) _ |
| _ = _ := by |
| rw [conditionalGroupLaw_expectation] |
| apply Finset.sum_congr rfl |
| intro g _ |
| rw [placement_hilbertBlock] |
| split_ifs <;> rfl |
| |
| omit [FiniteDimensional ℂ W] in |
| theorem augmented_block_average (z : ℝ) |
| (hp : conditionalNormalizer (augmentByChoice H x g₀) z≠0) : |
| hilbertBlock (groupAverage (coindHilbertRep (blockPerm e) σ |
| (fun x => (placementSection e x)⁻¹) (placement_cosets_bijective e)) |
| (fun a => (conditionalGroupLaw H z a:ℂ))) (augmentedOutputIndex H x g₀) x = |
| (conditionalNormalizer (augmentByChoice H x g₀) z / conditionalNormalizer H z) • |
| (σ (augmentedFrame H x g₀ e) * |
| groupAverage (augmentedRepresentation H x g₀ e σ) |
| (fun a => (conditionalGroupLaw (augmentByChoice H x g₀) z a:ℂ))) := by |
| rw [placementBlock_expectation] |
| have he := (augmentByChoice_extension H x g₀).conditional_sum z hp |
| (fun g => σ ((cosetEquiv (blockPerm e) (fun x => (placementSection e x)⁻¹) |
| (placement_cosets_bijective e)).symm ((placementSection e (augmentedOutputIndex H x g₀))⁻¹* |
| remainingPerm H g.val ((augmentByChoice_extension H x g₀).pathEvent_mono g.property))).1) |
| refine Eq.trans ?_ (Eq.trans he ?_) |
| · apply Finset.sum_congr rfl |
| intro g _ |
| by_cases hg : pathEvent (augmentByChoice H x g₀) g.val |
| · rw [dite_eq_left hg,ite_eq_left ((augmentedOutputIndex_event H x g₀ g).mpr hg)] |
| · rw [dite_eq_right hg,ite_eq_right (fun he => hg ((augmentedOutputIndex_event H x g₀ g).mp he)),smul_zero] |
| · congr 1 |
| simp_rw [augmented_subgroup_perm,map_mul] |
| rw [groupAverage] |
| have hh := conditionalGroupLaw_expectation (augmentByChoice H x g₀) z |
| (fun a => augmentedRepresentation H x g₀ e σ a) |
| have hc : (∑a, (conditionalGroupLaw (augmentByChoice H x g₀) z a:ℂ) • |
| augmentedRepresentation H x g₀ e σ a) = |
| ∑g : ConditionalChoices (augmentByChoice H x g₀), |
| conditionalChoiceWeight (augmentByChoice H x g₀) z g • |
| augmentedRepresentation H x g₀ e σ (remainingPerm (augmentByChoice H x g₀) g.val g.property) := by |
| refine Eq.trans ?_ hh |
| apply Finset.sum_congr rfl |
| intro a _ |
| exact algebraMap_smul ℂ (conditionalGroupLaw (augmentByChoice H x g₀) z a) |
| (augmentedRepresentation H x g₀ e σ a) |
| rw [hc,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro g _ |
| rw [mul_smul_comm] |
| rfl |
|
|
| end BinaryCoordinateSweeps |
|
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| end |
|
|
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| open Irrep Representation |
|
|
| variable {b h k : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) |
|
|
| lemma placementBijection_index (x : Placement H k 0) (g : ConditionalChoices H) : |
| placementBijection H k g x = placementIdentification H k (augmentedOutputIndex H x g) := by |
| apply Function.Embedding.ext |
| intro i |
| change remainingBijection H g.val g.property (x i) = |
| freeIdentification H ((freeIdentification H).symm (remainingBijection H g.val g.property (x i))) |
| exact (Equiv.apply_symm_apply _ _).symm |
|
|
| lemma placementKernel_ratio (z : ℝ) (x : Placement H k 0) (g : ConditionalChoices H) : |
| placementKernel H z x (augmentedOutputIndex H x g) = |
| conditionalNormalizer (augmentByChoice H x g) z / conditionalNormalizer H z := by |
| unfold placementKernel |
| rw [← placementBijection_index H x g,placementProbability_augmentation, |
| pathExtension_event_ratio (augmentByChoice_extension H x g)] |
|
|
| lemma placementKernel_output_pos {z : ℝ} (hz : 0≤z) (hz1 : z<1) |
| (x : Placement H k 0) (g : ConditionalChoices H) : |
| 0<placementKernel H z x (augmentedOutputIndex H x g) := by |
| rw [placementKernel_ratio] |
| exact div_pos (conditionalNormalizer_pos _ hz hz1) (conditionalNormalizer_pos _ hz hz1) |
|
|
| variable {A W : Type*} [Fintype A] [NormedAddCommGroup W] [InnerProductSpace ℂ W] |
| [FiniteDimensional ℂ W] (e : A ⊕ Fin k ≃ FreeSlot H 0) |
| (σ : Representation ℂ (Equiv.Perm A) W) |
|
|
| omit [FiniteDimensional ℂ W] in |
| lemma placementBlock_zero {z : ℝ} (hz : 0≤z) (hz1 : z<1) |
| (x y : Placement H k 0) (hp : placementKernel H z x y=0) : |
| hilbertBlock (groupAverage (coindHilbertRep (blockPerm e) σ |
| (fun x => (placementSection e x)⁻¹) (placement_cosets_bijective e)) |
| (fun a => (conditionalGroupLaw H z a:ℂ))) y x = 0 := by |
| rw [placementBlock_expectation] |
| apply Finset.sum_eq_zero |
| intro g _ |
| have hn : x.trans (remainingPerm H g.val g.property).toEmbedding ≠ y := by |
| intro he |
| have ht := placementKernel_output_pos H hz hz1 x g |
| change 0<placementKernel H z x (x.trans (remainingPerm H g.val g.property).toEmbedding) at ht |
| rw [he,hp] at ht |
| exact (lt_irrefl 0 ht) |
| rw [ite_eq_right hn,smul_zero] |
|
|
| lemma placementBlock_moment {z : ℝ} (hz : 0≤z) (hz1 : z<1) |
| (hσ : ∀ g v, ‖σ g v‖=‖v‖) (q : ℕ) |
| (x : Placement H k 0) (g : ConditionalChoices H) : |
| evenMoment q (hilbertBlock (groupAverage (coindHilbertRep (blockPerm e) σ |
| (fun x => (placementSection e x)⁻¹) (placement_cosets_bijective e)) |
| (fun a => (conditionalGroupLaw H z a:ℂ))) (augmentedOutputIndex H x g) x) = |
| (placementKernel H z x (augmentedOutputIndex H x g))^(2*q) * |
| evenMoment q (groupAverage (augmentedRepresentation H x g e σ) |
| (fun a => (conditionalGroupLaw (augmentByChoice H x g) z a:ℂ))) := by |
| rw [augmented_block_average H x g e σ z (conditionalNormalizer_pos _ hz hz1).ne', |
| evenMoment_smul,evenMoment_unitary_left] |
| · rw [placementKernel_ratio] |
| · rw [unitary_rep_adjoint σ hσ,←map_mul,inv_mul_cancel,map_one] |
|
|
| lemma placementBlock_moment_root {z : ℝ} (hz : 0≤z) (hz1 : z<1) |
| (hσ : ∀ g v, ‖σ g v‖=‖v‖) {q : ℕ} (hq : 0<q) |
| (x : Placement H k 0) (g : ConditionalChoices H) : |
| evenMoment q (hilbertBlock (groupAverage (coindHilbertRep (blockPerm e) σ |
| (fun x => (placementSection e x)⁻¹) (placement_cosets_bijective e)) |
| (fun a => (conditionalGroupLaw H z a:ℂ))) (augmentedOutputIndex H x g) x)^(((2*q:ℕ):ℝ)⁻¹) = |
| placementKernel H z x (augmentedOutputIndex H x g) * |
| evenMoment q (groupAverage (augmentedRepresentation H x g e σ) |
| (fun a => (conditionalGroupLaw (augmentByChoice H x g) z a:ℂ)))^(((2*q:ℕ):ℝ)⁻¹) := by |
| rw [placementBlock_moment H e σ hz hz1 hσ] |
| have hp := (placementKernel_output_pos H hz hz1 x g).le |
| rw [Real.mul_rpow (pow_nonneg hp _) (evenMoment_nonneg _ _), |
| ← Real.rpow_natCast,← Real.rpow_mul hp] |
| have hn : ((2*q:ℕ):ℝ)≠0 := by positivity |
| rw [mul_inv_cancel₀ hn,Real.rpow_one] |
|
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| end BinaryCoordinateSweeps |
|
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| end |
|
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| end OAI |
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