| import Mathlib |
| import OAI.Probability.IsingPerceptron.JointGGClosure |
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| namespace OAI |
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| /-! Enriched G G Finite. -/ |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
|
|
| def enrichedMonomialResidual {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (P : Measure A) (r : ℝ≥0) (n : ℕ) (b h : ℕ → ℝ) |
| (u : Fin N → ℝ) (ν : Measure (Spin N)) (φ : A → Spin N → ℝ) |
| (p d : ℕ) {m : ℕ} (D : (Fin (m+1) → Spin N × LabeledLeaf n) → ℝ) : ℝ := |
| let κ := fun x y : Spin N × LabeledLeaf n => spinOverlap x.1 y.1 ^ p * treeOverlap n x.2 y.2 ^ d |
| (m+1 : ℕ)*enrichedReplicaAverage P r n b h u ν φ |
| (fun τ : Fin (m+1+1) → Spin N × LabeledLeaf n => D (Fin.tail τ)*κ ((Fin.tail τ) 0) (τ 0)) - |
| enrichedReplicaAverage P r n b h u ν φ D * enrichedReplicaAverage P r n b h u ν φ |
| (fun τ : Fin 2 → Spin N × LabeledLeaf n => κ (τ 1) (τ 0)) - |
| enrichedReplicaAverage P r n b h u ν φ |
| (fun σ => D σ*(∑ l : Fin m, κ (σ 0) (σ l.succ))) |
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| lemma enrichedMonomialResidual_bound {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ) |
| (hb : CascadeExponents n b) {h : ℕ → ℝ} (hh : Monotone h) (h0 : 0 ≤ h 0) |
| {H : ℝ} (hH : h n ≤ H) (u : Fin N → ℝ) (hu : ∀ j, u j ∈ Icc (1:ℝ) 2) (j : Fin N) |
| (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {A : Type*} [MeasurableSpace A] (P : Measure A) [IsProbabilityMeasure P] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) {K : ℝ} (hK : 0 ≤ K) |
| (hφ : ∀ y x, |φ y x| ≤ K) (r : ℝ≥0) {α : ℝ} (hr : (r:ℝ) ≤ α*N) |
| (hmin : ∀ v ∈ Icc (1:ℝ) 2, enrichedCoordinateObjective P r n b h u j ν φ (u j) ≤ |
| enrichedCoordinateObjective P r n b h u j ν φ v) |
| (he : perturbationScale N ≤ 1/8) (hs : contactStep N ≤ 1/4) |
| {m : ℕ} (D : (Fin (m+1) → Spin N × LabeledLeaf n) → ℝ) |
| {B : ℝ} (hB : 0 ≤ B) (hD : ∀ σ, |D σ| ≤ B) : |
| |enrichedMonomialResidual P r n b h u ν φ (monomialIndex j).1 (monomialIndex j).2 D| ≤ |
| contactGGRate N j |
| (4*(∫ T, (Real.log (rawTreeTotal n T).toReal)^2 ∂(rawCascadeLaw n b : Measure (RawTree n)))+ |
| H+4+5*K^2*α) B := by |
| have hua : ∀ i, |u i| ≤ 2 := fun i => abs_le.mpr ⟨by linarith [(hu i).1],(hu i).2⟩ |
| have hint := enriched_coordinate_minimum_interior hN n b hb hh h0 u hua j ν P hm hK hφ r hmin he hs |
| have hn : (0:ℝ) < N := by exact_mod_cast hN |
| have hstep : 0 < contactStep N := Real.rpow_pos_of_pos hn _ |
| have hepos : 0 < perturbationScale N := Real.rpow_pos_of_pos hn _ |
| have ha := perturbationAmplitude_pos hN j |
| have hg := enriched_gg_at_coordinate_minimum hN n b hb hh h0 hH u hua j ν P hm hK hφ r hr |
| hstep (one_div_pos.mpr hepos) hint.1 hint.2 hmin D hB hD |
| change |perturbationAmplitude N j*u j| * |
| |enrichedMonomialResidual P r n b h u ν φ (monomialIndex j).1 (monomialIndex j).2 D| ≤ _ at hg |
| have hau : perturbationAmplitude N j ≤ |perturbationAmplitude N j*u j| := by |
| rw [abs_of_pos (mul_pos ha (lt_of_lt_of_le zero_lt_one (hu j).1))] |
| nlinarith [(hu j).1] |
| apply (le_div_iff₀ ha).2 |
| calc |
| _ = perturbationAmplitude N j* |
| |enrichedMonomialResidual P r n b h u ν φ (monomialIndex j).1 (monomialIndex j).2 D| := by ring |
| _ ≤ |perturbationAmplitude N j*u j| * |
| |enrichedMonomialResidual P r n b h u ν φ (monomialIndex j).1 (monomialIndex j).2 D| := |
| mul_le_mul_of_nonneg_right hau (abs_nonneg _) |
| _ ≤ _ := hg |
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| |
| def enrichedPerturbationObjective {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (P : Measure A) (r : ℝ≥0) (n : ℕ) (b h : ℕ → ℝ) |
| (ν : Measure (Spin N)) (φ : A → Spin N → ℝ) (u : Fin N → ℝ) : ℝ := |
| -enrichedMeanPressure P r n b h u ν φ + |
| ∑ j : Fin N, perturbationWeight j*(u j-3/2)^2 |
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| lemma enriched_minimum_coordinate {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (P : Measure A) (r : ℝ≥0) (n : ℕ) (b h : ℕ → ℝ) |
| (ν : Measure (Spin N)) (φ : A → Spin N → ℝ) (u : Fin N → ℝ) |
| (hu : ∀ j, u j ∈ Icc (1:ℝ) 2) |
| (hmin : ∀ v : Fin N → ℝ, (∀ j, v j ∈ Icc (1:ℝ) 2) → |
| enrichedPerturbationObjective P r n b h ν φ u ≤ enrichedPerturbationObjective P r n b h ν φ v) |
| (j : Fin N) (v : ℝ) (hv : v ∈ Icc (1:ℝ) 2) : |
| enrichedCoordinateObjective P r n b h u j ν φ (u j) ≤ |
| enrichedCoordinateObjective P r n b h u j ν φ v := by |
| classical |
| have huv : ∀ i, Function.update u j v i ∈ Icc (1:ℝ) 2 := by |
| intro i |
| by_cases hi : i = j |
| · subst i; simpa only [Function.update_self] using hv |
| · simpa only [Function.update_of_ne hi] using hu i |
| have hc := hmin (Function.update u j v) huv |
| unfold enrichedPerturbationObjective at hc |
| rw [← Finset.add_sum_erase _ _ (Finset.mem_univ j), |
| ← Finset.add_sum_erase _ _ (Finset.mem_univ j)] at hc |
| have he : (∑ i ∈ Finset.univ.erase j, perturbationWeight i*(Function.update u j v i-3/2)^2) = |
| ∑ i ∈ Finset.univ.erase j, perturbationWeight i*(u i-3/2)^2 := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [Function.update_of_ne (Finset.mem_erase.mp hi).1] |
| rw [he,Function.update_self] at hc |
| unfold enrichedCoordinateObjective |
| rw [Function.update_eq_self] |
| linarith |
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| end IsingPerceptron |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal BoundedContinuousFunction |
| namespace IsingPerceptron |
|
|
| def newReplicaEmbedding (k : ℕ) (e : Equiv.Perm (Fin k)) : Fin (k+1) → ℕ := |
| Fin.cases k (fun j => (e j).val) |
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| lemma newReplicaEmbedding_injective (k : ℕ) (e : Equiv.Perm (Fin k)) : |
| Function.Injective (newReplicaEmbedding k e) := by |
| intro a |
| refine Fin.cases ?_ (fun a => ?_) a |
| · intro b |
| refine Fin.cases ?_ (fun b => ?_) b |
| · exact fun _ => rfl |
| · intro hab |
| have hlt := (e b).isLt |
| simp only [newReplicaEmbedding,Fin.cases_zero,Fin.cases_succ] at hab |
| omega |
| · intro b |
| refine Fin.cases ?_ (fun b => ?_) b |
| · intro hab |
| have hlt := (e a).isLt |
| simp only [newReplicaEmbedding,Fin.cases_zero,Fin.cases_succ] at hab |
| omega |
| · intro hab |
| apply congrArg Fin.succ |
| apply e.injective |
| apply Fin.val_injective |
| exact hab |
|
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| lemma sum_nonzero_permutation {m : ℕ} (e : Equiv.Perm (Fin (m+1))) (f : Fin (m+1) → ℝ) : |
| (∑ l : Fin m, f (e l.succ)) = ∑ j ∈ Finset.univ.erase (e 0), f j := by |
| classical |
| have he := Equiv.sum_comp e f |
| rw [Fin.sum_univ_succ,← Finset.add_sum_erase _ _ (Finset.mem_univ (e 0))] at he |
| exact add_left_cancel he |
|
|
| lemma jointEntry_monomial_le (x : JointEntry) (p d : ℕ) : |x.1.1^p*x.2.1^d| ≤ 1 := by |
| rw [abs_mul,abs_pow,abs_pow] |
| have h1 : |x.1.1| ≤ 1 := abs_le.mpr x.1.2 |
| have h2 : |x.2.1| ≤ 1 := abs_le.mpr ⟨by linarith [x.2.2.1],x.2.2.2⟩ |
| exact (mul_le_mul (pow_le_one₀ (abs_nonneg _) h1) |
| (pow_le_one₀ (abs_nonneg _) h2) (pow_nonneg (abs_nonneg _) _) zero_le_one).trans_eq |
| (one_mul 1) |
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| def monomialPair (n p d : ℕ) {N : ℕ} (x y : Spin N × LabeledLeaf n) : ℝ := |
| spinOverlap x.1 y.1^p*treeOverlap n x.2 y.2^d |
|
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| lemma abs_monomialPair_le_one (n p d : ℕ) {N : ℕ} (x y : Spin N × LabeledLeaf n) : |
| |monomialPair n p d x y| ≤ 1 := jointEntry_monomial_le (enrichedJointEntry n x y) p d |
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| def finiteJointBlock {N m : ℕ} (n : ℕ) (σ : Fin m → Spin N × LabeledLeaf n) : JointBlock m := |
| fun i j => enrichedJointEntry n (σ i) (σ j) |
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| def permutedBlockTest {N m : ℕ} (n : ℕ) (e : Equiv.Perm (Fin m)) |
| (D : JointBlock m → ℝ) (σ : Fin m → Spin N × LabeledLeaf n) : ℝ := |
| D (finiteJointBlock n (σ ∘ e.symm)) |
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| lemma enrichedArrayLawTest_integral {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ) |
| {h : ℕ → ℝ} (hh : Monotone h) (h0 : 0 ≤ h 0) |
| (u : Fin N → ℝ) (hu : ∀ j, |u j| ≤ 2) |
| (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {A : Type*} [MeasurableSpace A] (P : Measure A) [IsProbabilityMeasure P] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) {K : ℝ} (hK : 0 ≤ K) |
| (hφ : ∀ y x, |φ y x| ≤ K) (r : ℝ≥0) |
| (F : JointArray → ℝ) (hF : Measurable F) |
| {m : ℕ} (D : (Fin m → Spin N × LabeledLeaf n) → ℝ) {C : ℝ} (hD : ∀ σ, |D σ| ≤ C) |
| (ι : Fin m → ℕ) (hi : Function.Injective ι) |
| (he : ∀ σ, F (sampledOverlapArray (enrichedJointEntry n) σ) = D (fun i => σ (ι i))) : |
| (∫ x, F x ∂(enrichedArrayLaw P r n b h u ν hm : Measure JointArray)) = |
| enrichedReplicaAverage P r n b h u ν φ D := by |
| rw [enrichedArrayLaw_integral _ _ _ _ _ _ _ _ _ hF] |
| simp_rw [he] |
| exact enrichedReplicaLaw_integral hN n b hh h0 u hu ν P hm hK hφ r D hD ι hi |
|
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| lemma permutedBlockTest_embedding {N m : ℕ} (n : ℕ) (e : Equiv.Perm (Fin m)) |
| (D : JointBlock m → ℝ) (σ : ℕ → Spin N × LabeledLeaf n) : |
| permutedBlockTest n e D (fun j => σ (e j)) = |
| D (jointBlockView m (sampledOverlapArray (enrichedJointEntry n) σ)) := by |
| unfold permutedBlockTest |
| apply congrArg D |
| funext i j |
| simp [finiteJointBlock,jointBlockView,sampledOverlapArray] |
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| lemma enrichedArrayLaw_residual {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ) |
| {h : ℕ → ℝ} (hh : Monotone h) (h0 : 0 ≤ h 0) |
| (u : Fin N → ℝ) (hu : ∀ j, |u j| ≤ 2) |
| (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {A : Type*} [MeasurableSpace A] (P : Measure A) [IsProbabilityMeasure P] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) {K : ℝ} (hK : 0 ≤ K) |
| (hφ : ∀ y x, |φ y x| ≤ K) (r : ℝ≥0) |
| {m : ℕ} (e : Equiv.Perm (Fin (m+1))) (D : JointBlock (m+1) → ℝ) |
| (hDc : Continuous D) {B : ℝ} (hB : 0 ≤ B) (hD : ∀ x, |D x| ≤ B) (p d : ℕ) : |
| jointMonomialResidual (enrichedArrayLaw P r n b h u ν hm) (m+1) (e 0) D p d = |
| enrichedMonomialResidual P r n b h u ν φ p d (permutedBlockTest n e D) := by |
| classical |
| let μ : Measure JointArray := enrichedArrayLaw P r n b h u ν hm |
| let D' := permutedBlockTest (N := N) n e D |
| have hD' : ∀ σ, |D' σ| ≤ B := fun σ => hD _ |
| have hcD : Continuous (fun x => D (jointBlockView (m+1) x)) := |
| hDc.comp (continuous_jointBlockView _) |
| have hmono (x y : Spin N × LabeledLeaf n) := abs_monomialPair_le_one n p d x y |
| have hnew : (∫ x, D (jointBlockView (m+1) x)*(x (e 0,m+1)).1.1^p*(x (e 0,m+1)).2.1^d ∂μ) = |
| enrichedReplicaAverage P r n b h u ν φ |
| (fun τ : Fin (m+1+1) → Spin N × LabeledLeaf n => |
| D' (Fin.tail τ)*monomialPair n p d ((Fin.tail τ) 0) (τ 0)) := by |
| refine enrichedArrayLawTest_integral hN n b hh h0 u hu ν P hm hK hφ r _ |
| (hcD.mul (by fun_prop) |>.mul (by fun_prop) |>.measurable) _ (C := B) |
| (fun τ => ?_) (newReplicaEmbedding (m+1) e) (newReplicaEmbedding_injective _ e) ?_ |
| · rw [abs_mul] |
| exact (mul_le_mul (hD' _) (hmono _ _) (abs_nonneg _) hB).trans_eq (mul_one B) |
| · intro σ |
| change D (jointBlockView (m+1) (sampledOverlapArray (enrichedJointEntry n) σ)) * |
| (enrichedJointEntry n (σ (e 0)) (σ (m+1))).1.1^p * |
| (enrichedJointEntry n (σ (e 0)) (σ (m+1))).2.1^d = |
| D' (fun i => σ ((e i).val)) * monomialPair n p d (σ (e 0)) (σ (m+1)) |
| rw [show D' (fun i => σ ((e i).val)) = |
| D (jointBlockView (m+1) (sampledOverlapArray (enrichedJointEntry n) σ)) from |
| permutedBlockTest_embedding n e D σ] |
| simp only [enrichedJointEntry,monomialPair,mul_assoc] |
| have hblock : (∫ x, D (jointBlockView (m+1) x) ∂μ) = |
| enrichedReplicaAverage P r n b h u ν φ D' := by |
| apply enrichedArrayLawTest_integral hN n b hh h0 u hu ν P hm hK hφ r _ hcD.measurable |
| D' hD' (fun j => (e j).val) (Fin.val_injective.comp e.injective) |
| intro σ |
| exact (permutedBlockTest_embedding n e D σ).symm |
| have hpair : (∫ x, (x (0,1)).1.1^p*(x (0,1)).2.1^d ∂μ) = |
| enrichedReplicaAverage P r n b h u ν φ |
| (fun τ : Fin 2 → Spin N × LabeledLeaf n => monomialPair n p d (τ 1) (τ 0)) := by |
| refine enrichedArrayLawTest_integral hN n b hh h0 u hu ν P hm hK hφ r _ (by fun_prop) |
| _ (C := 1) (fun τ => hmono _ _) ![1,0] ?_ ?_ |
| · intro a b hab |
| fin_cases a <;> fin_cases b <;> simp_all |
| · intro σ |
| rfl |
| have hsumBound (σ : Fin (m+1) → Spin N × LabeledLeaf n) : |
| |D' σ*(∑ l : Fin m, monomialPair n p d (σ 0) (σ l.succ))| ≤ B*m := by |
| rw [abs_mul] |
| apply mul_le_mul (hD' _) _ (abs_nonneg _) hB |
| calc |
| _ ≤ ∑ l : Fin m, |monomialPair n p d (σ 0) (σ l.succ)| := Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ _ : Fin m, (1:ℝ) := Finset.sum_le_sum (fun l _ => hmono _ _) |
| _ = _ := by simp |
| have hsum : (∑ j ∈ Finset.univ.erase (e 0), |
| ∫ x, D (jointBlockView (m+1) x)*(x (e 0,j)).1.1^p*(x (e 0,j)).2.1^d ∂μ) = |
| enrichedReplicaAverage P r n b h u ν φ |
| (fun σ => D' σ*(∑ l : Fin m, monomialPair n p d (σ 0) (σ l.succ))) := by |
| have hI (j : Fin (m+1)) : Integrable |
| (fun x : JointArray => D (jointBlockView (m+1) x)*(x (e 0,j)).1.1^p*(x (e 0,j)).2.1^d) μ := |
| compact_integrable (hcD.mul (by fun_prop) |>.mul (by fun_prop)) |
| rw [← integral_finsetSum _ (fun j _ => hI j)] |
| apply enrichedArrayLawTest_integral hN n b hh h0 u hu ν P hm hK hφ r _ |
| (by apply Finset.measurable_sum; intro j hj; exact (hcD.mul (by fun_prop) |>.mul (by fun_prop)).measurable) |
| _ hsumBound (fun j => (e j).val) (Fin.val_injective.comp e.injective) |
| intro σ |
| rw [show D' (fun j => σ ((e j).val)) = |
| D (jointBlockView (m+1) (sampledOverlapArray (enrichedJointEntry n) σ)) from |
| permutedBlockTest_embedding n e D σ] |
| rw [sum_nonzero_permutation e (fun j => monomialPair n p d (σ (e 0)) (σ j)),Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j hj |
| simp only [sampledOverlapArray,enrichedJointEntry,monomialPair,mul_assoc] |
| unfold jointMonomialResidual |
| rw [hnew,hblock,hpair,hsum] |
| rfl |
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| end IsingPerceptron |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal BoundedContinuousFunction |
| namespace IsingPerceptron |
|
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| lemma jointMonomialResidual_zero (μ : Measure JointArray) [IsProbabilityMeasure μ] |
| (m : ℕ) (i : Fin (m+1)) (D : JointBlock (m+1) → ℝ) : |
| jointMonomialResidual μ (m+1) i D 0 0 = 0 := by |
| classical |
| simp only [jointMonomialResidual,pow_zero,mul_one,integral_const,probReal_univ, |
| smul_eq_mul,Finset.sum_const,Finset.card_erase_of_mem (Finset.mem_univ i), |
| Finset.card_univ,Fintype.card_fin,Nat.add_sub_cancel,nsmul_eq_mul,Nat.cast_add,Nat.cast_one] |
| ring |
|
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| lemma enrichedArrayLaw_residual_bound {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ) |
| (hb : CascadeExponents n b) {h : ℕ → ℝ} (hh : Monotone h) (h0 : 0 ≤ h 0) |
| {H : ℝ} (hH : h n ≤ H) (u : Fin N → ℝ) (hu : ∀ j, u j ∈ Icc (1:ℝ) 2) (j : Fin N) |
| (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {A : Type*} [MeasurableSpace A] (P : Measure A) [IsProbabilityMeasure P] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) {K : ℝ} (hK : 0 ≤ K) |
| (hφ : ∀ y x, |φ y x| ≤ K) (r : ℝ≥0) {α : ℝ} (hr : (r:ℝ) ≤ α*N) |
| (hmin : ∀ v : Fin N → ℝ, (∀ j, v j ∈ Icc (1:ℝ) 2) → |
| enrichedPerturbationObjective P r n b h ν φ u ≤ enrichedPerturbationObjective P r n b h ν φ v) |
| (he : perturbationScale N ≤ 1/8) (hs : contactStep N ≤ 1/4) |
| {m : ℕ} (i : Fin (m+1)) (D : JointBlock (m+1) → ℝ) (hDc : Continuous D) |
| {B : ℝ} (hB : 0 ≤ B) (hD : ∀ x, |D x| ≤ B) : |
| |jointMonomialResidual (enrichedArrayLaw P r n b h u ν hm) (m+1) i D |
| (monomialIndex j).1 (monomialIndex j).2| ≤ |
| contactGGRate N j |
| (4*(∫ T, (Real.log (rawTreeTotal n T).toReal)^2 ∂(rawCascadeLaw n b : Measure (RawTree n)))+ |
| H+4+5*K^2*α) B := by |
| classical |
| have hua : ∀ l, |u l| ≤ 2 := fun l => abs_le.mpr ⟨by linarith [(hu l).1],(hu l).2⟩ |
| have hg := enrichedMonomialResidual_bound hN n b hb hh h0 hH u hu j ν P hm hK hφ r hr |
| (enriched_minimum_coordinate P r n b h ν φ u hu hmin j) he hs |
| (permutedBlockTest n (Equiv.swap 0 i) D) hB (fun σ => hD _) |
| have ht := enrichedArrayLaw_residual hN n b hh h0 u hua ν P hm hK hφ r |
| (Equiv.swap 0 i) D hDc hB hD (monomialIndex j).1 (monomialIndex j).2 |
| rw [Equiv.swap_apply_left] at ht |
| rw [ht] |
| exact hg |
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| theorem enriched_minimizers_jointGG |
| (N : ℕ → ℕ) (hN : ∀ k, 0 < N k) (hNlim : Tendsto N atTop atTop) |
| (n : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (h : ℕ → ℕ → ℝ) (hh : ∀ k, Monotone (h k)) (h0 : ∀ k, 0 ≤ h k 0) |
| (H : ℝ) (hH : ∀ k, h k n ≤ H) |
| (u : (k : ℕ) → Fin (N k) → ℝ) (hu : ∀ k j, u k j ∈ Icc (1:ℝ) 2) |
| (ν : (k : ℕ) → Measure (Spin (N k))) [∀ k, IsProbabilityMeasure (ν k)] |
| (A : ℕ → Type*) [∀ k, MeasurableSpace (A k)] |
| (P : (k : ℕ) → Measure (A k)) [∀ k, IsProbabilityMeasure (P k)] |
| (φ : (k : ℕ) → A k → Spin (N k) → ℝ) (hm : ∀ k, Measurable (φ k)) |
| (K : ℝ) (hK : 0 ≤ K) (hφ : ∀ k y x, |φ k y x| ≤ K) |
| (r : ℕ → ℝ≥0) (α : ℝ) (hr : ∀ k, (r k:ℝ) ≤ α*N k) |
| (hmin : ∀ k (v : Fin (N k) → ℝ), (∀ j, v j ∈ Icc (1:ℝ) 2) → |
| enrichedPerturbationObjective (P k) (r k) n b (h k) (ν k) (φ k) (u k) ≤ |
| enrichedPerturbationObjective (P k) (r k) n b (h k) (ν k) (φ k) v) |
| (μ : ProbabilityMeasure JointArray) |
| (hL : Tendsto (fun k => enrichedArrayLaw (P k) (r k) n b (h k) (u k) (ν k) (hm k)) atTop (𝓝 μ)) : |
| HasEntryGhirlandaGuerra (fun x i j => x (i,j)) (μ : Measure JointArray) := by |
| apply jointGG_of_weak_monomial_residuals hL |
| intro m hm2 i D hDc p d |
| cases m with |
| | zero => omega |
| | succ m => |
| by_cases hpd : p+d = 0 |
| · have hp : p = 0 := by omega |
| have hd : d = 0 := by omega |
| subst p d |
| simpa only [jointMonomialResidual_zero] using |
| (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0)) |
| · obtain ⟨j,hj⟩ := monomialIndex_surjective p d (by omega) |
| let BD := BoundedContinuousFunction.mkOfCompact ⟨D,hDc⟩ |
| have hD : ∀ x, |D x| ≤ ‖BD‖ := fun x => by |
| simpa only [BD,BoundedContinuousFunction.mkOfCompact_apply,ContinuousMap.coe_mk,Real.norm_eq_abs] using BD.norm_coe_le_norm x |
| let L := 4*(∫ T, (Real.log (rawTreeTotal n T).toReal)^2 |
| ∂(rawCascadeLaw n b : Measure (RawTree n)))+H+4+5*K^2*α |
| have he : ∀ᶠ k in atTop, perturbationScale (N k) ≤ 1/8 := |
| (perturbationScale_tendsto.comp hNlim).eventually (eventually_le_nhds (by norm_num)) |
| have hs : ∀ᶠ k in atTop, contactStep (N k) ≤ 1/4 := |
| (contactStep_tendsto.comp hNlim).eventually (eventually_le_nhds (by norm_num)) |
| have hjN : ∀ᶠ k in atTop, j < N k := hNlim.eventually (eventually_gt_atTop j) |
| apply squeeze_zero_norm' _ ((contactGGRate_tendsto j L ‖BD‖).comp hNlim) |
| filter_upwards [he,hs,hjN] with k he hs hjN |
| let j' : Fin (N k) := ⟨j,hjN⟩ |
| have ht := enrichedArrayLaw_residual_bound (hN k) n b hb (hh k) (h0 k) (hH k) |
| (u k) (hu k) j' (ν k) (P k) (hm k) hK (hφ k) (r k) (hr k) (hmin k) |
| he hs i D hDc (norm_nonneg BD) hD |
| simpa only [j',hj,Prod.fst,Prod.snd,Real.norm_eq_abs,Function.comp_def,L] using ht |
|
|
| end IsingPerceptron |
|
|
| end |
|
|
| end OAI |
|
|
