| import Mathlib |
| import OAI.Probability.IsingPerceptron.TiltedLabelLaw |
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| namespace OAI |
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| /-! Joint G G Closure. -/ |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal BoundedContinuousFunction |
| namespace IsingPerceptron |
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| |
| def jointMonomialResidual (μ : Measure JointArray) (n : ℕ) (i : Fin n) |
| (D : JointBlock n → ℝ) (p d : ℕ) : ℝ := |
| (n : ℝ)*(∫ x, D (jointBlockView n x)*(x (i,n)).1.1^p*(x (i,n)).2.1^d ∂μ) - |
| (∫ x, D (jointBlockView n x) ∂μ)*(∫ x, (x (0,1)).1.1^p*(x (0,1)).2.1^d ∂μ) - |
| ∑ j ∈ Finset.univ.erase i, |
| ∫ x, D (jointBlockView n x)*(x (i,j)).1.1^p*(x (i,j)).2.1^d ∂μ |
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| lemma jointMonomialResidual_weak_limit {L : ℕ → ProbabilityMeasure JointArray} |
| {μ : ProbabilityMeasure JointArray} (hL : Tendsto L atTop (𝓝 μ)) |
| (n : ℕ) (i : Fin n) (D : JointBlock n → ℝ) (hD : Continuous D) (p d : ℕ) : |
| Tendsto (fun N => jointMonomialResidual (L N) n i D p d) atTop |
| (𝓝 (jointMonomialResidual μ n i D p d)) := by |
| have hI : ∀ F : JointArray → ℝ, Continuous F → |
| Tendsto (fun N => ∫ x, F x ∂(L N : Measure JointArray)) atTop |
| (𝓝 (∫ x, F x ∂(μ : Measure JointArray))) := by |
| intro F hF |
| exact (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hL) |
| (BoundedContinuousFunction.mkOfCompact ⟨F,hF⟩) |
| have hcD : Continuous (fun x => D (jointBlockView n x)) := hD.comp (continuous_jointBlockView n) |
| have ht (j : ℕ) := hI (fun x => D (jointBlockView n x)*(x (i,j)).1.1^p*(x (i,j)).2.1^d) |
| (hcD.mul (by fun_prop) |>.mul (by fun_prop)) |
| exact (((ht n).const_mul (n : ℝ)).sub ((hI _ hcD).mul (hI _ (by fun_prop)))).sub |
| (tendsto_finsetSum _ (fun j _ => ht j)) |
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| theorem jointGG_of_weak_monomial_residuals {L : ℕ → ProbabilityMeasure JointArray} |
| {μ : ProbabilityMeasure JointArray} (hL : Tendsto L atTop (𝓝 μ)) |
| (hg : ∀ n, 2 ≤ n → ∀ i : Fin n, ∀ D : JointBlock n → ℝ, Continuous D → ∀ p d : ℕ, |
| Tendsto (fun N => jointMonomialResidual (L N) n i D p d) atTop (𝓝 0)) : |
| HasEntryGhirlandaGuerra (fun x i j => x (i,j)) (μ : Measure JointArray) := by |
| apply joint_monomialGG_measurable |
| intro n hn i D hD p d |
| have he := tendsto_nhds_unique (jointMonomialResidual_weak_limit hL n i D hD p d) |
| (hg n hn i D hD p d) |
| dsimp only [jointMonomialResidual] at he |
| 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 |
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|
| lemma replica_injective_map {X : Type*} [MeasurableSpace X] (ν : Measure X) |
| [IsProbabilityMeasure ν] {r : ℕ} (ι : Fin r → ℕ) (hi : Function.Injective ι) : |
| (Measure.infinitePi (fun _ : ℕ => ν)).map (fun σ => fun i : Fin r => σ (ι i)) = |
| Measure.pi (fun _ : Fin r => ν) := by |
| have h0 : iIndepFun (fun i : ℕ => fun σ : ℕ → X => σ i) |
| (Measure.infinitePi (fun _ : ℕ => ν)) := iIndepFun_infinitePi (fun _ => measurable_id) |
| have h := h0.precomp hi |
| have he := h.map_fun_eq_pi_map (fun _ => (by fun_prop : Measurable _).aemeasurable) |
| simpa only [Measure.infinitePi_map_eval] using he |
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| lemma replicaLaw_integral_injective {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] |
| [Countable X] [MeasurableSingletonClass X] (P : Measure Ω) [IsProbabilityMeasure P] |
| (ν : Ω → Measure X) (hν : Measurable ν) [∀ ω, IsProbabilityMeasure (ν ω)] |
| {r : ℕ} (D : (Fin r → X) → ℝ) {C : ℝ} (hD : ∀ σ, |D σ| ≤ C) |
| (ι : Fin r → ℕ) (hi : Function.Injective ι) : |
| (∫ σ, D (fun i : Fin r => σ (ι i)) ∂replicaLaw P ν hν) = |
| ∫ ω, ∫ σ, D σ ∂Measure.pi (fun _ : Fin r => ν ω) ∂P := by |
| rw [replicaLaw_integral_bounded P ν hν (F := fun σ => D (fun i : Fin r => σ (ι i))) |
| ((measurable_of_countable D).comp (by fun_prop)) (fun _ => hD _)] |
| apply integral_congr_ae |
| exact ae_of_all _ (fun ω => ((HasLaw.mk |
| ((Measurable.of_eval (fun coordinate : Fin r => measurable_pi_apply (ι coordinate))).aemeasurable) |
| (replica_injective_map (ν ω) ι hi)).integral_comp (measurable_of_countable D).aestronglyMeasurable)) |
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| def enrichedGibbs {N : ℕ} {A : Type*} (n : ℕ) (h : ℕ → ℝ) (u : Fin N → ℝ) |
| (ν : Measure (Spin N)) (φ : A → Spin N → ℝ) (p : EnrichedCylinderData n A) : |
| Measure (Spin N × LabeledLeaf n) := |
| gibbsProbability (labeledSpinReference n ν p.1.2) (enrichedHamiltonian n h u φ p) |
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| instance enrichedGibbs_probability {N : ℕ} {A : Type*} (n : ℕ) (h : ℕ → ℝ) (u : Fin N → ℝ) |
| (ν : Measure (Spin N)) [IsProbabilityMeasure ν] (φ : A → Spin N → ℝ) |
| (p : EnrichedCylinderData n A) : IsProbabilityMeasure (enrichedGibbs n h u ν φ p) := |
| inferInstanceAs (IsProbabilityMeasure (gibbsProbability _ _)) |
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| lemma measurable_enrichedGibbs {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (n : ℕ) (h : ℕ → ℝ) (u : Fin N → ℝ) (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) : |
| Measurable (enrichedGibbs n h u ν φ) := |
| measurable_gibbsProbability |
| (ν := fun p : EnrichedCylinderData n A => labeledSpinReference n ν p.1.2) |
| ((measurable_labeledSpinReference n ν).comp measurable_fst.snd) |
| (measurable_enrichedHamiltonian n h u hm) |
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| lemma spinOverlap_mem {N : ℕ} (x y : Spin N) : spinOverlap x y ∈ Icc (-1:ℝ) 1 := by |
| by_cases hn : N = 0 |
| · subst N; simp [spinOverlap] |
| · exact abs_le.mp (abs_spinOverlap_le (Nat.pos_of_ne_zero hn) x y) |
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| lemma treeOverlap_mem (n : ℕ) (x y : LabeledLeaf n) : treeOverlap n x y ∈ Icc (0:ℝ) 1 := by |
| constructor |
| · unfold treeOverlap; positivity |
| · unfold treeOverlap |
| apply (div_le_one (by positivity : (0:ℝ) < (n+1:ℕ))).mpr |
| exact_mod_cast (labeledCommonDepth_le n x y).trans (Nat.le_succ n) |
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| def enrichedJointEntry {N : ℕ} (n : ℕ) (x y : Spin N × LabeledLeaf n) : JointEntry := |
| (⟨spinOverlap x.1 y.1,spinOverlap_mem x.1 y.1⟩,⟨treeOverlap n x.2 y.2,treeOverlap_mem n x.2 y.2⟩) |
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| def enrichedReplicaLaw {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (P : Measure A) [IsProbabilityMeasure P] (r : ℝ≥0) (n : ℕ) (b h : ℕ → ℝ) |
| (u : Fin N → ℝ) (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) : Measure (ℕ → Spin N × LabeledLeaf n) := |
| replicaLaw (enrichedCylinderLaw P r n b) (enrichedGibbs n h u ν φ) (measurable_enrichedGibbs n h u ν hm) |
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| instance enrichedReplicaLaw_probability {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (P : Measure A) [IsProbabilityMeasure P] (r : ℝ≥0) (n : ℕ) (b h : ℕ → ℝ) |
| (u : Fin N → ℝ) (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) : IsProbabilityMeasure (enrichedReplicaLaw P r n b h u ν hm) := |
| inferInstanceAs (IsProbabilityMeasure (replicaLaw _ _ _)) |
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| def enrichedArrayLaw {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (P : Measure A) [IsProbabilityMeasure P] (r : ℝ≥0) (n : ℕ) (b h : ℕ → ℝ) |
| (u : Fin N → ℝ) (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) : ProbabilityMeasure JointArray := |
| overlapArrayLaw (enrichedCylinderLaw P r n b) (enrichedGibbs n h u ν φ) |
| (measurable_enrichedGibbs n h u ν hm) (enrichedJointEntry n) |
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| lemma enrichedReplicaLaw_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) |
| {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) : |
| (∫ σ, D (fun i => σ (ι i)) ∂enrichedReplicaLaw P r n b h u ν hm) = |
| enrichedReplicaAverage P r n b h u ν φ D := by |
| rw [enrichedReplicaLaw,replicaLaw_integral_injective _ _ _ D hD ι hi] |
| unfold enrichedReplicaAverage |
| apply integral_congr_ae |
| filter_upwards [enrichedCylinder_exp_ae hN n b hh h0 u hu ν P hK hφ r] with z hz |
| change Integrable (fun s => Real.exp (enrichedHamiltonian n h u φ z s)) |
| (labeledSpinReference n ν z.1.2) at hz |
| rw [referenceReplicaMean_eq_tilted _ _ hz] |
| simp only [enrichedGibbs,gibbsProbability_eq_tilted _ _ hz] |
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| lemma enrichedArrayLaw_integral {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (P : Measure A) [IsProbabilityMeasure P] (r : ℝ≥0) (n : ℕ) (b h : ℕ → ℝ) |
| (u : Fin N → ℝ) (ν : Measure (Spin N)) [IsProbabilityMeasure ν] |
| {φ : A → Spin N → ℝ} (hm : Measurable φ) |
| (F : JointArray → ℝ) (hF : Measurable F) : |
| (∫ x, F x ∂(enrichedArrayLaw P r n b h u ν hm : Measure JointArray)) = |
| ∫ σ, F (sampledOverlapArray (enrichedJointEntry n) σ) ∂enrichedReplicaLaw P r n b h u ν hm := by |
| exact integral_map (measurable_sampledOverlapArray (enrichedJointEntry n)).aemeasurable hF.aestronglyMeasurable |
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| end IsingPerceptron |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
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| |
| def enrichedCoordinateObjective {N : ℕ} {A : Type*} [MeasurableSpace A] |
| (P : Measure A) (r : ℝ≥0) (n : ℕ) (b h : ℕ → ℝ) |
| (u : Fin N → ℝ) (j : Fin N) (ν : Measure (Spin N)) (φ : A → Spin N → ℝ) |
| (v : ℝ) : ℝ := |
| -enrichedMeanPressure P r n b h (Function.update u j v) ν φ + |
| perturbationWeight j*(v-3/2)^2 |
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| lemma enriched_contact_cgf {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ) |
| (hb : CascadeExponents n b) {h : ℕ → ℝ} (hh : Monotone h) (h0 : 0 ≤ h 0) |
| (u : Fin N → ℝ) (hu : ∀ j, |u j| ≤ 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) |
| {s : ℝ} (hlo : 1 ≤ u j-s) (hhi : u j+s ≤ 2) (hs : 0 < s) |
| (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) : |
| let a := perturbationAmplitude N j |
| let Q := (enrichedFrozenLaw P r n b j).prod gaussianCoordinates |
| let L := fun t => ∫ z, cgf (fun x => cylinderField |
| (monomialCoefficients n (monomialIndex j).1 (monomialIndex j).2 x) z.2) |
| (enrichedFrozenReference n h u j ν φ z.1) t ∂Q |
| ∀ t ∈ Icc (a*u j-a*s) (a*u j+a*s), |
| -L (a*u j)+(N*perturbationWeight j/a^2)*(a*u j-a*(3/2))^2 ≤ |
| -L t+(N*perturbationWeight j/a^2)*(t-a*(3/2))^2 := by |
| let a := perturbationAmplitude N j |
| have ha : 0 < a := perturbationAmplitude_pos hN j |
| have hn : (0:ℝ) ≤ N := Nat.cast_nonneg N |
| dsimp only |
| intro t ht |
| have ht' : t/a ∈ Icc (1:ℝ) 2 := by |
| constructor |
| · apply (le_div_iff₀ ha).2 |
| nlinarith [ht.1] |
| · apply (div_le_iff₀ ha).2 |
| nlinarith [ht.2] |
| have htabs : |t/a| ≤ 2 := abs_le.mpr ⟨by linarith [ht'.1],ht'.2⟩ |
| have htmean := (enrichedCGF_integral hN n b hb hh h0 u hu j htabs ν P hm hK hφ r).2 |
| have hvmean := (enrichedCGF_integral hN n b hb hh h0 u hu j (hu j) ν P hm hK hφ r).2 |
| change (∫ z, cgf (fun x => cylinderField |
| (monomialCoefficients n (monomialIndex j).1 (monomialIndex j).2 x) z.2) |
| (enrichedFrozenReference n h u j ν φ z.1) (a*(t/a)) |
| ∂(enrichedFrozenLaw P r n b j).prod gaussianCoordinates) = _ at htmean |
| rw [mul_div_cancel₀ _ ha.ne'] at htmean |
| rw [show perturbationAmplitude N j = a from rfl] |
| change (∫ z, cgf (fun x => cylinderField |
| (monomialCoefficients n (monomialIndex j).1 (monomialIndex j).2 x) z.2) |
| (enrichedFrozenReference n h u j ν φ z.1) (a*u j) |
| ∂(enrichedFrozenLaw P r n b j).prod gaussianCoordinates) = _ at hvmean |
| rw [htmean,hvmean] |
| have hc := mul_le_mul_of_nonneg_left (hmin (t/a) ht') hn |
| unfold enrichedCoordinateObjective at hc |
| have he1 : (N*perturbationWeight j/a^2)*(a*u j-a*(3/2))^2 = |
| N*(perturbationWeight j*(u j-3/2)^2) := by field_simp [ha.ne'] |
| have he2 : (N*perturbationWeight j/a^2)*(t-a*(3/2))^2 = |
| N*(perturbationWeight j*(t/a-3/2)^2) := by field_simp [ha.ne'] |
| rw [he1,he2] |
| linarith |
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| |
| theorem enriched_gg_at_coordinate_minimum {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| ≤ 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) |
| {s θ : ℝ} (hs : 0 < s) (hθ : 0 < θ) (hlo : 1 ≤ u j-s) (hhi : u j+s ≤ 2) |
| (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) |
| {m : ℕ} (D : (Fin (m+1) → Spin N × LabeledLeaf n) → ℝ) |
| {B : ℝ} (hB : 0 ≤ B) (hD : ∀ σ, |D σ| ≤ B) : |
| let a := perturbationAmplitude N j |
| let c := N*perturbationWeight j/a^2 |
| let C := 4*∫ T, (Real.log (rawTreeTotal n T).toReal)^2 ∂(rawCascadeLaw n b : Measure (RawTree n)) |
| let δ := 2*N*Real.sqrt ((C+H+4+5*K^2*α)/N) |
| let κ := fun x y : Spin N × LabeledLeaf n => |
| spinOverlap x.1 y.1 ^ (monomialIndex j).1 * treeOverlap n x.2 y.2 ^ (monomialIndex j).2 |
| |a*u j| * |(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)))| ≤ |
| 2*B*((2*c+θ^2)/(2*θ)+c*(a*s)+4*δ/(a*s)) := by |
| let a := perturbationAmplitude N j |
| have ha : 0 < a := perturbationAmplitude_pos hN j |
| let δ := 2*N*Real.sqrt ((4*(∫ T, (Real.log (rawTreeTotal n T).toReal)^2 |
| ∂(rawCascadeLaw n b : Measure (RawTree n)))+H+4+5*K^2*α)/N) |
| have hν := measurable_enrichedFrozenReference n h u j ν hm |
| have hcontact := enriched_contact_cgf hN n b hb hh h0 u hu j ν P hm hK hφ r hlo hhi hs hmin |
| have hconc : ∀ t ∈ ({a*u j-a*s,a*u j,a*u j+a*s} : Set ℝ), |
| (∫ z : EnrichedFrozenData n j A × (ℕ → ℝ), |
| |cgf (fun x => cylinderField (monomialCoefficients n (monomialIndex j).1 (monomialIndex j).2 x) z.2) |
| (enrichedFrozenReference n h u j ν φ z.1) t - |
| (∫ z' : EnrichedFrozenData n j A × (ℕ → ℝ), |
| cgf (fun x => cylinderField (monomialCoefficients n (monomialIndex j).1 (monomialIndex j).2 x) z'.2) |
| (enrichedFrozenReference n h u j ν φ z'.1) t |
| ∂(enrichedFrozenLaw P r n b j).prod gaussianCoordinates)| |
| ∂(enrichedFrozenLaw P r n b j).prod gaussianCoordinates) ≤ δ := by |
| intro t ht |
| have hti : t/a ∈ Icc (1:ℝ) 2 := by |
| rcases ht with ht | ht | ht |
| · rw [ht] |
| constructor <;> (first | apply (le_div_iff₀ ha).2 | apply (div_le_iff₀ ha).2) <;> nlinarith |
| · rw [ht] |
| rw [mul_div_cancel_left₀ _ ha.ne'] |
| constructor <;> linarith |
| · rw [Set.mem_singleton_iff.mp ht] |
| constructor <;> (first | apply (le_div_iff₀ ha).2 | apply (div_le_iff₀ ha).2) <;> nlinarith |
| have hbnd := enrichedCGF_center_bound hN n b hb hh h0 hH u hu j |
| (v := t/a) (abs_le.mpr ⟨by linarith [hti.1],hti.2⟩) ν P hm hK hφ r hr |
| simpa only [enrichedCGF,show perturbationAmplitude N j = a from rfl,mul_div_cancel₀ _ ha.ne'] using hbnd |
| have hg := monomial_gg_at_minimum (P := enrichedFrozenLaw P r n b j) hN hν |
| (monomialIndex j).1 (monomialIndex j).2 (mul_pos ha hs) hθ hcontact hconc D hB hD |
| have hrep {l : ℕ} (E : (Fin l → Spin N × LabeledLeaf n) → ℝ) : |
| randomReplicaAverage (enrichedFrozenLaw P r n b j) (enrichedFrozenReference n h u j ν φ) |
| (monomialCoefficients n (monomialIndex j).1 (monomialIndex j).2) (a*u j) E = |
| enrichedReplicaAverage P r n b h u ν φ E := by |
| have he := enrichedReplicaAverage_eq_random hN n b hh h0 u hu j (hu j) ν P hm hK hφ r E |
| simpa only [Function.update_eq_self] using he.symm |
| dsimp only at hg ⊢ |
| rw [hrep,hrep,hrep,hrep] at hg |
| exact hg |
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| end IsingPerceptron |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
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| lemma gaussian_log_partition_mean_bounds {Ω X : Type*} |
| [MeasurableSpace Ω] [MeasurableSpace X] |
| {μ : Measure Ω} {ν : Measure X} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {H : Ω × X → ℝ} (hm : Measurable H) {v : X → NNReal} {K : ℝ} |
| (hg : ∀ x, μ.map (fun ω => H (ω,x)) = gaussianReal 0 (v x)) |
| (hv : ∀ x, (v x : ℝ) ≤ K) : |
| 0 ≤ (∫ ω, Real.log (∫ x, Real.exp (H (ω,x)) ∂ν) ∂μ) ∧ |
| (∫ ω, Real.log (∫ x, Real.exp (H (ω,x)) ∂ν) ∂μ) ≤ K/2 := by |
| let Z := fun ω => ∫ x, Real.exp (H (ω,x)) ∂ν |
| have hmZ : Measurable Z := hm.exp.stronglyMeasurable.integral_prod_right'.measurable |
| have hiH := gaussian_field_integrable (ν := ν) hm hg hv |
| have hexp := gaussian_field_exp_integrable (ν := ν) hm hg hv 1 |
| simp only [one_mul,one_pow] at hexp |
| have hZ : Integrable Z μ := hexp.1.integral_prod_left |
| have hpos : ∀ᵐ ω ∂μ, 0 < Z ω := by |
| filter_upwards [hexp.1.prod_right_ae] with ω hω |
| exact MeasureTheory.integral_exp_pos hω |
| have hiLog : Integrable (fun ω => Real.log (Z ω)) μ := |
| ((memLp_two_iff_integrable_sq hmZ.log.aestronglyMeasurable).2 |
| (gaussian_log_partition_second hm hg hv).1).integrable (by norm_num) |
| have hexplog : Integrable (fun ω => Real.exp (Real.log (Z ω))) μ := by |
| apply hZ.congr |
| filter_upwards [hpos] with ω hω |
| exact (Real.exp_log hω).symm |
| have hmean : (∫ ω, ∫ x, H (ω,x) ∂ν ∂μ) = 0 := by |
| rw [integral_integral_swap (f := fun ω x => H (ω,x)) hiH] |
| have hz : ∀ x, (∫ ω, H (ω,x) ∂μ) = 0 := by |
| intro x |
| have hmx : Measurable (fun ω => H (ω,x)) := hm.comp (measurable_id.prodMk measurable_const) |
| have he := integral_map hmx.aemeasurable (measurable_id.aestronglyMeasurable : |
| AEStronglyMeasurable (fun y : ℝ => y) (μ.map (fun ω => H (ω,x)))) |
| simp only [hg x,id_eq,integral_id_gaussianReal] at he |
| exact he.symm |
| simp_rw [hz] |
| simp |
| refine ⟨?_, ?_⟩ |
| · rw [← hmean] |
| apply integral_mono_ae hiH.integral_prod_left hiLog |
| filter_upwards [hiH.prod_right_ae,hexp.1.prod_right_ae,hpos] with ω hω heω hpω |
| exact Real.exp_le_exp.mp (by |
| rw [Real.exp_log hpω] |
| exact exp_integral_le_partition hω heω) |
| · apply Real.exp_le_exp.mp |
| calc |
| _ ≤ ∫ ω, Real.exp (Real.log (Z ω)) ∂μ := exp_integral_le_partition hiLog hexplog |
| _ = ∫ ω, Z ω ∂μ := integral_congr_ae (hpos.mono (fun _ h => Real.exp_log h)) |
| _ ≤ _ := hexp.2 |
| |
| lemma cylinder_cgf_mean_bounds {X : Type*} [MeasurableSpace X] |
| [Countable X] [MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν] |
| (A : X → ℕ →₀ ℝ) {K : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c^2) ≤ K) (t : ℝ) : |
| 0 ≤ (∫ g, cgf (fun x => cylinderField (A x) g) ν t ∂gaussianCoordinates) ∧ |
| (∫ g, cgf (fun x => cylinderField (A x) g) ν t ∂gaussianCoordinates) ≤ t^2*K/2 := by |
| have hm : Measurable (fun z : (ℕ → ℝ) × X => t*cylinderField (A z.2) z.1) := |
| (measurable_cylinderFields A).const_mul t |
| have hv (x : X) : ((⟨t^2,sq_nonneg t⟩ : NNReal)*(((A x).sum (fun _ c => c^2)).toNNReal) : ℝ) ≤ t^2*K := by |
| change t^2*((((A x).sum (fun _ c => c^2)).toNNReal : NNReal) : ℝ) ≤ t^2*K |
| rw [Real.coe_toNNReal _ (show 0 ≤ (A x).sum (fun _ c => c^2) from |
| Finset.sum_nonneg (fun _ _ => sq_nonneg _))] |
| exact mul_le_mul_of_nonneg_left (hA x) (sq_nonneg _) |
| exact gaussian_log_partition_mean_bounds (ν := ν) hm (fun x => cylinder_scaled_law A x t) hv |
| |
| lemma random_cylinder_cgf_mean_bounds {Ω X : Type*} |
| [MeasurableSpace Ω] [MeasurableSpace X] [Countable X] [MeasurableSingletonClass X] |
| {P : Measure Ω} [IsProbabilityMeasure P] {ν : Ω → Measure X} |
| (hν : Measurable ν) [∀ ω, IsProbabilityMeasure (ν ω)] |
| (A : X → ℕ →₀ ℝ) {K : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c^2) ≤ K) (t : ℝ) : |
| let F := fun z : Ω × (ℕ → ℝ) => cgf (fun x => cylinderField (A x) z.2) (ν z.1) t |
| 0 ≤ (∫ z, F z ∂P.prod gaussianCoordinates) ∧ |
| (∫ z, F z ∂P.prod gaussianCoordinates) ≤ t^2*K/2 := by |
| have hi := integrable_random_cylinder_cgf (P := P) hν A hA t |
| dsimp only |
| rw [integral_prod _ hi] |
| refine ⟨integral_nonneg (fun ω => (cylinder_cgf_mean_bounds (ν ω) A hA t).1), ?_⟩ |
| calc |
| _ ≤ ∫ _ : Ω, t^2*K/2 ∂P := integral_mono hi.integral_prod_left |
| (integrable_const _) (fun ω => (cylinder_cgf_mean_bounds (ν ω) A hA t).2) |
| _ = _ := by simp |
| |
| |
| |
| lemma enriched_coordinate_pressure_cost {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ) |
| (hb : CascadeExponents n b) {h : ℕ → ℝ} (hh : Monotone h) (h0 : 0 ≤ h 0) |
| (u : Fin N → ℝ) (hu : ∀ j, |u j| ≤ 2) (j : Fin N) {v : ℝ} (hv : |v| ≤ 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) : |
| let d := enrichedMeanPressure P r n b h (Function.update u j v) ν φ - |
| enrichedMeanPressure P r n b h (Function.update u j 0) ν φ |
| 0 ≤ d ∧ d ≤ (perturbationAmplitude N j*v)^2/(2*N) := by |
| have he := (enrichedCGF_integral hN n b hb hh h0 u hu j hv ν P hm hK hφ r).2 |
| have hbnd := random_cylinder_cgf_mean_bounds (P := enrichedFrozenLaw P r n b j) |
| (measurable_enrichedFrozenReference n h u j ν hm) |
| (monomialCoefficients n (monomialIndex j).1 (monomialIndex j).2) |
| (monomialCoefficients_variance_le hN n (monomialIndex j).1 (monomialIndex j).2) |
| (perturbationAmplitude N j*v) |
| change 0 ≤ (∫ z, enrichedCGF n h u j ν φ v z ∂(enrichedFrozenLaw P r n b j).prod gaussianCoordinates) ∧ |
| (∫ z, enrichedCGF n h u j ν φ v z ∂(enrichedFrozenLaw P r n b j).prod gaussianCoordinates) ≤ |
| (perturbationAmplitude N j*v)^2*1/2 at hbnd |
| rw [he] at hbnd |
| have hn : (0:ℝ) < N := by exact_mod_cast hN |
| dsimp only |
| constructor |
| · exact nonneg_of_mul_nonneg_right hbnd.1 hn |
| · apply (le_div_iff₀ (by positivity : 0 < 2*(N:ℝ))).2 |
| nlinarith [hbnd.2] |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
| |
| lemma perturbationWeight_pos (j : ℕ) : 0 < perturbationWeight j := by |
| unfold perturbationWeight |
| positivity |
| |
| lemma perturbationAmplitude_sq (N j : ℕ) : |
| (perturbationAmplitude N j)^2 = N*(perturbationScale N)^2*(perturbationWeight j)^2 := by |
| unfold perturbationAmplitude |
| rw [mul_pow,mul_pow,Real.sq_sqrt (Nat.cast_nonneg N)] |
| |
| lemma enriched_coordinate_minimum_penalty {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ) |
| (hb : CascadeExponents n b) {h : ℕ → ℝ} (hh : Monotone h) (h0 : 0 ≤ h 0) |
| (u : Fin N → ℝ) (hu : ∀ j, |u j| ≤ 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) |
| (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) : |
| (u j-3/2)^2 ≤ 2*(perturbationScale N)^2*perturbationWeight j := by |
| have hc := hmin (3/2) (by norm_num) |
| unfold enrichedCoordinateObjective at hc |
| have hv := enriched_coordinate_pressure_cost hN n b hb hh h0 u hu j (hu j) ν P hm hK hφ r |
| have hz := (enriched_coordinate_pressure_cost hN n b hb hh h0 u hu j |
| (v := 3/2) (by norm_num) ν P hm hK hφ r).1 |
| dsimp only at hv hz |
| have hn : (0:ℝ) < N := by exact_mod_cast hN |
| have hw := perturbationWeight_pos j |
| have ha := perturbationAmplitude_sq N j |
| have hanu : (perturbationAmplitude N j*u j)^2/(2*N) ≤ |
| 2*(perturbationScale N)^2*(perturbationWeight j)^2 := by |
| apply (div_le_iff₀ (by positivity : 0 < 2*(N:ℝ))).2 |
| rw [mul_pow,ha] |
| have hu2 : (u j)^2 ≤ 4 := by |
| have ha := (abs_le.mp (hu j)) |
| nlinarith [sq_nonneg (u j+2),sq_nonneg (u j-2)] |
| nlinarith [mul_le_mul_of_nonneg_left hu2 |
| (show 0 ≤ (N:ℝ)*(perturbationScale N)^2*(perturbationWeight j)^2 by positivity)] |
| have hmul : perturbationWeight j*(u j-3/2)^2 ≤ |
| perturbationWeight j*(2*(perturbationScale N)^2*perturbationWeight j) := by |
| nlinarith [hv.2] |
| exact (mul_le_mul_iff_right₀ hw).mp hmul |
| |
| lemma enriched_coordinate_minimum_interior {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ) |
| (hb : CascadeExponents n b) {h : ℕ → ℝ} (hh : Monotone h) (h0 : 0 ≤ h 0) |
| (u : Fin N → ℝ) (hu : ∀ j, |u j| ≤ 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) |
| (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) {s : ℝ} (hs : s ≤ 1/4) : |
| 1 ≤ u j-s ∧ u j+s ≤ 2 := by |
| have hp := enriched_coordinate_minimum_penalty hN n b hb hh h0 u hu j ν P hm hK hφ r hmin |
| have he0 := perturbationScale_nonneg N |
| have hw := perturbationWeight_le_one j |
| have hbnd : (u j-3/2)^2 ≤ 1/32 := by |
| have hmul := mul_le_mul_of_nonneg_left hw (show 0 ≤ 2*(perturbationScale N)^2 by positivity) |
| nlinarith [sq_nonneg (perturbationScale N-1/8)] |
| constructor <;> nlinarith [sq_nonneg (u j-3/2-1/4),sq_nonneg (u j-3/2+1/4)] |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
| |
| def contactStep (N : ℕ) : ℝ := (N:ℝ)^(-1/4:ℝ) |
| |
| def contactGGRate (N j : ℕ) (L B : ℝ) : ℝ := |
| let a := perturbationAmplitude N j |
| let c := N*perturbationWeight j/a^2 |
| let θ := 1/perturbationScale N |
| let δ := 2*N*Real.sqrt (L/N) |
| 2*B*((2*c+θ^2)/(2*θ)+c*(a*contactStep N)+4*δ/(a*contactStep N))/a |
| |
| lemma contactGGRate_eq {N : ℕ} (hN : 0 < N) (j : ℕ) (L B : ℝ) : |
| contactGGRate N j L B = |
| (B*(2/perturbationWeight j+1)/perturbationWeight j)*(N:ℝ)^(-3/8:ℝ) + |
| (2*B/perturbationWeight j)*(N:ℝ)^(-1/8:ℝ) + |
| (16*B*Real.sqrt L/(perturbationWeight j)^2)*(N:ℝ)^(-1/8:ℝ) := by |
| have hn : (0:ℝ) < N := by exact_mod_cast hN |
| have he : 0 < perturbationScale N := Real.rpow_pos_of_pos hn _ |
| have hs : 0 < contactStep N := Real.rpow_pos_of_pos hn _ |
| have hw := perturbationWeight_pos j |
| have hr : 0 < Real.sqrt N := Real.sqrt_pos.2 hn |
| have he2 : (perturbationScale N)^2 = (N:ℝ)^(-1/8:ℝ) := by |
| unfold perturbationScale |
| rw [← Real.rpow_mul_natCast hn.le] |
| congr 1 |
| norm_num |
| have h1 : 1/(Real.sqrt N*(perturbationScale N)^2) = (N:ℝ)^(-3/8:ℝ) := by |
| rw [Real.sqrt_eq_rpow,he2,← Real.rpow_add hn,one_div,← Real.rpow_neg hn.le] |
| norm_num |
| have h2 : contactStep N/(perturbationScale N)^2 = (N:ℝ)^(-1/8:ℝ) := by |
| rw [he2] |
| unfold contactStep |
| rw [← Real.rpow_sub hn] |
| norm_num |
| have h3 : 1/(Real.sqrt N*(perturbationScale N)^2*contactStep N) = (N:ℝ)^(-1/8:ℝ) := by |
| rw [Real.sqrt_eq_rpow,he2] |
| unfold contactStep |
| rw [← Real.rpow_add hn,← Real.rpow_add hn,one_div,← Real.rpow_neg hn.le] |
| norm_num |
| calc |
| contactGGRate N j L B = |
| (B*(2/perturbationWeight j+1)/perturbationWeight j)*(1/(Real.sqrt N*(perturbationScale N)^2)) + |
| (2*B/perturbationWeight j)*(contactStep N/(perturbationScale N)^2) + |
| (16*B*Real.sqrt L/(perturbationWeight j)^2)*(1/(Real.sqrt N*(perturbationScale N)^2*contactStep N)) := by |
| unfold contactGGRate perturbationAmplitude |
| rw [Real.sqrt_div' L hn.le] |
| have hn' : (N:ℝ) = (Real.sqrt N)^2 := (Real.sq_sqrt hn.le).symm |
| conv_lhs => rw [hn'] |
| simp only [Real.sqrt_sq hr.le] |
| field_simp |
| ring |
| _ = _ := by rw [h1,h2,h3] |
|
|
| lemma tendsto_nat_rpow_neg {c : ℝ} (hc : 0 < c) : |
| Tendsto (fun N : ℕ => (N:ℝ)^(-c)) atTop (𝓝 0) := |
| (tendsto_rpow_neg_atTop hc).comp tendsto_natCast_atTop_atTop |
|
|
| lemma perturbationScale_tendsto : Tendsto perturbationScale atTop (𝓝 0) := by |
| change Tendsto (fun N : ℕ => (N:ℝ)^(-1/16:ℝ)) atTop (𝓝 0) |
| simpa only [neg_div] using tendsto_nat_rpow_neg (c := 1/16) (by norm_num) |
|
|
| lemma contactStep_tendsto : Tendsto contactStep atTop (𝓝 0) := by |
| change Tendsto (fun N : ℕ => (N:ℝ)^(-1/4:ℝ)) atTop (𝓝 0) |
| simpa only [neg_div] using tendsto_nat_rpow_neg (c := 1/4) (by norm_num) |
|
|
| lemma contactGGRate_tendsto (j : ℕ) (L B : ℝ) : |
| Tendsto (fun N => contactGGRate N j L B) atTop (𝓝 0) := by |
| have h1 := (tendsto_nat_rpow_neg (c := 3/8) (by norm_num)).const_mul |
| (B*(2/perturbationWeight j+1)/perturbationWeight j) |
| have h2 := (tendsto_nat_rpow_neg (c := 1/8) (by norm_num)).const_mul (2*B/perturbationWeight j) |
| have h3 := (tendsto_nat_rpow_neg (c := 1/8) (by norm_num)).const_mul |
| (16*B*Real.sqrt L/(perturbationWeight j)^2) |
| apply Tendsto.congr' _ (show Tendsto _ _ (𝓝 (0:ℝ)) from by simpa using (h1.add h2).add h3) |
| filter_upwards [eventually_gt_atTop 0] with N hN |
| simpa only [neg_div,one_div] using (contactGGRate_eq hN j L B).symm |
| |
| end IsingPerceptron |
| |
| end |
| |
| end OAI |
| |
