| import OAI.Probability.InvariantIsing.Pressure.RandomCoefficientGG |
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| namespace OAI |
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| /-! Control of the finite spectral GG diagonal error by its actual L1 deviation. -/ |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators |
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| namespace InvariantIsing |
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| variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] |
| [Countable X] [MeasurableSingletonClass X] |
| {P : Measure Ω} [IsProbabilityMeasure P] {ν : Ω → Measure X} |
| [∀ ω, IsProbabilityMeasure (ν ω)] |
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| lemma randomCoefficient_bounded_observable_covariance (hν : Measurable ν) |
| (A : Ω → X → ℕ →₀ ℝ) |
| (hAm : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => cylinderField (A p.1.1 p.2) p.1.2)) |
| {B : ℝ} (hA : ∀ ω x, (A ω x).sum (fun _ c => c ^ 2) ≤ B) |
| (t b : ℝ) (Y : Ω → X → ℝ) (hYm : Measurable (Function.uncurry Y)) |
| {K : ℝ} (hK : 0 ≤ K) (hYb : ∀ ω x, |Y ω x| ≤ K) |
| {r : ℕ} (D : Ω → (Fin (r + 1) → X) → ℝ) |
| (hDm : Measurable (Function.uncurry D)) {c : ℝ} (hc : 0 ≤ c) |
| (hD : ∀ ω σ, |D ω σ| ≤ c) : |
| let E := fun z : Ω × (ℕ → ℝ) => ∫ x, |Y z.1 x - b| |
| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2) |
| Integrable E (P.prod gaussianCoordinates) ∧ |
| |randomCoefficientAverage P ν A t (fun ω σ => Y ω (σ 0) * D ω σ) - |
| b * randomCoefficientAverage P ν A t D| ≤ |
| c * ∫ z, E z ∂P.prod gaussianCoordinates := by |
| have hYm' : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => Y p.1.1 p.2) := |
| hYm.comp (measurable_fst.fst.prodMk measurable_snd) |
| have he := randomCoefficient_all_exp_ae (P := P) hν A hAm hA |
| have hdev (ω : Ω) (x : X) : |Y ω x - b| ≤ K + |b| := |
| (abs_sub _ _).trans (add_le_add (hYb ω x) le_rfl) |
| have hEm := measurable_random_tilted_integral (ν := fun z : Ω × (ℕ → ℝ) => ν z.1) |
| (hν.comp measurable_fst) (hAm.const_mul t) (hYm'.sub_const b |>.abs) |
| have hEi : Integrable (fun z : Ω × (ℕ → ℝ) => ∫ x, |Y z.1 x - b| |
| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)) |
| (P.prod gaussianCoordinates) := by |
| apply (integrable_const (K + |b|)).mono' hEm.aestronglyMeasurable |
| filter_upwards [he] with z hz |
| have : IsProbabilityMeasure ((ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)) := |
| isProbabilityMeasure_tilted (hz t) |
| rw [Real.norm_of_nonneg (integral_nonneg (fun _ => abs_nonneg _))] |
| calc |
| _ ≤ ∫ _x, K + |b| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2) := |
| integral_mono |
| (integrable_of_measurable_abs_le (measurable_of_countable _) |
| (fun x => by simpa only [abs_abs] using hdev z.1 x)) |
| (integrable_const _) (hdev z.1) |
| _ = _ := by simp |
| have hYi : ∀ᵐ z : Ω × (ℕ → ℝ) ∂P.prod gaussianCoordinates, |
| Integrable (Y z.1) ((ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)) := by |
| filter_upwards [he] with z hz |
| have : IsProbabilityMeasure ((ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)) := |
| isProbabilityMeasure_tilted (hz t) |
| exact integrable_of_measurable_abs_le (measurable_of_countable _) (hYb z.1) |
| have ht : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => |
| cylinderField (t • A p.1.1 p.2) p.1.2) := by |
| simpa only [cylinderField_smul] using hAm.const_mul t |
| have hYDm : Measurable (Function.uncurry (fun ω (σ : Fin (r + 1) → X) => |
| Y ω (σ 0) * D ω σ)) := by |
| have hp : Measurable (fun p : Ω × (Fin (r + 1) → X) => (p.1, p.2 0)) := |
| measurable_fst.prodMk ((measurable_pi_apply 0).comp measurable_snd) |
| exact (hYm.comp hp).mul hDm |
| have hiU := integrable_randomCoefficient_mean (P := P) hν (fun ω x => t • A ω x) ht |
| (fun ω (σ : Fin (r + 1) → X) => Y ω (σ 0) * D ω σ) hYDm (mul_nonneg hK hc) |
| (fun ω σ => by rw [abs_mul]; exact mul_le_mul (hYb ω _) (hD ω σ) (abs_nonneg _) hK) |
| have hiV := integrable_randomCoefficient_mean (P := P) hν (fun ω x => t • A ω x) ht D hDm hc hD |
| simp only [cylinderField_smul] at hiU hiV |
| refine ⟨hEi, ?_⟩ |
| exact random_replica_energy_covariance_bound (ν := fun z : Ω × (ℕ → ℝ) => ν z.1) |
| (H := fun p => t * cylinderField (A p.1.1 p.2) p.1.2) (Y := fun p => Y p.1.1 p.2) |
| (fun p => D p.1.1 p.2) |
| (hDm.comp (measurable_fst.fst.prodMk measurable_snd)) hc (fun z => hD z.1) b |
| (he.mono fun _ hz => hz t) hYi hiU hiV hEi |
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| /-- No constant self-overlap is imposed at finite N. Its actual Gibbs L1 |
| fluctuation controls precisely the extra covariance in the GG estimate. -/ |
| theorem randomCoefficient_diagonal_covariance_le (hν : Measurable ν) |
| (A : Ω → X → ℕ →₀ ℝ) |
| (hAm : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => cylinderField (A p.1.1 p.2) p.1.2)) |
| (hcross : Measurable (fun p : Ω × (X × X) => cylinderCross (A p.1 p.2.1) (A p.1 p.2.2))) |
| {B : ℝ} (hA : ∀ ω x, (A ω x).sum (fun _ c => c ^ 2) ≤ B) |
| (t d : ℝ) {r : ℕ} (D : Ω → (Fin (r + 1) → X) → ℝ) |
| (hDm : Measurable (Function.uncurry D)) {c : ℝ} (hc : 0 ≤ c) |
| (hD : ∀ ω σ, |D ω σ| ≤ c) : |
| |randomCoefficientAverage P ν A t |
| (fun ω σ => D ω σ * cylinderCross (A ω (σ 0)) (A ω (σ 0))) - |
| randomCoefficientAverage P ν A t D * randomCoefficientAverage P ν A t |
| (fun ω (σ : Fin 1 → X) => cylinderCross (A ω (σ 0)) (A ω (σ 0)))| ≤ |
| 2 * c * (∫ z : Ω × (ℕ → ℝ), ∫ x, |cylinderCross (A z.1 x) (A z.1 x) - d| |
| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2) |
| ∂P.prod gaussianCoordinates) := by |
| let Y := fun (ω : Ω) (x : X) => cylinderCross (A ω x) (A ω x) |
| have hYm : Measurable (Function.uncurry Y) := by |
| have hp : Measurable (fun p : Ω × X => (p.1, (p.2, p.2))) := |
| measurable_fst.prodMk (measurable_snd.prodMk measurable_snd) |
| exact hcross.comp hp |
| have hYb (ω : Ω) (x : X) : |Y ω x| ≤ |B| := cylinderCross_bound (A ω) (hA ω) x x |
| let E := ∫ z : Ω × (ℕ → ℝ), ∫ x, |Y z.1 x - d| |
| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2) |
| ∂P.prod gaussianCoordinates |
| let V := randomCoefficientAverage P ν A t D |
| let L := randomCoefficientAverage P ν A t (fun ω σ => D ω σ * Y ω (σ 0)) |
| let L₀ := randomCoefficientAverage P ν A t (fun ω (σ : Fin 1 → X) => Y ω (σ 0)) |
| have hL : |L - d * V| ≤ c * E := by |
| have h := (randomCoefficient_bounded_observable_covariance (P := P) hν A hAm hA t d |
| Y hYm (abs_nonneg B) hYb D hDm hc hD).2 |
| have heq : (fun ω σ => Y ω (σ 0) * D ω σ) = (fun ω σ => D ω σ * Y ω (σ 0)) := by |
| funext ω σ |
| exact mul_comm _ _ |
| rw [heq] at h |
| exact h |
| have hL₀ : |L₀ - d| ≤ E := by |
| have h := (randomCoefficient_bounded_observable_covariance (P := P) (r := 0) |
| hν A hAm hA t d Y hYm (abs_nonneg B) hYb |
| (fun (_ : Ω) (_ : Fin 1 → X) => 1) measurable_const (c := 1) (by norm_num) |
| (fun _ _ => by norm_num)).2 |
| simpa only [mul_one, randomCoefficientAverage_one hν A hAm hA, one_mul] using h |
| have hV : |V| ≤ c := randomCoefficientAverage_abs_le hν A hAm t D hDm hc hD |
| change |L - V * L₀| ≤ 2 * c * E |
| calc |
| _ = |(L - d * V) - V * (L₀ - d)| := by congr 1; ring |
| _ ≤ |L - d * V| + |V| * |L₀ - d| := by |
| simpa only [abs_mul] using (abs_sub (L - d * V) (V * (L₀ - d))) |
| _ ≤ c * E + c * E := add_le_add hL (mul_le_mul hV hL₀ (abs_nonneg _) hc) |
| _ = _ := by ring |
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| /-- Quantitative GG with both errors expressed as actual Gibbs L1 deviations. |
| The energy fluctuation is divided by the perturbation scale when passing to |
| the limiting GG identities; the diagonal fluctuation is not. -/ |
| theorem randomCoefficient_gaussian_gg_L1_bound (hν : Measurable ν) |
| (A : Ω → X → ℕ →₀ ℝ) |
| (hAm : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => cylinderField (A p.1.1 p.2) p.1.2)) |
| (hcross : Measurable (fun p : Ω × (X × X) => cylinderCross (A p.1 p.2.1) (A p.1 p.2.2))) |
| {B : ℝ} (hA : ∀ ω x, (A ω x).sum (fun _ c => c ^ 2) ≤ B) |
| (t b d : ℝ) {r : ℕ} (D : Ω → (Fin (r + 1) → X) → ℝ) |
| (hDm : Measurable (Function.uncurry D)) {c : ℝ} (hc : 0 ≤ c) |
| (hD : ∀ ω σ, |D ω σ| ≤ c) |
| (hE : Integrable (fun z : Ω × (ℕ → ℝ) => ∫ x, |
| |cylinderField (A z.1 x) z.2 - b| |
| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)) |
| (P.prod gaussianCoordinates)) : |
| |t| * |(r + 1 : ℕ) * randomCoefficientAverage P ν A t |
| (fun ω (τ : Fin (r + 1 + 1) → X) => D ω (Fin.tail τ) * |
| cylinderCross (A ω ((Fin.tail τ) 0)) (A ω (τ 0))) - |
| randomCoefficientAverage P ν A t D * randomCoefficientAverage P ν A t |
| (fun ω (τ : Fin 2 → X) => cylinderCross (A ω (τ 1)) (A ω (τ 0))) - |
| randomCoefficientAverage P ν A t |
| (fun ω σ => D ω σ * ∑ l : Fin r, cylinderCross (A ω (σ 0)) (A ω (σ l.succ)))| ≤ |
| 2 * c * ((∫ z : Ω × (ℕ → ℝ), ∫ x, |cylinderField (A z.1 x) z.2 - b| |
| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2) |
| ∂P.prod gaussianCoordinates) + |
| |t| * (∫ z : Ω × (ℕ → ℝ), ∫ x, |cylinderCross (A z.1 x) (A z.1 x) - d| |
| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2) |
| ∂P.prod gaussianCoordinates)) := by |
| have hg := randomCoefficient_gaussian_gg_bound (P := P) hν A hAm hcross hA t b D hDm hc hD hE |
| have hd := randomCoefficient_diagonal_covariance_le (P := P) hν A hAm hcross hA t d D hDm hc hD |
| refine hg.trans ((add_le_add le_rfl (mul_le_mul_of_nonneg_left hd (abs_nonneg t))).trans_eq ?_) |
| ring |
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| end InvariantIsing |
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| end |
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| end OAI |
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