| import OAI.Probability.FreeIsing.Trees.ResponseRowEnergy |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Set |
|
|
| namespace Problem367.ResponseRows |
|
|
| section Deterministic |
|
|
| variable {Ω ι : Type*} |
|
|
| theorem norm_tree_coefficient_le |
| {d : ℕ} {theta : ℝ} (hθ0 : 0 ≤ theta) (hθ : theta ^ 2 ≤ 1) |
| (s : TreeVertex d → Finset ι) (X : TreeVertex d → ι → Ω → ℝ) |
| (v : TreeVertex d) (ω : Ω) : |
| ‖coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X v ω‖ ≤ |
| theta ^ v.1.length := by |
| have hp := prod_responseAttenuation_pos (s v) (fun i => X v i ω) hθ |
| have hle := prod_attenuation_le_one (s v) (fun i => X v i ω) hθ |
| have hw : 0 ≤ theta ^ v.1.length := pow_nonneg hθ0 _ |
| rw [coefficient, pathProduct, Real.norm_eq_abs, abs_mul, abs_of_nonneg hw, |
| abs_of_pos hp] |
| exact mul_le_of_le_one_right hw hle |
|
|
| theorem summable_tree_coefficient_sq |
| {d : ℕ} {theta : ℝ} (hθ0 : 0 ≤ theta) (hθ : theta ^ 2 ≤ 1) |
| (hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) |
| (s : TreeVertex d → Finset ι) (X : TreeVertex d → ι → Ω → ℝ) (ω : Ω) : |
| Summable (fun v : TreeVertex d => |
| (coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X v ω) ^ 2) := by |
| simpa only [Real.norm_eq_abs, sq_abs] using |
| summable_norm_sq_of_radial_bound d hsub |
| (fun v => coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X v ω) |
| (fun v => norm_tree_coefficient_le hθ0 hθ s X v ω) |
|
|
| theorem summable_regularPathWeight {d : ℕ} {theta : ℝ} |
| (hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) : |
| Summable (regularPathWeight d theta) := by |
| have hs := summable_regularPathWeight_mul_length_sq hsub |
| apply Summable.of_nonneg_of_le (regularPathWeight_nonneg d theta) _ hs |
| intro n |
| have hr : (1 : ℝ) ≤ ((n + 1 : ℕ) : ℝ) := by exact_mod_cast Nat.succ_le_succ (Nat.zero_le n) |
| have hr2 : (1 : ℝ) ≤ ((n + 1 : ℕ) : ℝ) ^ 2 := by nlinarith |
| simpa only [mul_one] using |
| mul_le_mul_of_nonneg_left hr2 (regularPathWeight_nonneg d theta n) |
|
|
| theorem summable_regularPathWeight_sqrt_length_sq |
| {d : ℕ} {theta A : ℝ} (hA : 0 ≤ A) |
| (hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) : |
| Summable (fun n => regularPathWeight d theta n * |
| Real.sqrt (A * ((n + 1 : ℕ) : ℝ) ^ 2)) := by |
| have hs := (summable_regularPathWeight_mul_length_sq hsub).mul_left (Real.sqrt A) |
| apply Summable.of_nonneg_of_le (fun n => mul_nonneg |
| (regularPathWeight_nonneg d theta n) (Real.sqrt_nonneg _)) _ hs |
| intro n |
| have hr : (1 : ℝ) ≤ ((n + 1 : ℕ) : ℝ) := by exact_mod_cast Nat.succ_le_succ (Nat.zero_le n) |
| rw [Real.sqrt_mul hA, Real.sqrt_sq (Nat.cast_nonneg _)] |
| have hr2 : ((n + 1 : ℕ) : ℝ) ≤ ((n + 1 : ℕ) : ℝ) ^ 2 := by nlinarith |
| calc |
| regularPathWeight d theta n * (Real.sqrt A * ((n + 1 : ℕ) : ℝ)) ≤ |
| regularPathWeight d theta n * (Real.sqrt A * ((n + 1 : ℕ) : ℝ) ^ 2) := by |
| exact mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_left hr2 (Real.sqrt_nonneg A)) |
| (regularPathWeight_nonneg d theta n) |
| _ = _ := by ring |
|
|
| end Deterministic |
|
|
| section Probability |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
|
|
| theorem integral_tree_density_le_subcritical |
| {d : ℕ} {ι : Type*} (s : TreeVertex d → Finset ι) |
| (hcard : ∀ v, (s v).card = v.1.length) |
| {theta C : ℝ} (hθ0 : 0 ≤ theta) (hθ : theta ^ 2 ≤ 1) |
| (hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) (hC : 0 ≤ C) |
| {X Y : TreeVertex d → ι → Ω → ℝ} {H H' : Ω → ℝ} |
| (hX : ∀ v i, i ∈ s v → Measurable (X v i)) |
| (hY : ∀ v i, i ∈ s v → Measurable (Y v i)) |
| (hInt : ∀ v i, i ∈ s v → Integrable (fun ω => (X v i ω - Y v i ω) ^ 2) μ) |
| (hH : Measurable H) (hH' : Measurable H') |
| (hU : MemLp (fun ω => H ω - H' ω) 2 μ) |
| (hlog : ∀ v i, i ∈ s v → |
| (∫ ω, (Real.log (responseAttenuation theta (X v i ω)) - |
| Real.log (responseAttenuation theta (Y v i ω))) ^ 2 ∂μ) ≤ |
| C * (∫ ω, (H ω - H' ω) ^ 2 ∂μ)) : |
| let c := coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X |
| let c' := coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s Y |
| Integrable (DiscrepancyEnergy.density c c' (fun ω => H ω - H' ω) |
| (fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω))) μ ∧ |
| (∫ ω, DiscrepancyEnergy.density c c' (fun ω => H ω - H' ω) |
| (fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω)) ω ∂μ) ≤ |
| subcriticalPathConstant d theta (2 * C) * (∫ ω, (H ω - H' ω) ^ 2 ∂μ) := by |
| dsimp only |
| let V := ∫ ω, (H ω - H' ω) ^ 2 ∂μ |
| let Q := fun n : ℕ => (2 * C) * ((n + 1 : ℕ) : ℝ) ^ 2 * V |
| let L := fun _ : ℕ => V |
| have hV : 0 ≤ V := integral_nonneg (fun ω => sq_nonneg _) |
| have hQ0 (n : ℕ) : 0 ≤ Q n := by dsimp [Q]; positivity |
| have hL0 (n : ℕ) : 0 ≤ L n := hV |
| have hQ (n : ℕ) (v : TreeVertex d) (hv : v.1.length = n + 1) : |
| pathNoiseEnergy μ theta (s v) (X v) (Y v) ≤ Q n := by |
| have h := integral_weighted_path_log_sq_le (s v) hθ (hX v) (hY v) (hInt v) (hlog v) |
| rw [hcard v, hv] at h |
| change _ ≤ _ at h |
| calc |
| pathNoiseEnergy μ theta (s v) (X v) (Y v) ≤ |
| 2 * ((n + 1 : ℕ) : ℝ) ^ 2 * (C * V) := h |
| _ = Q n := by dsimp [Q]; ring |
| have hL (n : ℕ) (v : TreeVertex d) (_hv : v.1.length = n + 1) : |
| pathLinearEnergy μ theta (s v) (X v) (fun ω => H ω - H' ω) ≤ L n := |
| (integral_mul_prod_responseAttenuation_sq_le (s v) hθ (hX v) hU).2 |
| have hQs : Summable (fun n => regularPathWeight d theta n * Q n) := by |
| convert! (summable_regularPathWeight_mul_length_sq hsub).mul_left (2 * C * V) using 1 |
| funext n |
| dsimp only [Q] |
| ring |
| have hLs : Summable (fun n => regularPathWeight d theta n * L n) := |
| (summable_regularPathWeight hsub).mul_right V |
| have hRs : Summable (fun n => regularPathWeight d theta n * Real.sqrt (Q n)) := by |
| convert! summable_regularPathWeight_sqrt_length_sq (A := 2 * C * V) |
| (by positivity) hsub using 1 |
| funext n |
| congr 2 |
| dsimp only [Q] |
| ring |
| have hrows : ∀ᵐ ω ∂μ, |
| Summable (fun v => (coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X v ω) ^ 2) ∧ |
| Summable (fun v => (coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s Y v ω) ^ 2) := by |
| filter_upwards [] with ω |
| exact ⟨summable_tree_coefficient_sq hθ0 hθ hsub s X ω, |
| summable_tree_coefficient_sq hθ0 hθ hsub s Y ω⟩ |
| have hb := integral_tree_density_le_shell_energies s hcard hθ hX hY hInt hH hH' hU |
| hQ0 hL0 hQ hL hQs hLs hRs hrows |
| refine ⟨hb.1, ?_⟩ |
| exact (subcritical_noise_drift_bound hsub (by positivity : 0 ≤ 2 * C) hV |
| hQ0 hL0 (fun n => le_rfl) (fun n => le_rfl) hb.2).2 |
| |
| theorem eq_zero_of_subcritical_response_variance_identity |
| {d : ℕ} {ι : Type*} (s : TreeVertex d → Finset ι) |
| (hcard : ∀ v, (s v).card = v.1.length) |
| {theta C T : ℝ} (hθ0 : 0 ≤ theta) (hθ : theta ^ 2 ≤ 1) |
| (hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) (hC : 0 ≤ C) |
| {X Y : ℝ → TreeVertex d → ι → Ω → ℝ} {H H' : ℝ → Ω → ℝ} |
| (hX : ∀ t ∈ Icc 0 T, ∀ v i, i ∈ s v → Measurable (X t v i)) |
| (hY : ∀ t ∈ Icc 0 T, ∀ v i, i ∈ s v → Measurable (Y t v i)) |
| (hInt : ∀ t ∈ Icc 0 T, ∀ v i, i ∈ s v → |
| Integrable (fun ω => (X t v i ω - Y t v i ω) ^ 2) μ) |
| (hH : ∀ t ∈ Icc 0 T, Measurable (H t)) |
| (hH' : ∀ t ∈ Icc 0 T, Measurable (H' t)) |
| (hU : ∀ t ∈ Icc 0 T, MemLp (fun ω => H t ω - H' t ω) 2 μ) |
| (hlog : ∀ t ∈ Icc 0 T, ∀ v i, i ∈ s v → |
| (∫ ω, (Real.log (responseAttenuation theta (X t v i ω)) - |
| Real.log (responseAttenuation theta (Y t v i ω))) ^ 2 ∂μ) ≤ |
| C * (∫ ω, (H t ω - H' t ω) ^ 2 ∂μ)) |
| (hAC : AbsolutelyContinuousOnInterval (fun t => ∫ ω, (H t ω - H' t ω) ^ 2 ∂μ) 0 T) |
| (hzero : (∫ ω, (H 0 ω - H' 0 ω) ^ 2 ∂μ) = 0) |
| (hderiv : ∀ᵐ t : ℝ, t ∈ Icc 0 T → |
| deriv (fun r => ∫ ω, (H r ω - H' r ω) ^ 2 ∂μ) t = |
| ∫ ω, DiscrepancyEnergy.density |
| (coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s (X t)) |
| (coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s (Y t)) |
| (fun ω => H t ω - H' t ω) |
| (fun ω => Real.tanh (H t ω)) (fun ω => Real.tanh (H' t ω)) ω ∂μ) : |
| ∀ t ∈ Icc 0 T, (∫ ω, (H t ω - H' t ω) ^ 2 ∂μ) = 0 := by |
| apply eq_zero_of_ae_deriv_le_mul_self |
| (C := subcriticalPathConstant d theta (2 * C)) hAC hzero |
| (fun t _ => integral_nonneg (fun ω => sq_nonneg _)) |
| filter_upwards [hderiv] with t ht hmem |
| rw [ht hmem] |
| exact (integral_tree_density_le_subcritical s hcard hθ0 hθ hsub hC |
| (hX t hmem) (hY t hmem) (hInt t hmem) (hH t hmem) (hH' t hmem) |
| (hU t hmem) (hlog t hmem)).2 |
| |
| end Probability |
| |
| end Problem367.ResponseRows |
| |
| end |
| |
| end OAI |
| |
