| import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy |
| import OAI.Probability.FreeIsing.Estimates.RadialSummability |
| import OAI.Probability.FreeIsing.Trees.CoupledPathDrift |
| import OAI.Probability.FreeIsing.Trees.SubcriticalPathSums |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
| open scoped ENNReal |
|
|
| namespace Problem367 |
|
|
| theorem abs_tanh_weight_drift_le (h h' p p' : ℝ) : |
| |(h - h') * (Real.tanh h * p ^ 2 - Real.tanh h' * p' ^ 2)| ≤ |
| (h - h') ^ 2 * p ^ 2 + |h - h'| * |p ^ 2 - p' ^ 2| := by |
| have hdiff : |Real.tanh h - Real.tanh h'| ≤ |h - h'| := by |
| simpa only [Real.dist_eq, NNReal.coe_one, one_mul] using |
| lipschitzWith_tanh_response.dist_le_mul h h' |
| have hm : |Real.tanh h'| ≤ 1 := by |
| have hsq := Real.tanh_sq_lt_one h' |
| nlinarith [sq_abs (Real.tanh h'), abs_nonneg (Real.tanh h')] |
| have hsplit : Real.tanh h * p ^ 2 - Real.tanh h' * p' ^ 2 = |
| (Real.tanh h - Real.tanh h') * p ^ 2 + |
| Real.tanh h' * (p ^ 2 - p' ^ 2) := by ring |
| rw [abs_mul, hsplit] |
| calc |
| |h - h'| * |(Real.tanh h - Real.tanh h') * p ^ 2 + |
| Real.tanh h' * (p ^ 2 - p' ^ 2)| ≤ |
| |h - h'| * (|(Real.tanh h - Real.tanh h') * p ^ 2| + |
| |Real.tanh h' * (p ^ 2 - p' ^ 2)|) := |
| mul_le_mul_of_nonneg_left (abs_add_le _ _) (abs_nonneg _) |
| _ = |h - h'| * (|Real.tanh h - Real.tanh h'| * p ^ 2 + |
| |Real.tanh h'| * |p ^ 2 - p' ^ 2|) := by |
| rw [abs_mul, abs_mul, abs_of_nonneg (sq_nonneg p)] |
| _ ≤ |h - h'| * (|h - h'| * p ^ 2 + |p ^ 2 - p' ^ 2|) := by |
| apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) |
| apply add_le_add (mul_le_mul_of_nonneg_right hdiff (sq_nonneg p)) |
| exact mul_le_of_le_one_left (abs_nonneg _) hm |
| _ = _ := by rw [mul_add, ← mul_assoc, ← sq, sq_abs] |
| |
| section Probability |
| |
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| |
| theorem integral_abs_tanh_path_drift_le |
| {ι : Type*} (s : Finset ι) {theta : ℝ} (hθ : theta ^ 2 ≤ 1) |
| {X Y : ι → Ω → ℝ} {H H' : Ω → ℝ} |
| (hX : ∀ i ∈ s, Measurable (X i)) (hY : ∀ i ∈ s, Measurable (Y i)) |
| (hInt : ∀ i ∈ s, Integrable (fun ω => (X i ω - Y i ω) ^ 2) μ) |
| (hH : Measurable H) (hH' : Measurable H') |
| (hU : MemLp (fun ω => H ω - H' ω) 2 μ) : |
| let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω) |
| let P' := fun ω => ∏ i ∈ s, responseAttenuation theta (Y i ω) |
| let Z := fun ω => ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) - |
| Real.log (responseAttenuation theta (Y i ω))) |
| Integrable (fun ω => (H ω - H' ω) * |
| (Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2)) μ ∧ |
| (∫ ω, |(H ω - H' ω) * |
| (Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2)| ∂μ) ≤ |
| (∫ ω, (H ω - H' ω) ^ 2 * P ω ^ 2 ∂μ) + |
| 2 * Real.sqrt (2 * ∫ ω, (H ω - H' ω) ^ 2 ∂μ) * |
| Real.sqrt (∫ ω, (P ω ^ 2 + P' ω ^ 2) * Z ω ^ 2 ∂μ) := by |
| dsimp only |
| let U := fun ω => H ω - H' ω |
| let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω) |
| let P' := fun ω => ∏ i ∈ s, responseAttenuation theta (Y i ω) |
| let Z := fun ω => ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) - |
| Real.log (responseAttenuation theta (Y i ω))) |
| let D := fun ω => U ω * |
| (Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2) |
| have hP : Measurable P := measurable_prod_responseAttenuation s theta hX |
| have hP' : Measurable P' := measurable_prod_responseAttenuation s theta hY |
| have hL := (integral_mul_prod_responseAttenuation_sq_le s hθ hX hU).1 |
| have hB := integral_abs_mul_prod_responseAttenuation_sq_sub_sq_le s hθ hX hY hInt hU |
| have htanh := lipschitzWith_tanh_response.continuous.measurable |
| have hDm : Measurable D := (hH.sub hH').mul |
| (((htanh.comp hH).mul (hP.pow_const 2)).sub |
| ((htanh.comp hH').mul (hP'.pow_const 2))) |
| have hpoint (ω : Ω) : |D ω| ≤ U ω ^ 2 * P ω ^ 2 + |
| |U ω| * |P ω ^ 2 - P' ω ^ 2| := |
| abs_tanh_weight_drift_le (H ω) (H' ω) (P ω) (P' ω) |
| have hD : Integrable D μ := by |
| apply (hL.add hB.1).mono' hDm.aestronglyMeasurable |
| filter_upwards [] with ω |
| exact (show ‖D ω‖ ≤ _ by simpa only [Real.norm_eq_abs, Pi.add_apply, U, P, P'] using hpoint ω) |
| refine ⟨hD, ?_⟩ |
| calc |
| (∫ ω, |D ω| ∂μ) ≤ ∫ ω, U ω ^ 2 * P ω ^ 2 + |U ω| * |P ω ^ 2 - P' ω ^ 2| ∂μ := |
| integral_mono hD.abs (hL.add hB.1) hpoint |
| _ = (∫ ω, U ω ^ 2 * P ω ^ 2 ∂μ) + |
| ∫ ω, |U ω| * |P ω ^ 2 - P' ω ^ 2| ∂μ := integral_add hL hB.1 |
| _ ≤ _ := add_le_add le_rfl hB.2 |
| |
| theorem integrable_series_of_summable_integral_abs |
| {ι : Type*} [Countable ι] {F : ι → Ω → ℝ} |
| (hF : ∀ i, Integrable (F i) μ) |
| (hs : Summable (fun i => ∫ ω, |F i ω| ∂μ)) : |
| (∀ᵐ ω ∂μ, Summable (fun i => |F i ω|)) ∧ |
| Integrable (fun ω => ∑' i, F i ω) μ ∧ |
| (∫ ω, (∑' i, F i ω) ∂μ) = ∑' i, ∫ ω, F i ω ∂μ := by |
| have hm (i : ι) : AEMeasurable (fun ω => ENNReal.ofReal |F i ω|) μ := |
| (hF i).abs.aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| have hfin : (∫⁻ ω, ∑' i, ENNReal.ofReal |F i ω| ∂μ) < ∞ := by |
| rw [lintegral_tsum hm] |
| calc |
| (∑' i, ∫⁻ ω, ENNReal.ofReal |F i ω| ∂μ) = |
| ∑' i, ENNReal.ofReal (∫ ω, |F i ω| ∂μ) := by |
| apply tsum_congr |
| intro i |
| exact (ofReal_integral_eq_lintegral_ofReal (hF i).abs |
| (Filter.Eventually.of_forall (fun ω => abs_nonneg (F i ω)))).symm |
| _ < ∞ := hs.tsum_ofReal_lt_top |
| have hAe : ∀ᵐ ω ∂μ, Summable (fun i => |F i ω|) := by |
| filter_upwards [ae_lt_top' (AEMeasurable.tsum hm) hfin.ne] with ω hω |
| simpa only [ENNReal.toReal_ofReal (abs_nonneg _)] using |
| ENNReal.summable_toReal hω.ne |
| have hIabs : Integrable (fun ω => ∑' i, |F i ω|) μ := by |
| have h := integrable_toReal_of_lintegral_ne_top (AEMeasurable.tsum hm) hfin.ne |
| simpa only [ENNReal.tsum_toReal_eq (fun i => ENNReal.ofReal_ne_top), |
| ENNReal.toReal_ofReal (abs_nonneg _)] using h |
| refine ⟨hAe, ?_, ?_⟩ |
| · apply hIabs.mono' (AEStronglyMeasurable.tsum (fun i => (hF i).aestronglyMeasurable)) |
| filter_upwards [hAe] with ω hω |
| simpa only [Real.norm_eq_abs] using |
| norm_tsum_le_tsum_norm (show Summable (fun i => ‖F i ω‖) by |
| simpa only [Real.norm_eq_abs] using hω) |
| · exact (integral_tsum_of_summable_integral_norm hF |
| (by simpa only [Real.norm_eq_abs] using hs)).symm |
|
|
| theorem integral_series_le_of_abs_majorant |
| {ι : Type*} [Countable ι] {F : ι → Ω → ℝ} {B : ι → ℝ} |
| (hF : ∀ i, Integrable (F i) μ) (hB : Summable B) |
| (hbound : ∀ i, (∫ ω, |F i ω| ∂μ) ≤ B i) : |
| Integrable (fun ω => ∑' i, F i ω) μ ∧ |
| |∫ ω, (∑' i, F i ω) ∂μ| ≤ ∑' i, B i := by |
| have hs : Summable (fun i => ∫ ω, |F i ω| ∂μ) := |
| Summable.of_nonneg_of_le (fun i => integral_nonneg (fun ω => abs_nonneg _)) hbound hB |
| have hseries := integrable_series_of_summable_integral_abs hF hs |
| refine ⟨hseries.2.1, ?_⟩ |
| rw [hseries.2.2] |
| have hnorm : Summable (fun i => |∫ ω, F i ω ∂μ|) := |
| Summable.of_nonneg_of_le (fun i => abs_nonneg _) |
| (fun i => abs_integral_le_integral_abs) hs |
| calc |
| |∑' i, ∫ ω, F i ω ∂μ| ≤ ∑' i, |∫ ω, F i ω ∂μ| := by |
| simpa only [Real.norm_eq_abs] using |
| norm_tsum_le_tsum_norm (show Summable (fun i => ‖∫ ω, F i ω ∂μ‖) by |
| simpa only [Real.norm_eq_abs] using hnorm) |
| _ ≤ ∑' i, B i := hnorm.tsum_le_tsum |
| (fun i => abs_integral_le_integral_abs.trans (hbound i)) hB |
|
|
| theorem integral_response_row_density_le |
| {ι : Type*} [Countable ι] |
| {c c' : ι → Ω → ℝ} {U m m' : Ω → ℝ} {N D : ι → ℝ} |
| (hnoise : ∀ i, Integrable (fun ω => (c i ω - c' i ω) ^ 2) μ) |
| (hdrift : ∀ i, Integrable (fun ω => U ω * |
| (m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2)) μ) |
| (hN : Summable N) (hD : Summable D) |
| (hNb : ∀ i, (∫ ω, (c i ω - c' i ω) ^ 2 ∂μ) ≤ N i) |
| (hDb : ∀ i, (∫ ω, |U ω * |
| (m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2)| ∂μ) ≤ D i) |
| (hrows : ∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2) ∧ |
| Summable (fun i => (c' i ω) ^ 2)) : |
| Integrable (DiscrepancyEnergy.density c c' U m m') μ ∧ |
| (∫ ω, DiscrepancyEnergy.density c c' U m m' ω ∂μ) ≤ |
| (∑' i, N i) + 2 * ∑' i, D i := by |
| let A := fun ω => ∑' i, (c i ω - c' i ω) ^ 2 |
| let B := fun ω => ∑' i, U ω * (m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2) |
| have hAb : ∀ i, (∫ ω, |(c i ω - c' i ω) ^ 2| ∂μ) ≤ N i := by |
| simpa only [abs_sq] using hNb |
| have hA := integral_series_le_of_abs_majorant hnoise hN hAb |
| have hB := integral_series_le_of_abs_majorant hdrift hD hDb |
| have heq : DiscrepancyEnergy.density c c' U m m' =ᵐ[μ] |
| (fun ω => A ω + 2 * B ω) := by |
| filter_upwards [hrows] with ω hω |
| have hs : Summable (fun i => (U ω * m ω) * (c i ω) ^ 2) := |
| hω.1.mul_left (U ω * m ω) |
| have hs' : Summable (fun i => (U ω * m' ω) * (c' i ω) ^ 2) := |
| hω.2.mul_left (U ω * m' ω) |
| have hBω : B ω = U ω * |
| (m ω * (∑' i, (c i ω) ^ 2) - m' ω * (∑' i, (c' i ω) ^ 2)) := by |
| calc |
| B ω = ∑' i, ((U ω * m ω) * (c i ω) ^ 2 - |
| (U ω * m' ω) * (c' i ω) ^ 2) := by |
| apply tsum_congr |
| intro i |
| ring |
| _ = _ := by rw [hs.tsum_sub hs', tsum_mul_left, tsum_mul_left]; ring |
| dsimp only [DiscrepancyEnergy.density, A] |
| rw [hBω] |
| ring |
| have hint : Integrable (DiscrepancyEnergy.density c c' U m m') μ := |
| (hA.1.add (hB.1.const_mul 2)).congr heq.symm |
| refine ⟨hint, ?_⟩ |
| rw [integral_congr_ae heq, integral_add hA.1 (hB.1.const_mul 2), integral_const_mul] |
| exact add_le_add ((le_abs_self _).trans hA.2) |
| (mul_le_mul_of_nonneg_left ((le_abs_self _).trans hB.2) (by norm_num)) |
| |
| namespace ResponseRows |
| |
| def pathProduct {ι : Type*} (theta : ℝ) (s : Finset ι) (X : ι → Ω → ℝ) (ω : Ω) : ℝ := |
| ∏ i ∈ s, responseAttenuation theta (X i ω) |
| |
| def pathLogDiff {ι : Type*} (theta : ℝ) (s : Finset ι) |
| (X Y : ι → Ω → ℝ) (ω : Ω) : ℝ := |
| ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) - |
| Real.log (responseAttenuation theta (Y i ω))) |
| |
| def pathNoiseEnergy {ι : Type*} (μ : Measure Ω) (theta : ℝ) (s : Finset ι) |
| (X Y : ι → Ω → ℝ) : ℝ := |
| ∫ ω, (pathProduct theta s X ω ^ 2 + pathProduct theta s Y ω ^ 2) * |
| pathLogDiff theta s X Y ω ^ 2 ∂μ |
| |
| def pathLinearEnergy {ι : Type*} (μ : Measure Ω) (theta : ℝ) (s : Finset ι) |
| (X : ι → Ω → ℝ) (U : Ω → ℝ) : ℝ := |
| ∫ ω, U ω ^ 2 * pathProduct theta s X ω ^ 2 ∂μ |
| |
| def coefficient {J ι : Type*} (theta : ℝ) (a : J → ℝ) (s : J → Finset ι) |
| (X : J → ι → Ω → ℝ) (j : J) (ω : Ω) : ℝ := |
| a j * pathProduct theta (s j) (X j) ω |
| |
| theorem integral_density_le_path_energies |
| {J ι : Type*} [Countable J] |
| (s : J → Finset ι) (a : J → ℝ) {theta : ℝ} (hθ : theta ^ 2 ≤ 1) |
| {X Y : J → ι → Ω → ℝ} {H H' : Ω → ℝ} |
| (hX : ∀ j i, i ∈ s j → Measurable (X j i)) |
| (hY : ∀ j i, i ∈ s j → Measurable (Y j i)) |
| (hInt : ∀ j i, i ∈ s j → Integrable (fun ω => (X j i ω - Y j i ω) ^ 2) μ) |
| (hH : Measurable H) (hH' : Measurable H') |
| (hU : MemLp (fun ω => H ω - H' ω) 2 μ) |
| (hQsum : Summable (fun j => (a j) ^ 2 * pathNoiseEnergy μ theta (s j) (X j) (Y j))) |
| (hLsum : Summable (fun j => (a j) ^ 2 * |
| pathLinearEnergy μ theta (s j) (X j) (fun ω => H ω - H' ω))) |
| (hRsum : Summable (fun j => (a j) ^ 2 * |
| Real.sqrt (pathNoiseEnergy μ theta (s j) (X j) (Y j)))) |
| (hrows : ∀ᵐ ω ∂μ, Summable (fun j => (coefficient theta a s X j ω) ^ 2) ∧ |
| Summable (fun j => (coefficient theta a s Y j ω) ^ 2)) : |
| Integrable (DiscrepancyEnergy.density (coefficient theta a s X) |
| (coefficient theta a s Y) (fun ω => H ω - H' ω) |
| (fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω))) μ ∧ |
| (∫ ω, DiscrepancyEnergy.density (coefficient theta a s X) |
| (coefficient theta a s Y) (fun ω => H ω - H' ω) |
| (fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω)) ω ∂μ) ≤ |
| (∑' j, (a j) ^ 2 * pathNoiseEnergy μ theta (s j) (X j) (Y j)) + |
| 2 * (∑' j, (a j) ^ 2 * pathLinearEnergy μ theta (s j) (X j) (fun ω => H ω - H' ω)) + |
| 4 * Real.sqrt (2 * ∫ ω, (H ω - H' ω) ^ 2 ∂μ) * |
| (∑' j, (a j) ^ 2 * Real.sqrt (pathNoiseEnergy μ theta (s j) (X j) (Y j))) := by |
| let U := fun ω => H ω - H' ω |
| let P := fun j => pathProduct theta (s j) (X j) |
| let P' := fun j => pathProduct theta (s j) (Y j) |
| let Q := fun j => pathNoiseEnergy μ theta (s j) (X j) (Y j) |
| let L := fun j => pathLinearEnergy μ theta (s j) (X j) U |
| let V := ∫ ω, U ω ^ 2 ∂μ |
| let K := 2 * Real.sqrt (2 * V) |
| let F := fun j ω => U ω * |
| (Real.tanh (H ω) * P j ω ^ 2 - Real.tanh (H' ω) * P' j ω ^ 2) |
| have hfiniteN (j : J) := integral_prod_responseAttenuation_sub_sq_le |
| (s j) hθ (hX j) (hY j) (hInt j) |
| have hfiniteD (j : J) := integral_abs_tanh_path_drift_le |
| (s j) hθ (hX j) (hY j) (hInt j) hH hH' hU |
| have hnoiseEq (j : J) (ω : Ω) : |
| (coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2 = |
| (a j) ^ 2 * (P j ω - P' j ω) ^ 2 := by |
| dsimp only [coefficient, P, P'] |
| ring |
| have hdriftEq (j : J) (ω : Ω) : U ω * |
| (Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 - |
| Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2) = |
| (a j) ^ 2 * F j ω := by |
| dsimp only [coefficient, F, P, P'] |
| ring |
| have hnoise (j : J) : Integrable (fun ω => |
| (coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2) μ := by |
| simp_rw [hnoiseEq] |
| exact (hfiniteN j).2.1.const_mul _ |
| have hdrift (j : J) : Integrable (fun ω => U ω * |
| (Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 - |
| Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2)) μ := by |
| simp_rw [hdriftEq] |
| exact (hfiniteD j).1.const_mul _ |
| have hNb (j : J) : (∫ ω, |
| (coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2 ∂μ) ≤ |
| (a j) ^ 2 * Q j := by |
| simp_rw [hnoiseEq] |
| rw [integral_const_mul] |
| exact mul_le_mul_of_nonneg_left (hfiniteN j).2.2 (sq_nonneg _) |
| have hDb (j : J) : (∫ ω, |U ω * |
| (Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 - |
| Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2)| ∂μ) ≤ |
| (a j) ^ 2 * (L j + K * Real.sqrt (Q j)) := by |
| simp_rw [hdriftEq, abs_mul, abs_sq] |
| rw [integral_const_mul] |
| exact mul_le_mul_of_nonneg_left (hfiniteD j).2 (sq_nonneg _) |
| have hDsum : Summable (fun j => (a j) ^ 2 * (L j + K * Real.sqrt (Q j))) := by |
| convert hLsum.add (hRsum.mul_left K) using 1 |
| funext j |
| dsimp only [L, Q, U] |
| ring |
| have hb := integral_response_row_density_le hnoise hdrift hQsum hDsum hNb hDb hrows |
| refine ⟨hb.1, hb.2.trans_eq ?_⟩ |
| have hsumD : (∑' j, (a j) ^ 2 * (L j + K * Real.sqrt (Q j))) = |
| (∑' j, (a j) ^ 2 * L j) + K * ∑' j, (a j) ^ 2 * Real.sqrt (Q j) := by |
| calc |
| _ = ∑' j, ((a j) ^ 2 * L j + K * ((a j) ^ 2 * Real.sqrt (Q j))) := by |
| apply tsum_congr |
| intro j |
| ring |
| _ = _ := by rw [hLsum.tsum_add (hRsum.mul_left K), tsum_mul_left] |
| rw [hsumD] |
| dsimp only [K, V, Q, L, U] |
| ring |
|
|
| end ResponseRows |
|
|
| end Probability |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
