| import Mathlib |
| import OAI.Probability.Perceptron.Variational.QuadraticContactRemainders |
| import OAI.Probability.Perceptron.Variational.IntegrableGibbs |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace SphericalPerceptronFreeEnergy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators |
|
|
| lemma nonnegative_limit_integral_bound {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) |
| {F : ℕ → Ω → ℝ} {f : Ω → ℝ} {C : ℝ} (hm : AEStronglyMeasurable f P) |
| (hF : ∀ n, Integrable (F n) P) (hF0 : ∀ n, 0 ≤ᵐ[P] F n) (hf0 : 0 ≤ᵐ[P] f) |
| (ht : ∀ᵐ ω ∂P, Tendsto (fun n => F n ω) atTop (nhds (f ω))) |
| (hb : ∀ᶠ n in atTop, (∫ ω, F n ω ∂P) ≤ C) : |
| Integrable f P ∧ (∫ ω, f ω ∂P) ≤ C := by |
| have hle : (∫⁻ ω, ENNReal.ofReal (f ω) ∂P) ≤ ENNReal.ofReal C := by |
| calc |
| _ = ∫⁻ ω, liminf (fun n => ENNReal.ofReal (F n ω)) atTop ∂P := by |
| apply lintegral_congr_ae |
| filter_upwards [ht] with ω hω |
| exact (ENNReal.continuous_ofReal.continuousAt.tendsto.comp hω).liminf_eq.symm |
| _ ≤ liminf (fun n => ∫⁻ ω, ENNReal.ofReal (F n ω) ∂P) atTop := |
| lintegral_liminf_le' (fun n => (hF n).aestronglyMeasurable.aemeasurable.ennreal_ofReal) |
| _ ≤ ENNReal.ofReal C := by |
| apply liminf_le_of_frequently_le' |
| apply Eventually.frequently |
| filter_upwards [hb] with n hn |
| rw [← ofReal_integral_eq_lintegral_ofReal (hF n) (hF0 n)] |
| exact ENNReal.ofReal_le_ofReal hn |
| have hi : Integrable f P := ⟨hm,(hasFiniteIntegral_iff_ofReal hf0).mpr (hle.trans_lt ENNReal.ofReal_lt_top)⟩ |
| refine ⟨hi,?_⟩ |
| have hC : 0 ≤ C := by |
| obtain ⟨n,hn⟩ := hb.exists |
| exact (integral_nonneg_of_ae (hF0 n)).trans hn |
| rw [← ofReal_integral_eq_lintegral_ofReal hi hf0] at hle |
| exact (ENNReal.ofReal_le_ofReal_iff hC).mp hle |
|
|
| lemma taylor_two_explicit (F : ℝ → ℝ) (u v : ℝ) : |
| taylorWithinEval F 2 univ u v = F u+(v-u)*deriv F u+(v-u)^2/2*iteratedDeriv 2 F u := by |
| rw [taylorWithinEval_succ F 1,taylorWithinEval_succ F 0,taylor_within_zero_eval] |
| simp only [iteratedDerivWithin_univ,iteratedDeriv_one,Nat.cast_zero,Nat.factorial_zero, |
| Nat.cast_one,zero_add,one_mul,inv_one,pow_one,smul_eq_mul,Nat.factorial_one] |
| ring |
|
|
| lemma second_symmetric_quotient_tendsto {F : ℝ → ℝ} (hc : ContDiff ℝ 2 F) (u : ℝ) |
| {s : ℕ → ℝ} (hs : ∀ n, s n ≠ 0) (hst : Tendsto s atTop (nhds 0)) : |
| Tendsto (fun n => (F (u+s n)+F (u-s n)-2*F u)/(s n)^2) |
| atTop (nhds (iteratedDeriv 2 F u)) := by |
| have he := (taylor_isLittleO_univ (x₀ := u) hc).tendsto_div_nhds_zero |
| have hup : Tendsto (fun n => u+s n) atTop (nhds u) := by simpa using tendsto_const_nhds.add hst |
| have hum : Tendsto (fun n => u-s n) atTop (nhds u) := by simpa using tendsto_const_nhds.sub hst |
| have hp := he.comp hup |
| have hm := he.comp hum |
| have hp' : Tendsto (fun n => |
| (F (u+s n)-(F u+s n*deriv F u+(s n)^2/2*iteratedDeriv 2 F u))/(s n)^2) |
| atTop (nhds (0:ℝ)) := by |
| simpa only [Function.comp_def,taylor_two_explicit,add_sub_cancel_left,add_zero] using hp |
| have hm' : Tendsto (fun n => |
| (F (u-s n)-(F u-s n*deriv F u+(s n)^2/2*iteratedDeriv 2 F u))/(s n)^2) |
| atTop (nhds (0:ℝ)) := by |
| convert hm using 1 |
| funext n |
| simp only [Function.comp_def,taylor_two_explicit,sub_sub_cancel_left,neg_sq,neg_mul] |
| ring |
| have h := (hp'.add hm').add_const (iteratedDeriv 2 F u) |
| simp only [zero_add] at h |
| apply h.congr |
| intro n |
| field_simp [hs n] |
| ring |
|
|
| lemma convex_symmetric_quotient_nonneg {F : ℝ → ℝ} (hc : ConvexOn ℝ univ F) (u s : ℝ) : |
| 0 ≤ (F (u+s)+F (u-s)-2*F u)/s^2 := by |
| have hh := hc.2 (mem_univ (u+s)) (mem_univ (u-s)) |
| (by norm_num : (0:ℝ) ≤ 1/2) (by norm_num : (0:ℝ) ≤ 1/2) (by norm_num : (1:ℝ)/2+1/2=1) |
| simp only [smul_eq_mul] at hh |
| have he : (1:ℝ)/2*(u+s)+1/2*(u-s)=u := by ring |
| rw [he] at hh |
| exact div_nonneg (by linarith) (sq_nonneg _) |
|
|
| lemma quadratic_contact_curvature_bound {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) |
| {F : ℝ → Ω → ℝ} {J : Ω → ℝ} (hJ : AEStronglyMeasurable J P) |
| (hi : ∀ t, Integrable (F t) P) |
| (hc : ∀ᵐ ω ∂P, ContDiff ℝ 2 (fun t => F t ω) ∧ ConvexOn ℝ univ (fun t => F t ω)) |
| {u a c : ℝ} (he : ∀ᵐ ω ∂P, iteratedDeriv 2 (fun t => F t ω) u = J ω) |
| (hmin : IsLocalMin (fun t => c*(t-a)^2-∫ ω, F t ω ∂P) u) : |
| Integrable J P ∧ (∫ ω, J ω ∂P) ≤ 2*c := by |
| let s : ℕ → ℝ := fun n => 1/(n+1:ℝ) |
| have hs (n) : 0 < s n := by dsimp [s]; positivity |
| have hst : Tendsto s atTop (nhds 0) := tendsto_one_div_add_atTop_nhds_zero_nat |
| let Q : ℕ → Ω → ℝ := fun n ω => (F (u+s n) ω+F (u-s n) ω-2*F u ω)/(s n)^2 |
| have hQi (n) : Integrable (Q n) P := |
| (((hi (u+s n)).add (hi (u-s n))).sub ((hi u).const_mul 2)).div_const _ |
| have hQ0 (n) : 0 ≤ᵐ[P] Q n := hc.mono fun ω hω => convex_symmetric_quotient_nonneg hω.2 u (s n) |
| have hQt : ∀ᵐ ω ∂P, Tendsto (fun n => Q n ω) atTop (nhds (J ω)) := by |
| filter_upwards [hc,he] with ω hω heω |
| rw [← heω] |
| exact second_symmetric_quotient_tendsto hω.1 u (fun n => (hs n).ne') hst |
| have hJ0 : 0 ≤ᵐ[P] J := by |
| filter_upwards [hQt,ae_all_iff.mpr hQ0] with ω hω h0 |
| exact ge_of_tendsto hω (Eventually.of_forall h0) |
| apply nonnegative_limit_integral_bound P hJ hQi hQ0 hJ0 hQt |
| have hup : Tendsto (fun n => u+s n) atTop (nhds u) := by simpa using tendsto_const_nhds.add hst |
| have hum : Tendsto (fun n => u-s n) atTop (nhds u) := by simpa using tendsto_const_nhds.sub hst |
| have hp := hup.eventually hmin |
| have hm := hum.eventually hmin |
| filter_upwards [hp,hm] with n hpn hmn |
| have hsum : Integrable (fun ω => F (u+s n) ω+F (u-s n) ω) P := (hi _).add (hi _) |
| have hmul : Integrable (fun ω => 2*F u ω) P := (hi u).const_mul 2 |
| rw [integral_div,integral_sub hsum hmul,integral_add (hi (u+s n)) (hi (u-s n)),integral_const_mul] |
| apply (div_le_iff₀ (sq_pos_of_pos (hs n))).mpr |
| nlinarith |
|
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| variable {S : Type*} [MeasurableSpace S] (μ : Measure S) |
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| lemma coupling_weighted_power_integrable {H Y : S → ℝ} (hH : Measurable H) (hY : Measurable Y) |
| (he : ∀ a : ℝ, Integrable (fun x => Real.exp (H x+a*Y x)) μ) (a : ℝ) (k : ℕ) : |
| Integrable (fun x => Real.exp (H x+a*Y x)*(Y x)^k) μ := by |
| have hi := ((he (a+1)).add (he (a-1))).const_mul (k.factorial:ℝ) |
| apply hi.mono' (((hH.add (hY.const_mul a)).exp).mul (hY.pow_const k)).aestronglyMeasurable |
| filter_upwards [] with x |
| change ‖Real.exp (H x+a*Y x)*(Y x)^k‖ ≤ (k.factorial:ℝ)*(Real.exp (H x+(a+1)*Y x)+Real.exp (H x+(a-1)*Y x)) |
| rw [Real.norm_eq_abs,abs_mul,abs_of_pos (Real.exp_pos _),abs_pow,mul_comm] |
| convert abs_pow_mul_exp_le (H x+a*Y x) (Y x) k using 1 |
| congr 2 <;> ring_nf |
|
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| variable [IsProbabilityMeasure μ] |
|
|
| lemma coupling_tilt_memLp_two {H Y : S → ℝ} (hH : Measurable H) (hY : Measurable Y) |
| (he : ∀ a : ℝ, Integrable (fun x => Real.exp (H x+a*Y x)) μ) (a : ℝ) : |
| MemLp Y 2 (tiltLaw μ (fun x => H x+a*Y x) 1) := by |
| apply (memLp_two_iff_integrable_sq hY.aestronglyMeasurable).mpr |
| have hp : 0 < tiltPartition μ (fun x => H x+a*Y x) 1 := |
| tilt_partition_pos_of_integrable μ (by simpa only [one_mul] using he a) |
| unfold tiltLaw |
| apply Integrable.smul_measure _ (ENNReal.inv_ne_top.mpr (ENNReal.ofReal_ne_zero_iff.mpr hp)) |
| apply (integrable_withDensity_iff_integrable_smul' |
| ((hH.add (hY.const_mul a)).const_mul 1 |>.exp.ennreal_ofReal) |
| (ae_of_all _ fun _ => ENNReal.ofReal_lt_top)).mpr |
| simpa only [ENNReal.toReal_ofReal (Real.exp_pos _).le,smul_eq_mul,one_mul,Pi.add_apply] using |
| coupling_weighted_power_integrable μ hH hY he a 2 |
|
|
| lemma coupling_thermal_abs_sq_le {H Y : S → ℝ} (hH : Measurable H) (hY : Measurable Y) |
| (he : ∀ a : ℝ, Integrable (fun x => Real.exp (H x+a*Y x)) μ) (a : ℝ) : |
| (tiltMean μ (fun x => H x+a*Y x) |
| (fun x => |Y x-tiltMean μ (fun x => H x+a*Y x) Y 1|) 1)^2 ≤ |
| tiltMean μ (fun x => H x+a*Y x) |
| (fun x => (Y x-tiltMean μ (fun x => H x+a*Y x) Y 1)^2) 1 := by |
| let T := fun x => H x+a*Y x |
| have hT : Measurable T := hH.add (hY.const_mul a) |
| have hi : Integrable (fun x => Real.exp (1*T x)) μ := by simpa only [one_mul] using he a |
| let := tilt_law_probability_of_integrable μ hi |
| have hY2 := coupling_tilt_memLp_two μ hH hY he a |
| have hh := integral_centered_abs_le_sqrt_variance (tiltLaw μ T 1) hY2 |
| have hs := pow_le_pow_left₀ (integral_nonneg (fun x => abs_nonneg _)) hh 2 |
| rw [Real.sq_sqrt (variance_nonneg _ _),variance_eq_integral hY.aemeasurable] at hs |
| simpa only [tilt_law_integral_of_integrable μ hT hi] using hs |
|
|
| lemma coupling_tilt_center_triangle {H Y : S → ℝ} (hH : Measurable H) (hY : Measurable Y) |
| (he : ∀ a : ℝ, Integrable (fun x => Real.exp (H x+a*Y x)) μ) (a b c : ℝ) : |
| tiltMean μ (fun x => H x+a*Y x) (fun x => |Y x-c|) 1 ≤ |
| tiltMean μ (fun x => H x+a*Y x) (fun x => |Y x-b|) 1+|b-c| := by |
| let T := fun x => H x+a*Y x |
| have hT : Measurable T := hH.add (hY.const_mul a) |
| have hi : Integrable (fun x => Real.exp (1*T x)) μ := by simpa only [one_mul] using he a |
| let := tilt_law_probability_of_integrable μ hi |
| have hYi := (coupling_tilt_memLp_two μ hH hY he a).integrable (by norm_num) |
| change tiltMean μ T (fun x => |Y x-c|) 1 ≤ tiltMean μ T (fun x => |Y x-b|) 1+|b-c| |
| simp_rw [← tilt_law_integral_of_integrable μ hT hi] |
| have hc : (∫ _ : S, |b-c| ∂tiltLaw μ T 1)=|b-c| := by simp |
| have hib : Integrable (fun x => |Y x-b|) (tiltLaw μ T 1) := (hYi.sub (integrable_const b)).abs |
| rw [← hc,← integral_add hib (integrable_const _)] |
| apply integral_mono ((hYi.sub (integrable_const c)).abs) |
| (((hYi.sub (integrable_const b)).abs).add (integrable_const _)) |
| intro x |
| change |Y x-c| ≤ |Y x-b|+|b-c| |
| calc |
| _ = |(Y x-b)+(b-c)| := by congr 1; ring |
| _ ≤ _ := abs_add_le _ _ |
|
|
| end SphericalPerceptronFreeEnergy |
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| end |
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| end OAI |
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