| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.Coercivity |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| namespace KernelPotential |
| open Filter |
| open scoped Topology NNReal |
| namespace Coercivity |
| open KernelPotential.Geometry Sphere |
| open EntropySupport |
| lemma log_ratio_lower {a b P : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) |
| (hP : 0 < P) (hz : b = 0 → a = 0) : |
| a*(Real.log P+1)-P*b ≤ a*Real.log (a/b) := by |
| by_cases hb0 : b = 0 |
| · simp [hb0,hz hb0] |
| have hbp : 0 < b := lt_of_le_of_ne hb (Ne.symm hb0) |
| have h := xlog_lower_tangent (div_nonneg ha hb) hP |
| have H := mul_le_mul_of_nonneg_left h hb |
| dsimp [xlog] at H |
| field_simp at H |
| nlinarith |
|
|
| lemma log_ratio_mul {a b : ℝ} (hb : b ≠ 0) : |
| a*Real.log (a/b) = xlog a-a*Real.log b := by |
| by_cases ha : a = 0 |
| · simp [ha,xlog_zero] |
| rw [Real.log_div ha hb] |
| simp [xlog,mul_sub] |
|
|
| lemma perspective_sum_le {ι : Type*} [Fintype ι] (w x y : ι → ℝ) |
| (hw : ∀ i,0 ≤ w i) (hx : ∀ i,0 ≤ x i) (hy : ∀ i,0 < y i) |
| (hQ : 0 < ∑ i,w i*y i) : |
| xlog (∑ i,w i*x i)-(∑ i,w i*x i)*Real.log (∑ i,w i*y i) ≤ |
| ∑ i,w i*(xlog (x i)-x i*Real.log (y i)) := by |
| have hF0 : 0 ≤ ∑ i,w i*x i := Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (hx i)) |
| by_cases hF : ∑ i,w i*x i = 0 |
| · have hz : ∀ i,w i*x i=0 := fun i => (Finset.sum_eq_zero_iff_of_nonneg (fun i _ => mul_nonneg (hw i) (hx i))).mp hF i (Finset.mem_univ i) |
| have hz' : ∀ i,w i*(xlog (x i)-x i*Real.log (y i))=0 := by |
| intro i |
| rcases mul_eq_zero.mp (hz i) with hi|hi <;> simp [hi,xlog_zero] |
| simp [hF,hz',xlog_zero] |
| have hFp : 0 < ∑ i,w i*x i := lt_of_le_of_ne hF0 (Ne.symm hF) |
| let P := (∑ i,w i*x i)/(∑ i,w i*y i) |
| have hP : 0 < P := div_pos hFp hQ |
| have hi (i : ι) : w i*(x i*(Real.log P+1)-P*y i) ≤ |
| w i*(xlog (x i)-x i*Real.log (y i)) := by |
| have h := log_ratio_lower (hx i) (hy i).le hP (fun hz => ((hy i).ne' hz).elim) |
| rw [log_ratio_mul (hy i).ne'] at h |
| exact mul_le_mul_of_nonneg_left h (hw i) |
| have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hi i) |
| have he : (∑ i,w i*(x i*(Real.log P+1)-P*y i)) = |
| (∑ i,w i*x i)*(Real.log P+1)-P*(∑ i,w i*y i) := by |
| simp_rw [mul_sub,← mul_assoc] |
| rw [Finset.sum_sub_distrib,← Finset.sum_mul,Finset.mul_sum] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| rw [he] at h |
| have hlog : Real.log P = Real.log (∑ i,w i*x i)-Real.log (∑ i,w i*y i) := |
| Real.log_div hFp.ne' hQ.ne' |
| have hmul : P*(∑ i,w i*y i)=(∑ i,w i*x i) := div_mul_cancel₀ _ hQ.ne' |
| rw [hlog,hmul] at h |
| dsimp [xlog] at h ⊢ |
| nlinarith |
| |
| lemma mixture_probability {α ι : Type*} [MeasurableSpace α] [Fintype ι] |
| (w : ι → ℝ) (hw0 : ∀ i,0 ≤ w i) (hw1 : ∑ i,w i=1) |
| (ν : ι → Measure α) (hν : ∀ i,IsProbabilityMeasure (ν i)) : |
| IsProbabilityMeasure (∑ i,ENNReal.ofReal (w i) • ν i) := by |
| have (i : ι) : IsProbabilityMeasure (ν i) := hν i |
| constructor |
| rw [Measure.finsetSum_apply] |
| simp only [Measure.smul_apply,smul_eq_mul,measure_univ,mul_one] |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun i _ => hw0 i),hw1,ENNReal.ofReal_one] |
| |
| lemma mixture_density {α ι : Type*} [MeasurableSpace α] [Fintype ι] |
| (μ : Measure α) (w : ι → ℝ) (hw0 : ∀ i,0 ≤ w i) |
| (f : ι → α → ℝ) (hf : ∀ i,Measurable (f i)) (hf0 : ∀ i x,0 ≤ f i x) : |
| (∑ i,ENNReal.ofReal (w i) • μ.withDensity (fun x => ENNReal.ofReal (f i x))) = |
| μ.withDensity (fun x => ENNReal.ofReal (∑ i,w i*f i x)) := by |
| ext s hs |
| rw [Measure.finsetSum_apply,withDensity_apply _ hs] |
| simp only [Measure.smul_apply,smul_eq_mul,withDensity_apply _ hs] |
| simp_rw [ENNReal.ofReal_sum_of_nonneg (fun i _ => mul_nonneg (hw0 i) (hf0 i _)), |
| ENNReal.ofReal_mul (hw0 _)] |
| rw [lintegral_finsetSum (f := fun i x => ENNReal.ofReal (w i)*ENNReal.ofReal (f i x)) Finset.univ |
| (fun i _ => measurable_const.mul (hf i).ennreal_ofReal)] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact (lintegral_const_mul _ (hf i).ennreal_ofReal).symm |
| |
| lemma density_bounded {α : Type*} [MeasurableSpace α] (μ : Measure α) [SFinite μ] |
| (f : α → ℝ) {H : ℝ} (hfH : ∀ x,f x ≤ H) : |
| μ.withDensity (fun x => ENNReal.ofReal (f x)) ≤ ENNReal.ofReal H • μ := by |
| intro s |
| rw [withDensity_apply' _ s,Measure.smul_apply,smul_eq_mul] |
| calc |
| _ ≤ ∫⁻ x in s, ENNReal.ofReal H ∂μ := lintegral_mono (fun x => ENNReal.ofReal_le_ofReal (hfH x)) |
| _ = _ := by simp |
|
|
| lemma J_mixture {d q : ℕ} (hd : 16 ≤ d) (w : Fin q → ℝ) (hw : ∀ i,0 ≤ w i) |
| (ν : Fin q → Measure (Ambient d)) (hνp : ∀ i,IsProbabilityMeasure (ν i)) |
| (hνb : ∀ i,BoundedDensity (ν i)) (s : Ambient d) : |
| (J (∑ i,ENNReal.ofReal (w i) • ν i) s).toReal = ∑ i,w i*(J (ν i) s).toReal := by |
| have (i : Fin q) : IsProbabilityMeasure (ν i) := hνp i |
| have ht (i : Fin q) : J (ν i) s ≠ ∞ := by |
| obtain ⟨H,hH,hνH⟩ := hνb i |
| exact ne_top_of_le_ne_top ENNReal.ofReal_ne_top (bounded_J hd (ν i) hH hνH s) |
| unfold J |
| rw [lintegral_finsetSum_measure] |
| simp_rw [lintegral_smul_measure,smul_eq_mul] |
| change (∑ i,ENNReal.ofReal (w i)*J (ν i) s).toReal = _ |
| rw [ENNReal.toReal_sum (f := fun i => ENNReal.ofReal (w i)*J (ν i) s) |
| (fun i _ => ENNReal.mul_ne_top ENNReal.ofReal_ne_top (ht i))] |
| simp only [ENNReal.toReal_mul,ENNReal.toReal_ofReal (hw _),J] |
|
|
| lemma finite_convexity {d : ℕ} (hd : 16 ≤ d) (q : ℕ) (w : Fin q → ℝ) |
| (hw0 : ∀ i,0 ≤ w i) (hw1 : ∑ i,w i=1) |
| (ν : Fin q → Measure (Ambient d)) (hνp : ∀ i,IsProbabilityMeasure (ν i)) |
| (hνb : ∀ i,BoundedDensity (ν i)) : |
| Phi (∑ i,ENNReal.ofReal (w i) • ν i) ≤ ∑ i,w i*Phi (ν i) := by |
| have (i : Fin q) : IsProbabilityMeasure (ν i) := hνp i |
| have hνb' := hνb |
| choose H hH hνH using hνb' |
| choose f hf hf0 hfH heq using fun i => bounded_density_representation (sphereProbability d) (ν i) (hH i) (hνH i) |
| let F : Ambient d → ℝ := fun x => ∑ i,w i*f i x |
| let M : ℝ := 1+∑ i,w i*H i |
| let τ := ∑ i,ENNReal.ofReal (w i) • ν i |
| have hτeq : τ = (sphereProbability d).withDensity (fun x => ENNReal.ofReal (F x)) := by |
| dsimp [τ,F] |
| simp_rw [heq] |
| exact mixture_density (sphereProbability d) w hw0 f hf hf0 |
| have hF : Measurable F := Finset.measurable_sum _ (fun i _ => measurable_const.mul (hf i)) |
| have hF0 : ∀ x,0 ≤ F x := fun x => Finset.sum_nonneg (fun i _ => mul_nonneg (hw0 i) (hf0 i x)) |
| have hFM : ∀ x,F x ≤ M := by |
| intro x |
| have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ => mul_le_mul_of_nonneg_left (hfH i x) (hw0 i)) |
| dsimp [F,M] |
| linarith |
| have hM : 0 < M := by |
| have h := Finset.sum_nonneg (s := Finset.univ) (fun i _ => mul_nonneg (hw0 i) (hH i).le) |
| dsimp [M] |
| linarith |
| have hτH : τ ≤ ENNReal.ofReal M • sphereProbability d := by |
| rw [hτeq]; exact density_bounded _ F hFM |
| have hτb : BoundedDensity τ := ⟨M,hM,hτH⟩ |
| have : IsProbabilityMeasure τ := mixture_probability w hw0 hw1 ν hνp |
| have hix : ∀ i, Integrable (fun s => xlog (f i s)-f i s*Real.log (J (ν i) s).toReal) (sphereProbability d) := by |
| intro i |
| apply (integrable_xlog_bounded _ _ (hf i) (hf0 i) (hfH i)).sub |
| exact (integrable_log_J hd (ν i) (hνb i) _ (ae_unit_norm (by omega))).bdd_mul |
| (hf i).aestronglyMeasurable (ae_of_all _ (fun x => by rw [Real.norm_eq_abs,abs_of_nonneg (hf0 i x)]; exact hfH i x)) |
| have hiF : Integrable (fun s => xlog (F s)-F s*Real.log (J τ s).toReal) (sphereProbability d) := by |
| apply (integrable_xlog_bounded _ _ hF hF0 hFM).sub |
| exact (integrable_log_J hd τ hτb _ (ae_unit_norm (by omega))).bdd_mul |
| hF.aestronglyMeasurable (ae_of_all _ (fun x => by rw [Real.norm_eq_abs,abs_of_nonneg (hF0 x)]; exact hFM x)) |
| have hpoint : ∀ᵐ s ∂sphereProbability d, |
| xlog (F s)-F s*Real.log (J τ s).toReal ≤ |
| ∑ i,w i*(xlog (f i s)-f i s*Real.log (J (ν i) s).toReal) := by |
| filter_upwards [ae_unit_norm (by omega : 0 < d)] with s hs |
| have hji : ∀ i,0 < (J (ν i) s).toReal := fun i => |
| lt_of_lt_of_le (by positivity) (J_real_bounds hd (ν i) (hH i) (hνH i) s hs).1 |
| have hjτ : 0 < (J τ s).toReal := lt_of_lt_of_le (by positivity) (J_real_bounds hd τ hM hτH s hs).1 |
| have hJ := J_mixture hd w hw0 ν hνp hνb s |
| change (J τ s).toReal = _ at hJ |
| rw [hJ] at hjτ ⊢ |
| exact perspective_sum_le w (fun i => f i s) (fun i => (J (ν i) s).toReal) hw0 (fun i => hf0 i s) hji hjτ |
| have h := integral_mono_ae hiF (integrable_finsetSum _ (fun i _ => (hix i).const_mul (w i))) hpoint |
| rw [integral_finsetSum _ (fun i _ => (hix i).const_mul (w i))] at h |
| simp only [integral_const_mul] at h |
| have hrhs : (∑ i,w i*Phi (ν i)) = |
| ∑ i,w i*(∫ s,xlog (f i s)-f i s*Real.log (J (ν i) s).toReal ∂sphereProbability d) := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [Phi_density_formula hd (f i) (hf i) (hf0 i) (hfH i) (ν i) (hνb i) (heq i)] |
| change Phi τ ≤ _ |
| rw [Phi_density_formula hd F hF hF0 hFM τ hτb hτeq,hrhs] |
| exact h |
|
|
| end Coercivity |
|
|
| theorem kernel_coercivity : CoercivityStatement := by |
| refine ⟨10,by norm_num,16,?_⟩ |
| intro d hd |
| refine ⟨?_,Coercivity.Phi_uniform_le hd,?_⟩ |
| · intro ν hν hνb |
| have := hν |
| exact Coercivity.coercivity hd ν hνb |
| · exact Coercivity.finite_convexity hd |
|
|
| namespace Coercivity |
|
|
| end Coercivity |
|
|
| end KernelPotential |
|
|
| end OAI |
|
|
