| import Mathlib |
| import OAI.Probability.Perceptron.Variational.StationaryExistence |
|
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| namespace OAI |
|
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma Trial.ext_fun {m n : Trial} (h : ∀ t, m t=n t) : m=n := by |
| cases m |
| cases n |
| congr 1 |
| exact funext h |
|
|
| lemma weightedStepTrial_coupling_fst {I J : Type*} [Fintype I] [Fintype J] |
| (c : I×J → ℝ) (w : I → ℝ) (q : I → Time) |
| (hc : ∀ a, 0 ≤ c a) (hc1 : ∑ a, c a=1) |
| (hw : ∀ i, 0 ≤ w i) (hw1 : ∑ i, w i=1) |
| (hrow : ∀ i, ∑ j, c (i,j)=w i) : |
| weightedStepTrial c (fun a => q a.1) hc hc1=weightedStepTrial w q hw hw1 := by |
| apply Trial.ext_fun |
| intro t |
| change (∑ a : I×J, if q a.1 ≤ t then c a else 0)=∑ i, if q i ≤ t then w i else 0 |
| rw [Fintype.sum_prod_type] |
| apply Finset.sum_congr rfl |
| intro i _ |
| by_cases hi : q i ≤ t < |
|
|
| lemma weightedStepTrial_coupling_snd {I J : Type*} [Fintype I] [Fintype J] |
| (c : I×J → ℝ) (v : J → ℝ) (p : J → Time) |
| (hc : ∀ a, 0 ≤ c a) (hc1 : ∑ a, c a=1) |
| (hv : ∀ i, 0 ≤ v i) (hv1 : ∑ i, v i=1) |
| (hcol : ∀ j, ∑ i, c (i,j)=v j) : |
| weightedStepTrial c (fun a => p a.2) hc hc1=weightedStepTrial v p hv hv1 := by |
| apply Trial.ext_fun |
| intro t |
| change (∑ a : I×J, if p a.2 ≤ t then c a else 0)=∑ j, if p j ≤ t then v j else 0 |
| rw [Fintype.sum_prod_type,Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| by_cases hj : p j ≤ t < |
|
|
| lemma coupling_sum_fst {I J : Type*} [Fintype I] [Fintype J] |
| (c : I×J → ℝ) (w f : I → ℝ) (hrow : ∀ i, ∑ j, c (i,j)=w i) : |
| (∑ a, c a*f a.1)=∑ i, w i*f i := by |
| rw [Fintype.sum_prod_type] |
| simp_rw [← Finset.sum_mul,hrow] |
|
|
| lemma coupling_sum_snd {I J : Type*} [Fintype I] [Fintype J] |
| (c : I×J → ℝ) (v f : J → ℝ) (hcol : ∀ j, ∑ i, c (i,j)=v j) : |
| (∑ a, c a*f a.2)=∑ j, v j*f j := by |
| rw [Fintype.sum_prod_type,Finset.sum_comm] |
| simp_rw [← Finset.sum_mul,hcol] |
|
|
| lemma weightedStepTail_coupling_fst {I J : Type*} [Fintype I] [Fintype J] |
| (c : I×J → ℝ) (w q : I → ℝ) (hrow : ∀ i, ∑ j, c (i,j)=w i) : |
| weightedStepTail c (fun a => q a.1)=weightedStepTail w q := by |
| funext t |
| exact coupling_sum_fst c w (fun i => 1-max (q i) t) hrow |
|
|
| lemma weightedStepA_coupling_fst {I J : Type*} [Fintype I] [Fintype J] |
| (c : I×J → ℝ) (w q : I → ℝ) (hrow : ∀ i, ∑ j, c (i,j)=w i) : |
| weightedStepA c (fun a => q a.1)=weightedStepA w q := by |
| unfold weightedStepA |
| rw [weightedStepTail_coupling_fst c w q hrow] |
|
|
| lemma entropy_coupled_stationary_comparison {I J : Type*} [Fintype I] [Fintype J] |
| (c : I×J → ℝ) (w h : I → ℝ) (v g : J → ℝ) (q : I → Time) (p : J → Time) |
| (hc : ∀ a, 0 ≤ c a) (hw : ∀ i, 0 ≤ w i) (hw1 : ∑ i, w i=1) |
| (hv : ∀ j, 0 ≤ v j) (hv1 : ∑ j, v j=1) |
| (hrow : ∀ i, ∑ j, c (i,j)=w i) (hcol : ∀ j, ∑ i, c (i,j)=v j) |
| (B : ℝ) (hB0 : 0 ≤ B) (hB1 : B < 1) |
| (hq : ∀ i, (q i:ℝ) ≤ B) (hp : ∀ j, (p j:ℝ) ≤ B) |
| (hstat : ∀ i, 2*h i=weightedStepA w (fun i => (q i:ℝ)) (q i)) : |
| (entropy (weightedStepTrial w q hw hw1)).toReal-∑ i, w i*h i*(q i:ℝ) ≤ |
| (entropy (weightedStepTrial v p hv hv1)).toReal-∑ j, v j*g j*(p j:ℝ)+ |
| ∑ a, c a*|h a.1-g a.2| := by |
| have hc1 : ∑ a, c a=1 := by rw [Fintype.sum_prod_type] |
| have hs := entropy_weightedStep_support c (fun a => p a.2) (fun a => q a.1) |
| hc hc1 B hB0 hB1 (fun a => hp a.2) (fun a => hq a.1) |
| rw [weightedStepTrial_coupling_fst c w q hc hc1 hw hw1 hrow, |
| weightedStepTrial_coupling_snd c v p hc hc1 hv hv1 hcol, |
| weightedStepA_coupling_fst c w (fun i => (q i:ℝ)) hrow] at hs |
| have he : (∑ a, c a*((p a.2:ℝ)-(q a.1:ℝ))*weightedStepA w (fun i => (q i:ℝ)) (q a.1))/2 = |
| (∑ a, c a*h a.1*(p a.2:ℝ))-(∑ i, w i*h i*(q i:ℝ)) := by |
| simp only [mul_assoc] |
| rw [← coupling_sum_fst c w (fun i => h i*(q i:ℝ)) hrow] |
| simp only [← mul_assoc] |
| rw [Finset.sum_div,← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro a _ |
| rw [← hstat a.1] |
| ring |
| rw [he] at hs |
| have hb : (∑ j, v j*g j*(p j:ℝ))-(∑ a, c a*h a.1*(p a.2:ℝ)) ≤ |
| ∑ a, c a*|h a.1-g a.2| := by |
| simp only [mul_assoc] |
| rw [← coupling_sum_snd c v (fun j => g j*(p j:ℝ)) hcol] |
| simp only [← mul_assoc] |
| rw [← Finset.sum_sub_distrib] |
| apply Finset.sum_le_sum |
| intro a _ |
| have h₁ : (g a.2-h a.1)*(p a.2:ℝ) ≤ |h a.1-g a.2| := calc |
| _ ≤ |g a.2-h a.1| *(p a.2:ℝ) := mul_le_mul_of_nonneg_right (le_abs_self _) (p a.2).2.1 |
| _ ≤ |g a.2-h a.1| *1 := mul_le_mul_of_nonneg_left (p a.2).2.2 (abs_nonneg _) |
| _ = _ := by rw [mul_one,abs_sub_comm] |
| have H := mul_le_mul_of_nonneg_left h₁ (hc a) |
| nlinarith [H] |
| linarith |
|
|
| lemma finiteSphericalDual_eq_stationary {k : ℕ} (w h : Fin (k+1) → ℝ) |
| (q : Fin (k+1) → Time) (hw : ∀ i, 0 ≤ w i) (hw1 : ∑ i, w i=1) |
| (hq : Monotone q) (hq1 : (q (Fin.last k):ℝ) < 1) |
| (hs : ∀ i, 2*h i=weightedStepA w (fun j => (q j:ℝ)) (q i)) : |
| sInf (finiteSphericalDualValues w h hw hw1)=finiteSphericalDualObjective w h hw hw1 q := by |
| apply IsLeast.csInf_eq |
| refine ⟨⟨q,⟨hq,hq1⟩,rfl⟩,?_⟩ |
| rintro a ⟨p,⟨hp,hp1⟩,rfl⟩ |
| exact stationary_finite_entropy_minimizes w h q p hw hw1 hq hp hq1 hp1 hs |
|
|
| theorem finiteSphericalDual_abs_sub_le {k : ℕ} (w h g : Fin (k+1) → ℝ) |
| (hw : ∀ i, 0 < w i) (hw1 : ∑ i, w i=1) |
| (hh : Monotone h) (hh0 : 0 ≤ h 0) (hg : Monotone g) (hg0 : 0 ≤ g 0) : |
| |sInf (finiteSphericalDualValues w h (fun i => (hw i).le) hw1)- |
| sInf (finiteSphericalDualValues w g (fun i => (hw i).le) hw1)| ≤ |
| ∑ i, w i*|h i-g i| := by |
| classical |
| obtain ⟨q,hq,hq1,hsq⟩ := exists_finite_stationary_quantile w h hw hw1 hh hh0 |
| obtain ⟨p,hp,hp1,hsp⟩ := exists_finite_stationary_quantile w g hw hw1 hg hg0 |
| rw [finiteSphericalDual_eq_stationary w h q (fun i => (hw i).le) hw1 hq hq1 hsq, |
| finiteSphericalDual_eq_stationary w g p (fun i => (hw i).le) hw1 hp hp1 hsp] |
| let c : Fin (k+1)×Fin (k+1) → ℝ := fun a => if a.1=a.2 then w a.1 else 0 |
| have hc : ∀ a, 0 ≤ c a := by intro a |
| have hr : ∀ i, ∑ j, c (i,j)=w i := by intro i |
| have he : ∀ j, ∑ i, c (i,j)=w j := by intro j; simp [c] |
| let B := max (q (Fin.last k):ℝ) (p (Fin.last k):ℝ) |
| have hB0 : 0 ≤ B := (q (Fin.last k)).2.1.trans (le_max_left _ _) |
| have hB1 : B < 1 := max_lt hq1 hp1 |
| have hQ (i : Fin (k+1)) : (q i:ℝ) ≤ B := |
| (show (q i:ℝ) ≤ (q (Fin.last k):ℝ) from hq (Fin.le_last i)).trans (le_max_left _ _) |
| have hP (i : Fin (k+1)) : (p i:ℝ) ≤ B := |
| (show (p i:ℝ) ≤ (p (Fin.last k):ℝ) from hp (Fin.le_last i)).trans (le_max_right _ _) |
| have H := entropy_coupled_stationary_comparison c w h w g q p hc |
| (fun i => (hw i).le) hw1 (fun i => (hw i).le) hw1 hr he B hB0 hB1 hQ hP hsq |
| have H' := entropy_coupled_stationary_comparison c w g w h p q hc |
| (fun i => (hw i).le) hw1 (fun i => (hw i).le) hw1 hr he B hB0 hB1 hP hQ hsp |
| have hsum : (∑ a, c a*|h a.1-g a.2|)=∑ i, w i*|h i-g i| := by |
| simp only [Fintype.sum_prod_type,c,ite_mul,zero_mul] |
| simp |
| have hsum' : (∑ a, c a*|g a.1-h a.2|)=∑ i, w i*|h i-g i| := by |
| simp only [Fintype.sum_prod_type,c,ite_mul,zero_mul] |
| simp [abs_sub_comm] |
| rw [hsum] at H |
| rw [hsum'] at H' |
| unfold finiteSphericalDualObjective |
| exact abs_le.mpr ⟨by linarith,by linarith⟩ |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
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| end OAI |
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