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import OAI.Probability.Perceptron.Variational.LabelValue
import OAI.Probability.Perceptron.Control.StepControl
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology ENNReal NNReal BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
lemma oneDim_inner (v : EuclideanSpace ℝ (Fin 1)) :
inner ℝ (WithLp.toLp 2 (fun _ : Fin 1 => (1:ℝ))) v=v 0 := by
simp [PiLp.inner_apply]
lemma oneDim_gaussian_average (s : ℝ≥0) (f : ℝ →ᵇ ℝ) (a : ℝ) :
(∫ v : EuclideanSpace ℝ (Fin 1), f (a+Real.sqrt s*v 0)
∂stdGaussian (EuclideanSpace ℝ (Fin 1)))=gaussianAverage s f a := by
rw [gaussianAverage_scaling]
have h : (∫ z, f (a+Real.sqrt s*z) ∂((stdGaussian (EuclideanSpace ℝ (Fin 1))).map
(fun v => v 0)))=∫ v : EuclideanSpace ℝ (Fin 1), f (a+Real.sqrt s*v 0)
∂stdGaussian (EuclideanSpace ℝ (Fin 1)) :=
integral_map (show Measurable (fun v : EuclideanSpace ℝ (Fin 1) => v 0) from by fun_prop).aemeasurable
(show Measurable (fun z : ℝ => f (a+Real.sqrt s*z)) from by fun_prop).aestronglyMeasurable
rw [stdGaussian_coordinate_law] at h
exact h.symm
lemma oneDim_gaussian_entropic (s b : ℝ≥0) (f : ℝ →ᵇ ℝ) (x : EuclideanSpace ℝ (Fin 1)) :
gaussianLinearEntropic (stdGaussian (EuclideanSpace ℝ (Fin 1))) b
(diagonalMark (fun _ => Real.sqrt s)) (fun v => f (v 0)) x=heatLog s b f (x 0) := by
have hX : ∀ a : ℝ, Integrable
(fun v : EuclideanSpace ℝ (Fin 1) => Real.exp (a*f (x 0+Real.sqrt s*v 0)))
(stdGaussian (EuclideanSpace ℝ (Fin 1))) := fun a =>
boundedContinuousFunction_integrable_comp (expBCF a f) (by fun_prop)
change entropicMean (stdGaussian (EuclideanSpace ℝ (Fin 1))) b
(fun v => f (x 0+Real.sqrt s*v 0))=heatLog s b f (x 0)
by_cases hb : b=0
· subst b
rw [NNReal.coe_zero,entropicMean_zero _ hX,heatLog,ite_eq_left rfl,oneDim_gaussian_average]
· rw [entropicMean_eq_div _ hX (by exact_mod_cast hb),heatLog,ite_eq_right hb]
unfold cgf mgf
change Real.log (∫ v : EuclideanSpace ℝ (Fin 1), expBCF b f (x 0+Real.sqrt s*v 0)
∂stdGaussian (EuclideanSpace ℝ (Fin 1)))/(b:ℝ)=_
rw [oneDim_gaussian_average]
ring
lemma linearCascadeWord_congr {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
(A B : ℕ→E→L[ℝ] E) (k : ℕ) (z : Fin k→ℝ) (h : ∀ j<k,A j=B j) :
linearCascadeWord A k z=linearCascadeWord B k z := by
induction k with
| zero => rfl
| succ k ih =>
simp only [linearCascadeWord]
rw [h k (Nat.lt_succ_self k),ih (fun i => z i.succ) (fun j hj => h j (Nat.lt_succ_of_lt hj))]
lemma profileGaussianStep_first {k : ℕ} (q : Fin (k+2)→Time) :
profileGaussianStep (fun l (_ : Fin 1) => (q l:ℝ)) k=
fun _ => Real.sqrt (timeSpan (q 0) (q 1)) := by
funext i
simp only [profileGaussianStep,dite_eq_left (Nat.lt_succ_self k)]
have h0 : (⟨k+1-k-1,by omega⟩ : Fin (k+2))=0 := by ext; simp
have h1 : (⟨k+1-k,by omega⟩ : Fin (k+2))=1 := by ext; simp
rw [h0,h1]
simp only [timeSpan,Real.coe_toNNReal']
by_cases h : 0≤(q 1:ℝ)-q 0
· rw [max_eq_left h]
· have hn : (q 1:ℝ)-q 0≤0 := by linarith
rw [max_eq_right hn,Real.sqrt_eq_zero_of_nonpos hn,Real.sqrt_zero]
lemma profileGaussianStep_tail {k : ℕ} (q : Fin (k+2)→Time) {j : ℕ} (hj : j<k) :
profileGaussianStep (fun l (_ : Fin 1) => (q l:ℝ)) j=
profileGaussianStep (fun l (_ : Fin 1) => (q l.succ:ℝ)) j := by
funext i
simp only [profileGaussianStep,dite_eq_left hj,dite_eq_left (Nat.lt_succ_of_lt hj)]
have ha : (⟨k+1-j,by omega⟩ : Fin (k+2))=(⟨k-j,by omega⟩ : Fin (k+1)).succ := by
ext; simp only [Fin.val_succ]; omega
have hb : (⟨k+1-j-1,by omega⟩ : Fin (k+2))=(⟨k-j-1,by omega⟩ : Fin (k+1)).succ := by
ext; simp only [Fin.val_succ]; omega
rw [ha,hb]
lemma oneDim_profile_backward (g : Jet3) (k : ℕ) (q : Fin (k+1)→Time)
(z : Fin k→ℝ≥0) (x : EuclideanSpace ℝ (Fin 1)) :
gaussianLinearBackward (stdGaussian (EuclideanSpace ℝ (Fin 1)))
(linearCascadeWord (fun j => diagonalMark (profileGaussianStep (fun l (_ : Fin 1) => (q l:ℝ)) j))
k (fun i => z i)) (fun v => g.f (v 0)) x=(finiteStepControlJet g k q z).f (x 0) := by
induction k generalizing x with
| zero => rfl
| succ k ih =>
simp only [linearCascadeWord,gaussianLinearBackward,finiteStepControlJet]
rw [profileGaussianStep_first]
have hw := linearCascadeWord_congr
(fun j => diagonalMark (profileGaussianStep (fun l (_ : Fin 1) => (q l:ℝ)) j))
(fun j => diagonalMark (profileGaussianStep (fun l (_ : Fin 1) => (q l.succ:ℝ)) j))
k (fun i => (z i.succ:ℝ)) (fun j hj => congrArg diagonalMark (profileGaussianStep_tail q hj))
rw [hw]
have hf := funext (ih (fun i => q i.succ) (fun i => z i.succ))
rw [hf,oneDim_gaussian_entropic,Jet3.heatLog_f]
change _=heatLog _ _ _ _
rfl
lemma HeatChain.prepend {m : Trial} {g h j : Jet3} {a b c : Time}
(H : HeatChain m g b c h) (J : HeatChain m h a b j) : HeatChain m g a c j := by
induction J with
| nil => exact H
| cons d hat htb hm tail ih =>
exact HeatChain.cons d hat (htb.trans H.left_le_right) hm (ih H)
lemma weightedStepTrial_controlValue_core (k : ℕ) (w : Fin (k+1)→ℝ) (q : Fin (k+1)→Time)
(hw : ∀ i, 0≤w i) (hw1 : ∑ i,w i=1) (hq : Monotone q) (g : Jet3)
(P : Measure BrownianPath) [IsProbabilityMeasure P] (hB : IsBrownianReal brownianEval P) :
controlValue P g.f (weightedStepTrial w q hw hw1)=
((finiteStepControlJet (g.heatLog (timeSpan (q (Fin.last k)) 1) 1) k q
(weightedControlCoefficient k w hw)).heatLog (timeSpan 0 (q 0)) 0).f 0 := by
let m := leftWeightedStepTrial w q hw hw1
have ht : HeatChain m g (q (Fin.last k)) 1 (g.heatLog (timeSpan (q (Fin.last k)) 1) 1) := by
rcases lt_or_eq_of_le (show q (Fin.last k)≤1 from le_top) with hl|he
· apply HeatChain.cons 1 hl le_rfl _ (HeatChain.nil 1)
intro r hr
change (∑ i, if q i<r then w i else 0)=(1:ℝ)
have hqr i : q i<r := (hq (Fin.le_last i)).trans_lt hr.1
simp only [hqr,ite_true,hw1]
· rw [he]
simp only [timeSpan,sub_self,Real.toNNReal_zero,Jet3.heatLog_zero_time]
exact HeatChain.nil 1
have hm := finiteStepControlJet_chain (g.heatLog (timeSpan (q (Fin.last k)) 1) 1)
m k q (weightedControlCoefficient k w hw) hq
(fun i r hr => leftWeightedStepTrial_interval k w q hw hw1 hq i hr)
have hm' := ht.prepend hm
have H : HeatChain m g 0 1 ((finiteStepControlJet (g.heatLog (timeSpan (q (Fin.last k)) 1) 1)
k q (weightedControlCoefficient k w hw)).heatLog (timeSpan 0 (q 0)) 0) := by
rcases lt_or_eq_of_le (show (0:Time)≤q 0 from bot_le) with hl|he
· apply HeatChain.cons 0 hl (show q 0≤1 from le_top) _ hm'
intro r hr
change (∑ i, if q i<r then w i else 0)=(0:ℝ)
have hqr i : ¬ q i<r := not_lt_of_ge (hr.2.trans (hq (Fin.zero_le i)))
simp only [hqr,ite_false,Finset.sum_const_zero]
· rw [←he] at hm' ⊢
simpa only [timeSpan,sub_self,Real.toNNReal_zero,Jet3.heatLog_zero_time] using hm'
have he := leftWeightedStepTrial_ae w q hw hw1
have hv := controlValue_trial_abs_sub_le P g.f (weightedStepTrial w q hw hw1) m ‖g.d1‖₊ g.lipschitz
have hz : (∫ t, |weightedStepTrial w q hw hw1 t-m t| ∂timeLaw)=0 := by
apply integral_eq_zero_of_ae
filter_upwards [he] with t ht
simp only [m,ht,sub_self,abs_zero,Pi.zero_apply]
rw [hz,mul_zero] at hv
rw [sub_eq_zero.mp (abs_eq_zero.mp (le_antisymm hv (abs_nonneg _)))]
exact H.controlValue_eq P hB
theorem labelProfileValue_control (k : ℕ) (w : Fin (k+1)→ℝ) (q : Fin (k+1)→Time)
(hw : ∀ i, 0≤w i) (hw1 : ∑ i,w i=1) (hq : Monotone q) (g : Jet3)
(P : Measure BrownianPath) [IsProbabilityMeasure P] (hB : IsBrownianReal brownianEval P) :
labelProfileValue k (stepCumulative w) (fun i => (q i:ℝ)) (q (Fin.last k)).prop.2 g=
controlValue P g.f (weightedStepTrial w q hw hw1) := by
rw [weightedStepTrial_controlValue_core k w q hw hw1 hq g P hB]
have hend : (⟨1-(q (Fin.last k):ℝ),sub_nonneg.mpr (q (Fin.last k)).prop.2⟩ : ℝ≥0)=
timeSpan (q (Fin.last k)) 1 := by
apply Subtype.ext
change 1-(q (Fin.last k):ℝ)=(timeSpan (q (Fin.last k)) 1:ℝ)
exact (timeSpan_coe (show q (Fin.last k)≤1 from le_top)).symm
have hstart : (timeSpan 0 (q 0):ℝ)=(q 0:ℝ) := by
rw [timeSpan_coe (show (0:Time)≤q 0 from bot_le)]; simp
have hterm : labelProfileTerminal g (timeSpan (q (Fin.last k)) 1)=
fun v : EuclideanSpace ℝ (Fin 1) => (g.heatLog (timeSpan (q (Fin.last k)) 1) 1).f (v 0) := by
funext v
rw [labelProfileTerminal,oneDim_inner,Jet3.heatLog_f]
rfl
have hc : stepCumulative w=fun i => (weightedControlCoefficient k w hw i:ℝ) := rfl
unfold labelProfileValue
rw [hend,hterm,hc]
simp_rw [oneDim_profile_backward]
simp only [diagonalMark_apply,profileGaussianRoot]
rw [Jet3.heatLog_f]
change (∫ x : EuclideanSpace ℝ (Fin 1),
(finiteStepControlJet (g.heatLog (timeSpan (q (Fin.last k)) 1) 1) k q
(weightedControlCoefficient k w hw)).f (Real.sqrt (q 0:ℝ)*x 0)
∂stdGaussian (EuclideanSpace ℝ (Fin 1)))=heatLog (timeSpan 0 (q 0)) 0 _ 0
rw [heatLog,ite_eq_left rfl]
simpa only [hstart,zero_add] using oneDim_gaussian_average (timeSpan 0 (q 0))
(finiteStepControlJet (g.heatLog (timeSpan (q (Fin.last k)) 1) 1) k q
(weightedControlCoefficient k w hw)).f 0
end SphericalPerceptronFreeEnergy
end
end OAI
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