| import Mathlib |
| 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 |
| |
