| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.PrimitiveExpectedField |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory |
| open scoped Classical BigOperators NNReal |
|
|
| inductive MeanTree (X : Type*) [MeasurableSpace X] (d : ℕ) where |
| | node (k : ℕ) (base : X → Point d) (measurable_base : Measurable base) |
| (coeff scale : Fin k → ℝ) (center : Fin k → MeanTree X d) |
|
|
| namespace MeanTree |
| variable {X : Type*} [MeasurableSpace X] {d : ℕ} |
|
|
| def eval (F : Point d → ℝ) : MeanTree X d → X → Point d |
| | .node _ b _ a r E,z => b z+∑i,a i • primitiveExpectedField F (eval F (E i) z) (r i) |
|
|
| def depth : MeanTree X d → ℕ |
| | .node _ _ _ _ _ E => Finset.univ.sup (fun i => depth (E i)+1) |
|
|
| def terms : MeanTree X d → ℕ |
| | .node k .. => k |
|
|
| def weight : MeanTree X d → ℝ |
| | .node _ _ _ a _ _ => ∑i,|a i| |
|
|
| def anchor (b : X → Point d) (hb : Measurable b) : MeanTree X d := |
| .node 0 b hb Fin.elim0 Fin.elim0 Fin.elim0 |
|
|
| def zero : MeanTree X d := anchor (fun _ => 0) measurable_const |
|
|
| def scale (s : ℝ) : MeanTree X d → MeanTree X d |
| | .node k b hb a r E => .node k (fun z => s • b z) (by fun_prop) (fun i => s*a i) r E |
|
|
| def add : MeanTree X d → MeanTree X d → MeanTree X d |
| | .node k b hb a r E,.node l c hc v t D => |
| .node (k+l) (fun z => b z+c z) (by fun_prop) |
| (Fin.addCases a v) (Fin.addCases r t) (Fin.addCases E D) |
|
|
| def mean (r : ℝ) (E : MeanTree X d) : MeanTree X d := |
| .node 1 (fun _ => 0) measurable_const (fun _ => 1) (fun _ => r) (fun _ => E) |
|
|
| @[simp] theorem eval_anchor (F : Point d → ℝ) (b : X → Point d) (hb : Measurable b) (z : X) : |
| eval F (anchor b hb) z=b z := by simp [eval,anchor] |
| @[simp] theorem eval_zero (F : Point d → ℝ) (z : X) : eval F zero z=0 := by simp [zero] |
| @[simp] theorem eval_scale (F : Point d → ℝ) (s : ℝ) (E : MeanTree X d) (z : X) : |
| eval F (scale s E) z=s • eval F E z := by |
| cases E |
| simp only [eval,scale,smul_add,Finset.smul_sum,smul_smul] |
| @[simp] theorem eval_add (F : Point d → ℝ) (E D : MeanTree X d) (z : X) : |
| eval F (add E D) z=eval F E z+eval F D z := by |
| cases E; cases D |
| simp only [eval,add,Fin.sum_univ_add,Fin.addCases_left,Fin.addCases_right] |
| abel |
| @[simp] theorem eval_mean (F : Point d → ℝ) (r : ℝ) (E : MeanTree X d) (z : X) : |
| eval F (mean r E) z=primitiveExpectedField F (eval F E z) r := by |
| simp [mean,eval] |
|
|
| @[simp] theorem depth_anchor (b : X → Point d) (hb : Measurable b) : depth (anchor b hb)=0 := by |
| simp [anchor,depth] |
| @[simp] theorem depth_zero : depth (zero : MeanTree X d)=0 := by simp [zero] |
| @[simp] theorem depth_scale (s : ℝ) (E : MeanTree X d) : depth (scale s E)=depth E := by |
| cases E; rfl |
| @[simp] theorem depth_mean (r : ℝ) (E : MeanTree X d) : depth (mean r E)=depth E+1 := by |
| simp [mean,depth] |
|
|
| theorem depth_add_le (E D : MeanTree X d) {N : ℕ} (hE : depth E≤N) (hD : depth D≤N) : |
| depth (add E D)≤N := by |
| cases E with | node k b hb a r C => |
| cases D with | node l c hc v t B => |
| simp only [depth,Finset.sup_le_iff,Finset.mem_univ,true_implies] at hE hD |
| simp only [add,depth,Finset.sup_le_iff,Finset.mem_univ,true_implies] |
| intro i |
| refine Fin.addCases (fun j => ?_) (fun j => ?_) i |
| · simpa only [Fin.addCases_left] using hE j |
| · simpa only [Fin.addCases_right] using hD j |
|
|
| def sumFin : (k : ℕ) → (Fin k → MeanTree X d) → MeanTree X d |
| | 0,_ => zero |
| | k+1,E => add (E 0) (sumFin k (fun i => E i.succ)) |
|
|
| @[simp] lemma eval_sumFin (F : Point d → ℝ) (k : ℕ) (E : Fin k → MeanTree X d) (z : X) : |
| eval F (sumFin k E) z=∑i,eval F (E i) z := by |
| induction k with |
| | zero => simp [sumFin] |
| | succ k ih => simp only [sumFin,eval_add,ih,Fin.sum_univ_succ] |
|
|
| lemma depth_sumFin_le (k : ℕ) (E : Fin k → MeanTree X d) {N : ℕ} |
| (hE : ∀i,depth (E i)≤N) : depth (sumFin k E)≤N := by |
| induction k with |
| | zero => simp [sumFin] |
| | succ k ih => exact depth_add_le (E 0) (sumFin k _) (hE 0) (ih _ (fun i => hE i.succ)) |
|
|
| def sumFamily {I : Type*} [Fintype I] (E : I → MeanTree X d) : MeanTree X d := |
| sumFin (Fintype.card I) (fun i => E ((Fintype.equivFin I).symm i)) |
|
|
| @[simp] lemma eval_sumFamily {I : Type*} [Fintype I] (F : Point d → ℝ) |
| (E : I → MeanTree X d) (z : X) : |
| eval F (sumFamily E) z=∑i,eval F (E i) z := by |
| rw [sumFamily,eval_sumFin] |
| exact (Fintype.sum_equiv (Fintype.equivFin I).symm _ _ (fun _ => rfl)) |
| lemma depth_sumFamily_le {I : Type*} [Fintype I] (E : I → MeanTree X d) {N : ℕ} |
| (hE : ∀i,depth (E i)≤N) : depth (sumFamily E)≤N := |
| depth_sumFin_le _ _ (fun _ => hE _) |
|
|
| def conditional (r ρ : ℝ) (x y : MeanTree X d) : MeanTree X d := |
| mean (r*Real.sqrt (1-ρ^2)) (add x (scale (r*ρ) y)) |
|
|
| lemma conditionalFieldMean_primitive {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) (r ρ : ℝ) (y : Point d) (ha : 0<1-ρ^2) : |
| conditionalFieldMean F x r ρ y= |
| primitiveExpectedField F (x+(r*ρ) • y) (r*Real.sqrt (1-ρ^2)) := by |
| rw [conditionalFieldMean,←conditionalLaw_map hF x r ρ y ha] |
| rw [integral_map (by fun_prop) (primitiveField_contDiff hF x r).continuous.aestronglyMeasurable] |
| unfold primitiveExpectedField |
| simp only [primitiveField,smul_add,smul_smul] |
| congr 1 |
| funext z |
| congr 1 |
| abel |
|
|
| @[simp] lemma eval_conditional {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| (r ρ : ℝ) (x y : MeanTree X d) (z : X) (ha : 0<1-ρ^2) : |
| eval F (conditional r ρ x y) z=conditionalFieldMean F (eval F x z) r ρ (eval F y z) := by |
| simp only [conditional,eval_mean,eval_add,eval_scale] |
| exact (conditionalFieldMean_primitive hF _ r ρ _ ha).symm |
|
|
| lemma depth_conditional_le (r ρ : ℝ) (x y : MeanTree X d) {N : ℕ} |
| (hx : depth x≤N) (hy : depth y≤N) : depth (conditional r ρ x y)≤N+1 := by |
| simp only [conditional,depth_mean] |
| exact Nat.add_le_add_right (depth_add_le _ _ hx (by simpa using hy)) 1 |
|
|
| end MeanTree |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
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| end OAI |
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