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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | import OAI.Geometry.PinnedDistances.EmpiricalCompactness
namespace OAI
universe u1 u2 u3 u4 u5 u6 u7
section
noncomputable section
open MeasureTheory Filter
open scoped BigOperators Topology ENNReal
namespace WeakPinned.Bounded
variable {I : Type u1} {V : Type u2} {X : Type u3} {Y : Type u4} [Fintype I] [Fintype V]
[MeasurableSpace X] [MeasurableSpace Y]
lemma finiteLaw_fst (w : I → ℝ) (p : I → V → ℝ) (U : I → X)
(hw : ∀ i,0≤w i) (hp : ∀ i x,0≤p i x) (hpsum : ∀ i,∑ x,p i x=1) :
finiteLaw (edgeWeight w p) (fun r => U r.1)=finiteLaw w U := by
unfold finiteLaw edgeWeight
rw [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro i _
dsimp only
rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun x _ => mul_nonneg (hw i) (hp i x)),
← Finset.mul_sum,hpsum,mul_one]
lemma tripleLaw_edge (w : I → ℝ) (p : I → V → ℝ) (U : I × V → X)
(hw : ∀ i,0≤w i) (hp : ∀ i x,0≤p i x) (hpsum : ∀ i,∑ x,p i x=1) :
finiteLaw (tripleWeight w p) (fun r => U (r.1,r.2.1))=finiteLaw (edgeWeight w p) U := by
unfold finiteLaw tripleWeight edgeWeight
simp only [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro x _
rw [show (∑ z,∑ t, ENNReal.ofReal (w i*p i x*p i z*p i t) • Measure.dirac (U (i,x))) =
∑ z,ENNReal.ofReal (w i*p i x*p i z) • Measure.dirac (U (i,x)) by
apply Finset.sum_congr rfl
intro z _
rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun t _ => mul_nonneg (mul_nonneg (mul_nonneg (hw i) (hp i x)) (hp i z)) (hp i t)),
← Finset.mul_sum,hpsum,mul_one]]
rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun z _ => mul_nonneg (mul_nonneg (hw i) (hp i x)) (hp i z)),
← Finset.mul_sum,hpsum,mul_one]
lemma finiteLaw_mono (w v : I → ℝ) (U : I → X) (hle : ∀ i,w i≤v i) :
finiteLaw w U≤finiteLaw v U := by
apply Measure.le_iff.mpr
intro s hs
simp only [finiteLaw,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul]
apply Finset.sum_le_sum
intro i _
gcongr
exact hle i
lemma finiteLaw_prod (w : I → ℝ) (v : V → ℝ) (U : I → X) (W : V → Y)
(hw : ∀ i,0≤w i) :
(finiteLaw w U).prod (finiteLaw v W)=
finiteLaw (fun r : I × V => w r.1*v r.2) (fun r => (U r.1,W r.2)) := by
simp only [finiteLaw,← Measure.sum_fintype,Measure.prod_sum,
Measure.prod_smul_left,Measure.prod_smul_right,Measure.dirac_prod_dirac,
smul_smul,ENNReal.ofReal_mul (hw _),mul_comm]
lemma finiteLaw_smul (w : I → ℝ) (U : I → X) (c : ℝ) (hc : 0≤c) :
finiteLaw (fun i => c*w i) U=ENNReal.ofReal c •finiteLaw w U := by
simp only [finiteLaw,ENNReal.ofReal_mul hc,Finset.smul_sum,smul_smul]
end WeakPinned.Bounded
end
end
section
noncomputable section
open MeasureTheory Filter
open scoped BigOperators Topology ENNReal
namespace WeakPinned.Bounded
open Variance
variable {V : Type u5} {I : Type u6} {X : Type u7} [Fintype V] [DecidableEq V] [Fintype I]
[MeasurableSpace X] [MeasurableSingletonClass X]
[TopologicalSpace X] [BorelSpace X] [HasOuterApproxClosed X]
variable (G : Core.Fibers V I)
lemma graph_edge_eval (F : V → V → ℝ) :
(∑ e,G.fiberWeight e*∑ y,prob (G.targets e) y*F (G.source e) y)=
∑ x,∑ y,G.edgeLaw x y*F x y := by
rw [← G.incoming_eval]
simp only [Finset.mul_sum,← mul_assoc,G.sampling_conditional]
exact Finset.sum_comm
lemma graph_edge_law (U : V → V → X) (hG : 0<G.edgeCount)
(hB : ∀ e,(G.targets e).Nonempty) :
finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e)))
(fun r => U (G.source r.1) r.2)=
finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => U r.1 r.2) := by
have hw (r : I × V) : 0≤edgeWeight G.fiberWeight (fun e => prob (G.targets e)) r :=
mul_nonneg (G.fiberWeight_nonneg _) (Core.Fibers.prob_nn _ _)
let : IsProbabilityMeasure (finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e)))
(fun r => U (G.source r.1) r.2)) := finiteLaw_probability _ _ hw
(sum_edgeWeight _ _ (G.sum_fiberWeight hG) (fun e => sum_prob _ (hB e)))
let : IsProbabilityMeasure (finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => U r.1 r.2)) :=
finiteLaw_probability _ _ (fun r => G.edgeLaw_nonneg _ _)
(by simpa only [Fintype.sum_prod_type] using G.sum_edgeLaw hG)
apply ext_of_forall_integral_eq_of_IsFiniteMeasure
intro f
rw [integral_finiteLaw _ _ hw,integral_finiteLaw _ _ (fun r => G.edgeLaw_nonneg _ _)]
simp only [edgeWeight,Fintype.sum_prod_type,mul_assoc,← Finset.mul_sum]
exact graph_edge_eval G (fun x y => f (U x y))
omit G in
lemma pairProb_le_uniform (hn : 2≤Fintype.card V) (x y : V) :
pairProb Finset.univ x y≤2*(Fintype.card V:ℝ)⁻¹*(Fintype.card V:ℝ)⁻¹ := by
have hnR : (2:ℝ)≤Fintype.card V := by exact_mod_cast hn
have hn0 : (0:ℝ)<Fintype.card V := by linarith
simp only [pairProb,Finset.mem_univ,true_and,Finset.card_univ]
split_ifs
· field_simp
apply (div_le_iff₀ (by linarith : 0<(Fintype.card V:ℝ)-1)).mpr
nlinarith
· positivity
omit [Fintype I] [MeasurableSingletonClass X] [TopologicalSpace X] [BorelSpace X] [HasOuterApproxClosed X] in
lemma graph_domination (U : V → X) (hn : 2≤Fintype.card V) (M : ℝ) (hM : 0≤M)
(hdom : ∀ x y,G.edgeLaw x y≤M*pairProb Finset.univ x y) :
finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => (U r.1,U r.2))≤
ENNReal.ofReal (2*M) • (uniformLaw U).prod (uniformLaw U) := by
rw [uniformLaw,finiteLaw_prod _ _ _ _ (fun _ => by positivity),← finiteLaw_smul _ _ _ (by positivity)]
apply finiteLaw_mono
intro r
calc
_ ≤ M*pairProb Finset.univ r.1 r.2 := hdom _ _
_ ≤ M*(2*(Fintype.card V:ℝ)⁻¹*(Fintype.card V:ℝ)⁻¹) :=
mul_le_mul_of_nonneg_left (pairProb_le_uniform hn _ _) hM
_ = _ := by ring
end WeakPinned.Bounded
end
end
end OAI
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