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a2ee19a | 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 Mathlib
import OAI.Probability.Perceptron.Cavity.CavityFullEvaluation
import OAI.Probability.Perceptron.Cascade.BulkRPCCoupling
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Classical ENNReal NNReal BigOperators Topology BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
def roundedCavityPair (η : Measure Time) [IsProbabilityMeasure η] (B : ℝ)
{K a : ℝ} (ha0 : 0 ≤ a) (haK : a ≤ K) (j : ℕ) :
Fin ((strictRoundModel (unitQuantileField (boundedQuantile η) (boundedQuantile_monotone η)) j).depth+1) → BulkPairRange K :=
fun i => continuousCavityProfile η B ha0 haK
(roundTimeValue (unitQuantileField (boundedQuantile η) (boundedQuantile_monotone η)) j i)
lemma roundedCavityPair_conditions (n d : ℕ) (η : Measure Time) [IsProbabilityMeasure η]
{B K a : ℝ} (hB : B < 1) (ha0 : 0 ≤ a) (haK : a ≤ K) (j : ℕ) :
(∀ i, 0 ≤ cavityPairProfile n d (roundedCavityPair η B ha0 haK j) 0 i) ∧
(∀ i, Monotone (fun l => cavityPairProfile n d (roundedCavityPair η B ha0 haK j) l i)) ∧
(∀ i, cavityPairProfile n d (roundedCavityPair η B ha0 haK j)
(Fin.last _) i ≤ cavityPairDiagonal n d (cavityTrueDiagonal ha0 haK) i) := by
have h0 := fun l => continuousCavityProfile_nonneg η B ha0 haK
(roundTimeValue (unitQuantileField (boundedQuantile η) (boundedQuantile_monotone η)) j l)
have hD := fun l => continuousCavityProfile_le_diagonal η B ha0 haK
(roundTimeValue (unitQuantileField (boundedQuantile η) (boundedQuantile_monotone η)) j l)
refine ⟨?_,?_,?_⟩
· intro i; cases i with
| inl i => exact (h0 0).2
| inr i => exact (h0 0).1
· intro i; cases i with
| inl i => exact (continuousCavityProfile_mono η hB ha0 haK).comp (roundTimeValue_mono _ j)
| inr i => exact fun l m h => roundTimeValue_mono _ j h
· intro i; cases i with
| inl i => exact (hD (Fin.last _)).2
| inr i => exact (hD (Fin.last _)).1
def roundedCavityLog (n d : ℕ) (f : ℝ →ᵇ ℝ) (Λ : ℝ) (hΛ : 1 ≤ Λ)
(η : Measure Time) [IsProbabilityMeasure η] (B : ℝ) {K a : ℝ}
(ha0 : 0 ≤ a) (haK : a ≤ K) (j : ℕ) : ℝ :=
let F := unitQuantileField (boundedQuantile η) (boundedQuantile_monotone η)
let k := (strictRoundModel F j).depth
let p := roundedCavityPair η B ha0 haK j
let D := cavityTrueDiagonal ha0 haK
∫ y,finiteCascadeLogRecursion (gaussianMarkLaw (E:=EuclideanSpace ℝ (Fin (n+1)⊕Fin d)))
(gaussianLinearMarkStep (fun l=>diagonalMark (profileGaussianStep (cavityPairProfile n d p) l))) k
(stepCumulative (strictRoundModel F j).weight)
(fun t=>Real.log (cavityDiagonalAverage
(fun i=>Real.sqrt (cavityPairDiagonal n d D i-cavityPairProfile n d p (Fin.last k) i))
(cavitySingleTest n d f Λ hΛ) (t 0)))
(fun _=>diagonalMark (profileGaussianRoot (cavityPairProfile n d p)) y)
∂stdGaussian (EuclideanSpace ℝ (Fin (n+1)⊕Fin d))
theorem bulk_cavity_rounded_recursion (M : ℕ→ℕ) (g : Jet3) (v : ℕ→ℕ→ℝ)
(hv : ∀ n p,1≤v n p) {C : ℝ} (hC : 0≤C) (hvC : ∀ n p,|v n (p+1)|≤C)
(hdO : ∀ p,Tendsto (fun n=>bulkDeviationAt n (M n) g.f p (v n)) atTop (𝓝 0))
(hdF : ∀ p,Tendsto (fun n=>bulkDeviationAt n (M n+2) g.f p (v n)) atTop (𝓝 0))
(s : ℕ→ℕ) (hs : StrictMono s) (α : ℝ) (hα : 0≤α)
(hd : Tendsto (fun n=>((M (s n)+2:ℕ):ℝ)/(s n+1:ℕ)) atTop (𝓝 α))
(ν : ProbabilityMeasure (CompactArray CompactOverlap))
(ho : Tendsto (fun n=>bulkGibbsArrayLaw (s n) (M (s n)) g.f (v (s n))) atTop (𝓝 ν))
(K : ℝ) (hK0 : 0≤K) (hK : ∀ n,((M n+2:ℕ):ℝ)/(n+1:ℕ)*‖g.d1‖^2≤K)
(μ : ProbabilityMeasure (CompactArray (BulkPairRange K)))
(hf : Tendsto (fun n=>bulkMarkedArrayLaw (s n) (M (s n)+2) g (v (s n)) K (hK (s n))) atTop (𝓝 μ)) :
∃ a : ℝ, ∃ ha0 : 0 ≤ a, ∃ haK : a ≤ K, a ≤ α*‖g.d1‖^2 ∧
∀ n d (Λ : ℝ) (hΛ : 1 ≤ Λ),
Tendsto (fun j => cavityBulkLog n d (s j) (M (s j)+2) g (v (s j)) Λ hΛ -
roundedCavityLog n d g.f Λ hΛ (markedTimePair K μ)
((α*‖g.d1‖^2)/(1+α*‖g.d1‖^2)) ha0 haK j) atTop (𝓝 0) := by
obtain ⟨a,ha0,haK,haA,hrpc⟩ := bulk_marked_rpc_coupling M g v hv hC hvC hdO hdF s hs
α hα hd ν ho K hK0 (fun j => hK (s j)) μ hf
refine ⟨a,ha0,haK,haA,?_⟩
intro n d Λ hΛ
let η := markedTimePair K μ
let F := unitQuantileField (boundedQuantile η) (boundedQuantile_monotone η)
let B := (α*‖g.d1‖^2)/(1+α*‖g.d1‖^2)
have hA := mul_nonneg hα (sq_nonneg ‖g.d1‖)
have hB : B < 1 := (div_lt_one (by linarith : 0 < 1+α*‖g.d1‖^2)).mpr (by linarith)
have hp j := roundedCavityPair_conditions n d η hB ha0 haK j
exact cavity_joint_array_recursion_limit n d (fun m => M m+2) g v Λ hΛ K hK s hf
(fun j => (strictRoundModel F j).depth) (roundedCavityPair η B ha0 haK)
(fun _ => cavityTrueDiagonal ha0 haK) (fun j => stepCumulative (strictRoundModel F j).weight)
(fun j => (hp j).1) (fun j => (hp j).2.1) (fun j => (hp j).2.2)
(fun j => stepCumulative_strictMono _ (strictRoundModel F j).weight_pos)
(fun j => stepCumulative_pos _ (strictRoundModel F j).weight_pos)
(fun j => stepCumulative_lt_one _ (strictRoundModel F j).weight_pos (strictRoundModel F j).weight_sum) hrpc
def roundedCavityValue (n d : ℕ) (g : Jet3) (P : Measure BrownianPath)
(η : Measure Time) [IsProbabilityMeasure η] (B : ℝ) {K a : ℝ}
(ha0 : 0≤a) (haK : a≤K) (j : ℕ) : ℝ :=
let F:=unitQuantileField (boundedQuantile η) (boundedQuantile_monotone η)
let M:=strictRoundModel F j
(n+1:ℕ)*(a/2+finiteSphericalFieldValue n M.depth M.weight
(fun l=>(roundedCavityPair η B ha0 haK j l).2.val/2))+
d*controlValue P g.f (weightedStepTrial M.weight (roundTimeValue F j)
(fun l=>(M.weight_pos l).le) M.weight_sum)
theorem roundedCavity_uncap (n d : ℕ) (g : Jet3) (P : Measure BrownianPath)
[IsProbabilityMeasure P] (hBrow : IsBrownianReal brownianEval P)
(η : Measure Time) [IsProbabilityMeasure η] {B K a : ℝ} (hB : B<1)
(ha0 : 0≤a) (haK : a≤K) (j : ℕ) {Λ : ℝ} (hΛ : 1≤Λ) :
0≤roundedCavityValue n d g P η B ha0 haK j-roundedCavityLog n d g.f Λ hΛ η B ha0 haK j ∧
roundedCavityValue n d g P η B ha0 haK j-roundedCavityLog n d g.f Λ hΛ η B ha0 haK j≤
Real.exp (2*((d:ℝ)*‖g.f‖))/Λ*Real.exp (2*(n+1:ℕ)*a) := by
let F:=unitQuantileField (boundedQuantile η) (boundedQuantile_monotone η)
let M:=strictRoundModel F j
let p:=roundedCavityPair η B ha0 haK j
let D:=cavityTrueDiagonal ha0 haK
let Q:=cavityPairProfile n d p
let σ:=fun i=>Real.sqrt (cavityPairDiagonal n d D i-Q (Fin.last M.depth) i)
let z:=stepCumulative M.weight
have hp:=roundedCavityPair_conditions n d η hB ha0 haK j
have hfull:=cavityLabel_full_value n d p D hp.1 hp.2.1 hp.2.2
(roundTimeValue F j) (fun l=>rfl) rfl M.weight M.weight_pos M.weight_sum
(roundTimeValue_mono F j) g P hBrow
have hcap:=cavityLabelPartitionLaw_log_recursion n d M.depth g.f Λ hΛ σ Q z
(stepCumulative_strictMono M.weight M.weight_pos)
(stepCumulative_pos M.weight M.weight_pos)
(stepCumulative_lt_one M.weight M.weight_pos M.weight_sum)
rw [cavityLabelPartitionLaw_integral _ _ _ _ _ _ _ _ _ Real.log Real.measurable_log] at hcap
have he:=cavityLabel_uncap_annealed n d p D hp.1 hp.2.1 hp.2.2 g.f z hΛ
have hi:=cavityLabel_full_log_integrable n d p D hp.1 hp.2.1 hp.2.2 g.f z
have hc:=cavityLabel_capped_log_integrable n d M.depth g.f Λ hΛ σ Q z
have hint : (∫ t,Real.log (cavityLabelFullPartition n d M.depth g.f σ Q t)-
Real.log (cavityLabelPartition n d M.depth g.f Λ hΛ σ Q t)
∂(indexedCascadeBaseLaw M.depth z : Measure (IndexedCascadeBase M.depth)).prod countableGaussianLaw)=
roundedCavityValue n d g P η B ha0 haK j-roundedCavityLog n d g.f Λ hΛ η B ha0 haK j := by
rw [integral_sub hi hc,hfull,hcap]
rfl
change _ ∧ _ at he
exact ⟨by rw [←hint]; exact he.2.1,by rw [←hint]; exact he.2.2⟩
end SphericalPerceptronFreeEnergy
end
end OAI
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