| 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) |
|
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| 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 |
|
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| 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)) |
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| 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 |
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| 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) |
|
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| 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 |
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| end |
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| end OAI |
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