| import Mathlib |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| namespace KernelPotential |
| local instance namedAtLeastTwoTwo : Nat.AtLeastTwo 2 := ⟨Nat.le_refl 2⟩ |
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| abbrev Ambient (d : ℕ) := EuclideanSpace ℝ (Fin d) |
| abbrev Rows (d : ℕ) := Fin (d/16) → Fin d → ℝ |
| noncomputable def radial {d : ℕ} (x : Ambient d) : Ambient d := ‖x‖⁻¹ • x |
| noncomputable def sphereProbability (d : ℕ) : Measure (Ambient d) := |
| ((volume (ball (0 : Ambient d) 1))⁻¹ • volume.restrict (ball (0 : Ambient d) 1)).map radial |
| noncomputable def J {d : ℕ} (ν : Measure (Ambient d)) (s : Ambient d) : ℝ≥0∞ := |
| ∫⁻ u,((ENNReal.ofReal ‖u-s‖)⁻¹)^(2*(d/16)) ∂ν |
| noncomputable def B {d : ℕ} (ν : Measure (Ambient d)) : ℝ := |
| ∫ s, Real.log ((J ν s).toReal) ∂ν |
| noncomputable def Phi {d : ℕ} (ν : Measure (Ambient d)) : ℝ := |
| (InformationTheory.klDiv ν (sphereProbability d)).toReal - B ν |
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| def BoundedDensity {d : ℕ} (ν : Measure (Ambient d)) : Prop := |
| ∃ H : ℝ, 0 < H ∧ ν ≤ ENNReal.ofReal H • sphereProbability d |
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| noncomputable def rowLaw (d : ℕ) : Measure (Rows d) := |
| Measure.pi (fun _ : Fin (d/16) => Measure.pi (fun _ : Fin d => gaussianReal 0 1)) |
| noncomputable def labels {d : ℕ} (a : Rows d) (s : Ambient d) : Fin (d/16) → ℝ := |
| fun i => ∑ j,a i j*s j |
| structure BlockRule (d N : ℕ) where |
| prob : (Fin N × (Rows d × (Fin (d/16) → ℝ))) → Fin N → ℝ |
| measurable_prob : ∀ w, Measurable (fun x => prob x w) |
| nonneg_prob : ∀ x w, 0 ≤ prob x w |
| sum_prob : ∀ x, ∑ w,prob x w=1 |
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| noncomputable def outgoingLaw {d N : ℕ} (μ : Measure (Ambient d × Fin N)) |
| (rule : BlockRule d N) : Measure (Ambient d × Fin N) := |
| ∑ w : Fin N, ((μ.prod (rowLaw d)).withDensity (fun x => |
| ENNReal.ofReal (rule.prob (x.1.2,x.2,labels x.2 x.1.1) w))).map (fun x => (x.1.1,w)) |
| noncomputable def component {d N : ℕ} (μ : Measure (Ambient d × Fin N)) (v : Fin N) : |
| Measure (Ambient d) := (μ.restrict {x|x.2=v}).map Prod.fst |
| noncomputable def posterior {d N : ℕ} (μ : Measure (Ambient d × Fin N)) (v : Fin N) : |
| Measure (Ambient d) := (component μ v univ)⁻¹ • component μ v |
| noncomputable def averagePhi {d N : ℕ} (μ : Measure (Ambient d × Fin N)) : ℝ := |
| ∑ v : Fin N, (component μ v univ).toReal * Phi (posterior μ v) |
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| def CoercivityStatement : Prop := |
| ∃ C : ℝ, 0 < C ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → |
| (∀ ν : Measure (Ambient d), IsProbabilityMeasure ν → BoundedDensity ν → |
| (1/2:ℝ)*(InformationTheory.klDiv ν (sphereProbability d)).toReal-C*d ≤ Phi ν) ∧ |
| Phi (sphereProbability d) ≤ (2*(d/16):ℕ)*Real.log 2 ∧ |
| (∀ (q : ℕ) (w : Fin q → ℝ), (∀ i,0 ≤ w i) → (∑ i,w i)=1 → |
| ∀ ν : Fin q → Measure (Ambient d), (∀ i,IsProbabilityMeasure (ν i)) → |
| (∀ i,BoundedDensity (ν i)) → |
| Phi (∑ i,ENNReal.ofReal (w i) • ν i) ≤ ∑ i,w i*Phi (ν i)) |
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| def DriftStatement : Prop := |
| ∃ C : ℝ, 0 < C ∧ ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ N : ℕ, 1 ≤ N → |
| Real.log N ≤ (d:ℝ)^2 → ∀ μ : Measure (Ambient d × Fin N), IsProbabilityMeasure μ → |
| μ.map Prod.fst=sphereProbability d → ∀ rule : BlockRule d N, |
| averagePhi (outgoingLaw μ rule) ≤ averagePhi μ+C*d |
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| open Filter |
| open scoped Topology NNReal |
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| end KernelPotential |
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| end OAI |
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