| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators BoundedContinuousFunction |
|
|
| namespace SphericalPerceptronFreeEnergy |
|
|
| abbrev Time := unitInterval |
| abbrev BrownianPath := C(ℝ≥0, ℝ) |
| |
| instance : MeasurableSpace BrownianPath := borel BrownianPath |
| instance : BorelSpace BrownianPath := ⟨rfl⟩ |
| |
| abbrev Spin (N : ℕ) := EuclideanSpace ℝ (Fin N) |
| |
| def patternCount (α : ℝ) (N : ℕ) : ℕ := ⌊α * (N : ℝ)⌋₊ |
| abbrev Patterns (α : ℝ) (N : ℕ) := Fin (patternCount α N) → Fin N → ℝ |
| |
| def patternLaw (α : ℝ) (N : ℕ) : Measure (Patterns α N) := |
| Measure.pi fun _ => Measure.pi fun _ => gaussianReal 0 1 |
| |
| def unitSphereLaw (N : ℕ) : Measure (Metric.sphere (0 : Spin N) 1) := |
| let s := (volume : Measure (Spin N)).toSphere |
| (s Set.univ)⁻¹ • s |
| |
| def sphereLaw (N : ℕ) : Measure (Spin N) := |
| (unitSphereLaw N).map fun x => Real.sqrt (N : ℝ) • x.val |
| |
| def patternField {α : ℝ} {N : ℕ} (g : Patterns α N) |
| (a : Fin (patternCount α N)) (x : Spin N) : ℝ := |
| (∑ i : Fin N, g a i * x i) / Real.sqrt (N : ℝ) |
| |
| def hamiltonian (α : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) (g : Patterns α N) |
| (x : Spin N) : ℝ := |
| ∑ a : Fin (patternCount α N), φ (patternField g a x) |
| |
| def pressure (α β : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) (g : Patterns α N) : ℝ := |
| Real.log (∫ x, Real.exp (β * hamiltonian α φ N g x) ∂sphereLaw N) / (N : ℝ) |
| |
| def expectedPressure (α β : ℝ) (φ : ℝ →ᵇ ℝ) (N : ℕ) : ℝ := |
| ∫ g, pressure α β φ N g ∂patternLaw α N |
| |
| structure Trial where |
| toFun : Time → ℝ |
| monotone : Monotone toFun |
| measurable : Measurable toFun |
| nonneg : ∀ t, 0 ≤ toFun t |
| le_one : ∀ t, toFun t ≤ 1 |
| |
| instance : CoeFun Trial (fun _ => Time → ℝ) := ⟨Trial.toFun⟩ |
| |
| def timeLaw : Measure Time := Measure.comap Subtype.val volume |
| |
| def tailIntegral (m : Trial) (t : Time) : ℝ := |
| ∫ s in Set.Ici t, m s ∂timeLaw |
| |
| def entropy (m : Trial) : ℝ≥0∞ := |
| (∫⁻ t, (ENNReal.ofReal (tailIntegral m t))⁻¹ - |
| (ENNReal.ofReal (1 - (t : ℝ)))⁻¹ ∂timeLaw) / 2 |
| |
| def brownianEval (t : ℝ≥0) (ω : BrownianPath) : ℝ := ω t |
| |
| @[instance_reducible] def nullSigma (P : Measure BrownianPath) : MeasurableSpace BrownianPath := |
| MeasurableSpace.generateFrom {s | P s = 0} |
| |
| @[instance_reducible] def usualBrownianSigma (P : Measure BrownianPath) (t : Time) : |
| MeasurableSpace BrownianPath := |
| ⨅ s : {s : ℝ≥0 // (t : ℝ) < s}, |
| (⨆ r : {r : ℝ≥0 // r ≤ s.val}, |
| MeasurableSpace.comap (brownianEval r.val) (borel ℝ)) ⊔ nullSigma P |
| |
| def Progressive (P : Measure BrownianPath) (v : Time → BrownianPath → ℝ) : Prop := |
| ∀ t : Time, |
| @Measurable (Set.Iic t × BrownianPath) ℝ |
| (MeasurableSpace.prod inferInstance (usualBrownianSigma P t)) (borel ℝ) |
| (fun p => v p.1.val p.2) |
| |
| def controlCost (P : Measure BrownianPath) (m : Trial) |
| (v : Time → BrownianPath → ℝ) : ℝ≥0∞ := |
| ∫⁻ ω, ∫⁻ t, ENNReal.ofReal (m t * v t ω ^ 2) ∂timeLaw ∂P |
| |
| def controlPayoff (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (m : Trial) |
| (v : Time → BrownianPath → ℝ) : ℝ := |
| (∫ ω, f (brownianEval 1 ω + ∫ t, m t * v t ω ∂timeLaw) ∂P) - |
| (controlCost P m v).toReal / 2 |
| |
| def controlValue (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (m : Trial) : ℝ := |
| sSup {a | ∃ v : Time → BrownianPath → ℝ, |
| Progressive P v ∧ controlCost P m v < ∞ ∧ a = controlPayoff P f m v} |
| |
| def variationalValue (P : Measure BrownianPath) (α β : ℝ) (φ : ℝ →ᵇ ℝ) : EReal := |
| ⨅ m : Trial, ((α * controlValue P (β • φ) m : ℝ) : EReal) + (entropy m).toEReal |
| |
| end SphericalPerceptronFreeEnergy |
| |
| end |
| |
| end OAI |
| |
