| import OAI.Probability.GaussianReplacement.FiberBlock |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace CurrentProjection |
| open InnerProductSpace |
| open InformationTheory |
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| def fiberObservationLaw (d k : ℕ) : Measure ((Rows k d×(Fin k→ℝ))×Sphere d) := |
| ((gaussianRows k d).prod (uniformSphere d)).map |
| (fun z => ((z.1,label z.1 (z.2:Vector d)),z.2)) |
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| lemma measurable_fiberObservationMap {d k : ℕ} : |
| Measurable (fun z : Rows k d×Sphere d => ((z.1,label z.1 (z.2:Vector d)),z.2)) := |
| (measurable_fst.prodMk measurable_own_label).prodMk measurable_snd |
| |
| instance uniformSphere_finite (d : ℕ) : IsFiniteMeasure (uniformSphere d) := by |
| constructor |
| unfold uniformSphere |
| rw [Measure.smul_apply, smul_eq_mul] |
| exact lt_of_le_of_lt (ENNReal.inv_mul_le_one _) ENNReal.one_lt_top |
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| instance fiberObservationLaw_finite (d k : ℕ) : IsFiniteMeasure (fiberObservationLaw d k) := by |
| unfold fiberObservationLaw |
| infer_instance |
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| lemma sphere_standardBorel (d : ℕ) : StandardBorelSpace (Sphere d) := |
| Metric.isClosed_sphere.measurableSet.standardBorel |
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| def sphericalFiberKernel (d k : ℕ) [sphereNonempty : Nonempty (Sphere d)] : |
| Kernel (Rows k d×(Fin k→ℝ)) (Sphere d) := |
| @Measure.condKernel (Rows k d×(Fin k→ℝ)) (Sphere d) _ _ |
| (sphere_standardBorel d) sphereNonempty |
| (fiberObservationLaw d k) (fiberObservationLaw_finite d k) |
| |
| instance sphericalFiberKernel_markov (d k : ℕ) [Nonempty (Sphere d)] : |
| IsMarkovKernel (sphericalFiberKernel d k) := by |
| unfold sphericalFiberKernel |
| infer_instance |
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| def fiberTwoPointLeft (d k : ℕ) [Nonempty (Sphere d)] : Measure (Rows k d×(Sphere d×Sphere d)) := |
| ((((gaussianRows k d).prod (uniformSphere d)).withDensity |
| (fun z => exactDensity (uniformSphere d) z.1 (label z.1 (z.2:Vector d)))) ⊗ₘ |
| ((sphericalFiberKernel d k).comap (fun z => (z.1,label z.1 (z.2:Vector d))) |
| (measurable_fst.prodMk measurable_own_label))).map |
| (fun z => (z.1.1,(z.1.2,z.2))) |
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| def perpendicularRow {d : ℕ} (v g : Vector d) : Vector d := |
| letI : NormedAddCommGroup (Vector d) := PiLp.normedAddCommGroup 2 (fun _ : Fin d => ℝ) |
| letI : InnerProductSpace ℝ (Vector d) := PiLp.innerProductSpace (fun _ : Fin d => ℝ) |
| g - (⟪v,g⟫ / ‖v‖^2) • v |
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| lemma measurable_perpendicularRow {d : ℕ} : |
| Measurable (fun z : Vector d×Vector d => perpendicularRow z.1 z.2) := by |
| unfold perpendicularRow |
| fun_prop |
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| def perpendicularRowsKernel (d k : ℕ) : Kernel (Sphere d×Sphere d) (Rows k d) := |
| dependentMapKernel (Kernel.const (Sphere d×Sphere d) (gaussianRows k d)) |
| (fun z => fun i => perpendicularRow ((z.1.1:Vector d)-(z.1.2:Vector d)) (z.2 i)) |
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| def fiberTwoPointConstant (k : ℕ) : ℝ≥0∞ := |
| letI : OfNat ℝ 2 := @instOfNatAtLeastTwo ℝ 2 Real.instNatCast inferInstance |
| letI : Mul ℝ := Real.instMul |
| letI : Inv ℝ := Real.instInv |
| letI : Pow ℝ ℕ := @NPow.toPow ℝ Real.instMonoid.toNPow |
| ENNReal.ofReal ((Real.sqrt (2*Real.pi))⁻¹ ^ k) |
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| def fiberDistanceWeight {d : ℕ} (k : ℕ) (z : Sphere d×Sphere d) : ℝ≥0∞ := |
| (ENNReal.ofReal ‖(z.1:Vector d)-(z.2:Vector d)‖)⁻¹ ^ k |
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| def fiberTwoPointRight (d k : ℕ) : Measure (Rows k d×(Sphere d×Sphere d)) := |
| letI : IsScalarTower ℝ≥0∞ ℝ≥0∞ ℝ≥0∞ := Semigroup.isScalarTower |
| fiberTwoPointConstant k • |
| Measure.map (Prod.swap : (Sphere d×Sphere d)×Rows k d → Rows k d×(Sphere d×Sphere d)) |
| (Measure.compProd |
| (Measure.withDensity (Measure.prod (uniformSphere d) (uniformSphere d)) |
| (fiberDistanceWeight k)) |
| (perpendicularRowsKernel d k)) |
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| def FiberTwoPointMain : Prop := |
| ∀ (d k : ℕ) (_hk : 1 ≤ k) (hkd : k < d-1), |
| letI : IsProbabilityMeasure (uniformSphere d) := |
| uniformSphere_probability_of_pos (by omega) |
| letI : Nonempty (Sphere d) := nonempty_of_isProbabilityMeasure (uniformSphere d) |
| IsFiniteMeasure (fiberTwoPointLeft d k) ∧ IsFiniteMeasure (fiberTwoPointRight d k) ∧ |
| fiberTwoPointLeft d k = fiberTwoPointRight d k |
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| end CurrentProjection |
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| end |
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| end OAI |
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