| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.KernelModel |
| import OAI.Probability.MemoryPrecision.WeightedImage |
| import OAI.Probability.MemoryPrecision.WeightedSelection |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
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| noncomputable section |
| universe u v |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision |
| abbrev Sphere (d : ℕ) := Metric.sphere (0 : Space d) 1 |
| def SpherePrior (d : ℕ) : Measure (Sphere d) := |
| MemoryPrecision.Kernel.UniformSphere d |
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| def SphereBallBound {d : ℕ} (h : Sphere d → ℝ≥0∞) (H α : ℝ) : Prop := |
| ∀ z : Space d, ∀ r : ℝ, 0 < r → |
| (∫⁻ s in {s : Sphere d | dist s.val z ≤ r}, h s ∂SpherePrior d) ≤ |
| ENNReal.ofReal (H * r^α) |
| def SphereBlock {d m N : ℕ} (h : Fin N → Sphere d → ℝ≥0∞) |
| (g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞) |
| (s : Sphere d) : ℝ≥0∞ := |
| WeightedBlock Subtype.val h g s |
| def CoefficientStepStatement : Prop := |
| let two : ℝ := @OfNat.ofNat ℝ 2 (@instOfNatAtLeastTwo ℝ 2 Real.commRing.toNatCast |
| (Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0))) |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ d m N : ℕ, 16 ≤ d → |
| 1 ≤ m → m ≤ d/8 → 1 ≤ N → |
| ∀ α H : ℝ, ((d:ℝ)-1)/two ≤ α → α ≤ (d:ℝ)/two → 0 < H → |
| ∀ h : Fin N → Sphere d → ℝ≥0∞, |
| (∀ w, Measurable (h w)) → (∀ w s, h w s ≤ 1) → |
| (∀ w, SphereBallBound (h w) H α) → |
| ∀ g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞, |
| (∀ w, Measurable (g w)) → (∀ x, ∑ w, g w x ≤ 1) → |
| SphereBallBound (SphereBlock h g) (C^d * (N:ℝ)^(1/((d/8:ℕ):ℝ)) * H) α |
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| theorem coefficient_step : CoefficientStepStatement := by |
| refine ⟨1344, by norm_num, ?_⟩ |
| intro d m N hd hm hmp hN α H hαlo hαhi hH h hh hh1 hb g hg hgsum z r hr |
| have : NeZero d := ⟨by omega⟩ |
| have : IsProbabilityMeasure (SpherePrior d) := |
| (show IsProbabilityMeasure (NoiselessRegression.sphereLaw (E := Space d)) from inferInstance) |
| let ν (w : Fin N) := ((SpherePrior d).withDensity (h w)).map Subtype.val |
| let (w : Fin N) : IsFiniteMeasure (ν w) := |
| weighted_image_finite (SpherePrior d) Subtype.val (h w) (hh1 w) |
| have hν (w) : AmbientBallBound (ν w) H α := by |
| intro x ρ hρ |
| rw [show ν w = ((SpherePrior d).withDensity (h w)).map Subtype.val from rfl, |
| weighted_image_ball _ measurable_subtype_coe] |
| exact hb w x ρ hρ |
| have hp : 2 ≤ d/8 := by omega |
| have hmd : m ≤ d := hmp.trans (Nat.div_le_self _ _) |
| have hgap : (m:ℝ)+((d/8:ℕ):ℝ)-2 < α := by |
| have hm' : (m:ℝ) ≤ (d/8:ℕ) := by exact_mod_cast hmp |
| have hp' : ((d/8:ℕ):ℝ)*8 ≤ d := by exact_mod_cast Nat.div_mul_le_self d 8 |
| have hd' : (16:ℝ) ≤ d := by exact_mod_cast hd |
| linarith |
| have hJ := fun j hj => (specialized_J_bound d m (d/8) hd rfl hm hmp α hαlo hαhi j hj).2.2 |
| have hsel := localized_weighted_selection ν hp hm hmd (by norm_num : (1:ℝ) ≤ 7) |
| hH hgap hJ hν g hg hgsum z hr |
| have he := weightedBlock_mass_identity (SpherePrior d) measurable_subtype_coe hh hh1 hg |
| (show MeasurableSet (Metric.closedBall z r) from Metric.isClosed_closedBall.measurableSet) |
| change (∫⁻ s in Subtype.val ⁻¹' Metric.closedBall z r, |
| WeightedBlock Subtype.val h g s ∂SpherePrior d) ≤ _ |
| rw [he] |
| convert hsel using 1 |
| norm_num [ν, mul_assoc] |
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| end MemoryPrecision |
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| end |
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| end OAI |
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