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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.KernelModel
import OAI.Probability.MemoryPrecision.WeightedImage
import OAI.Probability.MemoryPrecision.WeightedSelection
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
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
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
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]
end MemoryPrecision
end
end OAI
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