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b317718 | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Probability.GaussianRegression.Stream
import OAI.Probability.GaussianRegression.Sphere
namespace OAI
/-!
Spherical cap mass bounds along sequential routes.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
open scoped Pointwise
noncomputable section
universe u
namespace NoiselessRegression
section StreamCaps
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E]
variable {J : Type*} [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] [Nonempty J]
def capWeight (y : E) (ε : ℝ) : E → ℝ≥0∞ :=
(Metric.closedBall y ε).indicator (fun _ => 1)
omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [Nontrivial E] in
lemma measurable_capWeight (y : E) (ε : ℝ) : Measurable (capWeight y ε) :=
measurable_const.indicator Metric.isClosed_closedBall.measurableSet
omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nontrivial E] in
lemma capWeight_le_one (y s : E) (ε : ℝ) : capWeight y ε s ≤ 1 := by
classical
simp only [capWeight, Set.indicator_apply]
split <;> simp
lemma capWeight_mass (y : E) {ε : ℝ} (hε : 0 < ε) (a : ℕ)
(ha : 2*a ≤ Module.finrank ℝ E - 1) :
PolynomialMass ((ambientSphereLaw (E := E)).withDensity (capWeight y ε)) a
(ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^a)) := by
change PolynomialMass ((ambientSphereLaw (E := E)).withDensity ((Metric.closedBall y ε).indicator 1)) a _
rw [withDensity_indicator_one Metric.isClosed_closedBall.measurableSet]
exact fun z r hr => terminal_cap_mass y z hε hr a ha
lemma ambient_mass_bound {f : E → ℝ≥0∞} {a : ℕ} {C : ℝ≥0∞}
(h : PolynomialMass ((ambientSphereLaw (E := E)).withDensity f) a C) :
(∫⁻ s, f s ∂ambientSphereLaw) ≤ C := by
rw [← setLIntegral_univ, ← withDensity_apply _ MeasurableSet.univ]
apply (measure_mono_ae ?_).trans (by simpa using h 0 1 zero_lt_one)
filter_upwards [(withDensity_absolutelyContinuous (ambientSphereLaw (E := E)) f).ae_le
(ambientSphereLaw_ae_norm (E := E))] with s hs
intro _
change dist s 0 ≤ 1
simp [dist_zero_right, hs]
lemma continuation_cap_integral_bound {δ : ℕ → J → (E × ℝ) → J}
(hδ : ∀ n j, Measurable (δ n j)) (y : ℕ → J → E)
{ε : ℝ} (hε : 0 < ε) {a q : ℕ} (ha : 2*a ≤ Module.finrank ℝ E - 1)
(hq : 2 ≤ q) (hqa : 2*q ≤ a) (n k : ℕ) (j : J) :
(∫⁻ s, continuation δ (fun n j => capWeight (y n j) ε) n k j s ∂ambientSphereLaw) ≤
(ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^a)) *
streamFactor (E := E) (J := J) q ^ (k/q+1) := by
apply ambient_mass_bound
apply continuation_mass_bound _ hδ (fun n j => measurable_capWeight _ _)
(fun n j s => capWeight_le_one _ _ _)
(mul_ne_zero (by positivity) (by positivity)) (ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top)
(fun n j => capWeight_mass (y n j) hε a ha)
(Module.finrank_pos (R := ℝ) (M := E)) (by omega) hq hqa
lemma route_cap_bound {δ : ℕ → J → (E × ℝ) → J}
(hδ : ∀ n j, Measurable (δ n j)) (y : J → E)
{ε : ℝ} (hε : 0 < ε) {a q : ℕ} (ha : 2*a ≤ Module.finrank ℝ E - 1)
(hq : 2 ≤ q) (hqa : 2*q ≤ a) (k : ℕ) (j : J) :
((sphereLaw (E := E)).prod (Measure.pi (fun _ : Fin k => stdGaussian E)))
{z | dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} ≤
(ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^a)) *
streamFactor (E := E) (J := J) q ^ (k/q+1) := by
let w : ℕ → J → E → ℝ≥0∞ := fun _ j => capWeight (y j) ε
have hV := measurable_continuation (w := w) hδ (fun _ j => measurable_capWeight (y j) ε) 0 k j
have he (s : E) : continuation δ w 0 k j s =
∫⁻ A : Fin k → E, capWeight (y (streamRoute δ 0 k j (A,fun i => ⟪s,A i⟫))) ε s
∂Measure.pi (fun _ => stdGaussian E) := by
simpa [w, blockValue, continuation] using continuation_unroll (w := w) hδ 0 k 0 j s
let S : Set ((Metric.sphere (0 : E) 1) × (Fin k → E)) :=
{z | dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε}
have hS : MeasurableSet S := by
apply measurableSet_le _ measurable_const
exact measurable_subtype_coe.comp measurable_fst |>.dist
((measurable_of_finite y).comp ((measurable_streamRoute hδ 0 k j).comp (by fun_prop)))
change ((sphereLaw (E := E)).prod _) S ≤ _
rw [← one_mul (((sphereLaw (E := E)).prod _) S), ← lintegral_indicator_const hS 1,
lintegral_prod _ (measurable_const.indicator hS).aemeasurable]
have hf : ∀ s : Metric.sphere (0 : E) 1,
(∫⁻ A, S.indicator (fun _ => (1 : ℝ≥0∞)) (s,A) ∂Measure.pi (fun _ => stdGaussian E)) =
continuation δ w 0 k j s.val := by
intro s
rw [he]
apply lintegral_congr
intro A
classical
change (if dist s.val (y (streamRoute δ 0 k j (A,fun i => ⟪s.val,A i⟫))) ≤ ε then 1 else 0) = _
unfold capWeight
rw [Set.indicator_apply]
rfl
simp_rw [hf]
rw [← lintegral_map hV measurable_subtype_coe]
exact continuation_cap_integral_bound hδ (fun _ => y) hε ha hq hqa 0 k j
end StreamCaps
end NoiselessRegression
end
end OAI
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