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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 | import OAI.Probability.GaussianReplacement.StreamModel
import OAI.Probability.GaussianRegression.Nets
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
universe u
namespace NoiselessRegression
open scoped Topology
section InverseMoments
variable {X : Type*} [MeasurableSpace X]
omit [MeasurableSpace X] in
lemma inverse_pow_dyadic_majorant (f : X → ℝ) (m : ℕ) (hf : ∀ x, 0 ≤ f x) (x : X) :
ENNReal.ofReal ((f x)⁻¹ ^ m) ≤
1 + ∑' k : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator
(fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1)))) x := by
classical
by_cases hzero : f x = 0
· simp only [hzero, inv_zero]
exact le_add_of_le_left (by cases m <;> simp)
have hxpos : 0 < f x := lt_of_le_of_ne (hf x) (Ne.symm hzero)
by_cases hxone : 1 ≤ f x
· apply le_add_of_le_left
have h : (f x)⁻¹ ^ m ≤ 1 :=
pow_le_one₀ (inv_nonneg.mpr (hf x)) (inv_le_one_of_one_le₀ hxone)
simpa using ENNReal.ofReal_le_ofReal h
obtain ⟨k, hklo, hkhi⟩ := exists_nat_pow_near_of_lt_one hxpos
(le_of_not_ge hxone) (by norm_num : (0 : ℝ) < 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1)
have hinv : (f x)⁻¹ ≤ (2 : ℝ) ^ (k + 1) := by
have h := (inv_le_inv₀ hxpos (pow_pos (by norm_num : (0 : ℝ) < 1 / 2) _)).2 hklo.le
simpa using h
have hpow : ENNReal.ofReal ((f x)⁻¹ ^ m) ≤
ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) := by
apply ENNReal.ofReal_le_ofReal
simpa [← pow_mul, Nat.mul_comm] using pow_le_pow_left₀ (inv_nonneg.mpr (hf x)) hinv m
calc
ENNReal.ofReal ((f x)⁻¹ ^ m) ≤ ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) := hpow
_ = ({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator
(fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1)))) x := by
rw [Set.indicator_of_mem (show x ∈ {x | f x ≤ (1 / 2 : ℝ) ^ k} from hkhi)]
_ ≤ ∑' j : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator
(fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x :=
ENNReal.le_tsum (f := fun j => ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator
(fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x) k
_ ≤ 1 + ∑' j : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator
(fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x := le_add_left le_rfl
lemma dyadic_power_cancel (m k : ℕ) :
(2 : ℝ) ^ (m * (k + 1)) * ((1 / 2 : ℝ) ^ k) ^ (m + 1) =
(2 : ℝ) ^ m * (1 / 2 : ℝ) ^ k := by
rw [Nat.mul_add, mul_one, pow_add, pow_succ]
have h : (2 : ℝ) ^ (m * k) * ((1 / 2 : ℝ) ^ k) ^ m = 1 := by
rw [Nat.mul_comm m k, pow_mul, ← mul_pow, ← mul_pow]
norm_num
calc
2 ^ (m * k) * 2 ^ m * (((1 / 2 : ℝ) ^ k) ^ m * (1 / 2 : ℝ) ^ k) =
(2 ^ (m * k) * ((1 / 2 : ℝ) ^ k) ^ m) * (2 ^ m * (1 / 2 : ℝ) ^ k) := by ring
_ = _ := by rw [h, one_mul]
theorem inverse_moment_of_tube_bound (μ : Measure X) (f : X → ℝ) (m : ℕ)
(hf : Measurable f) (hnonneg : ∀ x, 0 ≤ f x) (C : ℝ≥0∞)
(htube : ∀ r : ℝ, 0 < r → r ≤ 1 →
μ {x | f x ≤ r} ≤ C * ENNReal.ofReal (r ^ (m + 1))) :
∫⁻ x, ENNReal.ofReal ((f x)⁻¹ ^ m) ∂μ ≤
μ Set.univ + C * (2 : ℝ≥0∞) ^ (m + 1) := by
classical
let g : ℕ → X → ℝ≥0∞ := fun k =>
({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator
(fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))))
have hg : ∀ k, Measurable (g k) := fun k =>
measurable_const.indicator (measurableSet_le hf measurable_const)
calc
∫⁻ x, ENNReal.ofReal ((f x)⁻¹ ^ m) ∂μ ≤ ∫⁻ x, 1 + ∑' k, g k x ∂μ :=
lintegral_mono (inverse_pow_dyadic_majorant f m hnonneg)
_ = μ Set.univ + ∑' k, ∫⁻ x, g k x ∂μ := by
rw [lintegral_add_left measurable_const, lintegral_const, one_mul, lintegral_tsum (fun k => (hg k).aemeasurable)]
_ ≤ μ Set.univ + ∑' k, C * ENNReal.ofReal ((2 : ℝ) ^ m * (1 / 2 : ℝ) ^ k) := by
apply add_le_add le_rfl
apply ENNReal.tsum_le_tsum
intro k
dsimp only [g]
rw [lintegral_indicator (measurableSet_le hf measurable_const), lintegral_const,
Measure.restrict_apply_univ]
calc
ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) * μ {x | f x ≤ (1 / 2 : ℝ) ^ k} ≤
ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) *
(C * ENNReal.ofReal (((1 / 2 : ℝ) ^ k) ^ (m + 1))) := by
gcongr
exact htube _ (by positivity) (pow_le_one₀ (by norm_num) (by norm_num))
_ = _ := by
rw [← mul_assoc, mul_comm _ C, mul_assoc, ← ENNReal.ofReal_mul (by positivity),
dyadic_power_cancel]
_ = μ Set.univ + C * (2 : ℝ≥0∞) ^ (m + 1) := by
simp_rw [ENNReal.ofReal_mul (by positivity : (0 : ℝ) ≤ 2 ^ m),
ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 1 / 2),
ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 2)]
rw [ENNReal.tsum_mul_left, ENNReal.tsum_mul_left, ENNReal.tsum_geometric]
norm_num [pow_succ, mul_assoc, ENNReal.ofReal_div_of_pos]
end InverseMoments
section FrostmanGeometry
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
end FrostmanGeometry
section FiniteNets
variable {E : Type*} [PseudoMetricSpace E]
end FiniteNets
section EuclideanNets
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
end EuclideanNets
section AffineTubes
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
open EuclideanGeometry
theorem exists_affine_tube_cover (L : AffineSubspace ℝ E) [Nonempty L]
{r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) :
∃ s : Finset L.direction,
(s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ L.direction ∧
∀ t : E, ‖t‖ ≤ 1 → Metric.infDist t L ≤ r →
∃ y ∈ s, dist t ((y : E) + (orthogonalProjection L 0 : E)) ≤ 2 * r := by
classical
obtain ⟨s, _, hcard, hnet⟩ := exists_unit_ball_net (E := L.direction) hr hr1
refine ⟨s, hcard, ?_⟩
intro t ht hdist
obtain ⟨y, hy, hty⟩ := hnet (L.direction.orthogonalProjectionOnto t)
((L.direction.norm_orthogonalProjectionOnto_apply_le t).trans ht)
refine ⟨y, hy, ?_⟩
have hproj : (orthogonalProjection L t : E) =
(L.direction.orthogonalProjectionOnto t : E) + (orthogonalProjection L 0 : E) := by
have h := (orthogonalProjection L).map_vadd (0 : E) t
simpa only [vadd_eq_add, add_zero, orthogonalProjection_contLinear,
AffineSubspace.coe_vadd] using congrArg (fun x : L => (x : E)) h
have hpdist : dist (orthogonalProjection L t : E)
((y : E) + (orthogonalProjection L 0 : E)) ≤ r := by
rw [hproj, dist_add_right]
exact hty
have hd : dist t (orthogonalProjection L t : E) ≤ r := by
rwa [dist_orthogonalProjection_eq_infDist]
calc
dist t ((y : E) + (orthogonalProjection L 0 : E)) ≤
dist t (orthogonalProjection L t : E) +
dist (orthogonalProjection L t : E) ((y : E) + (orthogonalProjection L 0 : E)) :=
dist_triangle _ _ _
_ ≤ r + r := add_le_add hd hpdist
_ = 2 * r := by ring
theorem HasLocalMassBound.affine_tube {ν : Measure E} {a : ℝ}
(hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1)
(L : AffineSubspace ℝ E) [Nonempty L] {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) :
ν {t | Metric.infDist t L ≤ r} ≤
ENNReal.ofReal ((3 / r) ^ Module.finrank ℝ L.direction * (4 * r) ^ a) := by
classical
obtain ⟨s, hcard, hcover⟩ := exists_affine_tube_cover L hr hr1
let c : L.direction → E := fun y => (y : E) + (orthogonalProjection L 0 : E)
have hsub : {t | Metric.infDist t L ≤ r} ≤ᵐ[ν]
⋃ y ∈ s, Metric.closedBall (c y) (2 * r) := by
filter_upwards [ν.support_mem_ae] with t ht
intro hdist
have htunit : ‖t‖ ≤ 1 := by simpa using hsupport ht
obtain ⟨y, hy, hty⟩ := hcover t htunit hdist
exact mem_iUnion₂.mpr ⟨y, hy, hty⟩
calc
ν {t | Metric.infDist t L ≤ r} ≤ ν (⋃ y ∈ s, Metric.closedBall (c y) (2 * r)) :=
measure_mono_ae hsub
_ ≤ ∑ y ∈ s, ν (Metric.closedBall (c y) (2 * r)) := measure_biUnion_finset_le _ _
_ ≤ ∑ _y ∈ s, ENNReal.ofReal ((4 * r) ^ a) := by
refine Finset.sum_le_sum fun y _ => ?_
simpa only [show (2 : ℝ) * (2 * r) = 4 * r by ring] using
hν.any_center (c y) (r := 2 * r) (by positivity)
_ = (s.card : ℝ≥0∞) * ENNReal.ofReal ((4 * r) ^ a) := by simp
_ ≤ ENNReal.ofReal ((3 / r) ^ Module.finrank ℝ L.direction) *
ENNReal.ofReal ((4 * r) ^ a) := by
gcongr
exact_mod_cast ENNReal.ofReal_le_ofReal hcard
_ = _ := (ENNReal.ofReal_mul (by positivity)).symm
lemma tube_radius_factors (ℓ : ℕ) (a : ℝ) {r : ℝ} (hr : 0 < r) :
(3 / r) ^ ℓ * (4 * r) ^ a =
(3 : ℝ) ^ ℓ * (4 : ℝ) ^ a * r ^ (a - ℓ) := by
rw [div_pow, Real.mul_rpow (by norm_num) hr.le,
Real.rpow_sub_natCast hr.ne']
ring
theorem HasLocalMassBound.affine_tube_pow {ν : Measure E} {a : ℝ}
(hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1)
(L : AffineSubspace ℝ E) [Nonempty L] (k : ℕ)
(hmargin : (Module.finrank ℝ L.direction : ℝ) + k ≤ a)
{r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) :
ν {t | Metric.infDist t L ≤ r} ≤
ENNReal.ofReal ((3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a) *
ENNReal.ofReal (r ^ k) := by
refine (hν.affine_tube hsupport L hr hr1).trans ?_
rw [tube_radius_factors _ _ hr, ← ENNReal.ofReal_mul (by positivity)]
apply ENNReal.ofReal_le_ofReal
apply mul_le_mul_of_nonneg_left _ (by positivity)
rw [← Real.rpow_natCast]
exact Real.rpow_le_rpow_of_exponent_ge hr hr1 (by linarith)
theorem HasLocalMassBound.affine_inverse_moment {ν : Measure E} {a : ℝ}
(hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1)
(L : AffineSubspace ℝ E) [Nonempty L] (m : ℕ)
(hmargin : (Module.finrank ℝ L.direction : ℝ) + (m + 1 : ℕ) ≤ a) :
∫⁻ t, ENNReal.ofReal ((Metric.infDist t L)⁻¹ ^ m) ∂ν ≤
ν Set.univ +
ENNReal.ofReal ((3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a) *
(2 : ℝ≥0∞) ^ (m + 1) := by
apply inverse_moment_of_tube_bound ν _ m (Metric.continuous_infDist_pt _).measurable
(fun _ => Metric.infDist_nonneg)
intro r hr hr1
exact hν.affine_tube_pow hsupport L (m + 1) hmargin hr hr1
theorem HasLocalMassBound.affine_null {ν : Measure E} {a : ℝ}
(hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1)
(L : AffineSubspace ℝ E) [Nonempty L]
(hmargin : (Module.finrank ℝ L.direction : ℝ) + 1 ≤ a) : ν L = 0 := by
let C : ℝ := (3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a
have hC : 0 ≤ C := by dsimp [C]; positivity
have hb (n : ℕ) : ν L ≤ ENNReal.ofReal (C * (1 / 2 : ℝ) ^ n) := by
have hn : 0 < (1 / 2 : ℝ) ^ n := by positivity
have hn1 : (1 / 2 : ℝ) ^ n ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
calc
ν L ≤ ν {t | Metric.infDist t L ≤ (1 / 2 : ℝ) ^ n} := by
apply measure_mono
intro t ht
simp only [mem_ofPred_eq, Metric.infDist_zero_of_mem ht]
exact hn.le
_ ≤ ENNReal.ofReal C * ENNReal.ofReal (((1 / 2 : ℝ) ^ n) ^ 1) :=
hν.affine_tube_pow hsupport L 1 (by simpa using hmargin) hn hn1
_ = _ := by rw [pow_one, ENNReal.ofReal_mul hC]
have hlim : Tendsto (fun n : ℕ => ENNReal.ofReal (C * (1 / 2 : ℝ) ^ n))
atTop (𝓝 0) := by
convert ENNReal.tendsto_ofReal ((tendsto_const_nhds (x := C)).mul
(tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2)
(by norm_num : (1 / 2 : ℝ) < 1))) using 1
simp
exact le_antisymm (ge_of_tendsto' hlim hb) (by positivity)
end AffineTubes
end NoiselessRegression
end
end OAI
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