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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 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.CubeGeometry
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
open NoiselessRegression
lemma cube_J_bound {d k : ℕ} (hd : 16 ≤ d) (hk : k ≤ (d/2)/4) :
∀ j : ℕ, j ≤ (d/2)/4-2 → J (3*(d/2:ℕ)/4:ℝ) k j ≤ (7:ℝ)^d := by
have hp2 : 2 ≤ (d/2)/4 := by omega
have hn4 : (((d/2)/4:ℕ):ℝ)*4 ≤ (d/2:ℕ) := by exact_mod_cast Nat.div_mul_le_self (d/2) 4
have hn : ((d/2:ℕ):ℝ) ≤ d := by exact_mod_cast Nat.div_le_self d 2
have hm : (k:ℝ) ≤ ((d/2)/4:ℕ) := by exact_mod_cast hk
intro j hj
have hj' : (j:ℝ)+2 ≤ ((d/2)/4:ℕ) := by exact_mod_cast (show j+2 ≤ (d/2)/4 by omega)
have hgap : 0 < (3*(d/2:ℕ)/4:ℝ)-(j:ℝ)-(k:ℝ) := by linarith
have hquot : (k:ℝ)/((3*(d/2:ℕ)/4:ℝ)-(j:ℝ)-(k:ℝ)) ≤ 1 := by
rw [div_le_one hgap]
linarith
have h3 : (3:ℝ)^j ≤ (3:ℝ)^d := pow_le_pow_right₀ (by norm_num) (by omega)
have h2 : (2:ℝ)^(3*(d/2:ℕ)/4:ℝ) ≤ (2:ℝ)^d := by
rw [← Real.rpow_natCast]
apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
linarith
calc
J (3*(d/2:ℕ)/4:ℝ) k j = 1+(3:ℝ)^j*(2:ℝ)^(3*(d/2:ℕ)/4:ℝ)*
((k:ℝ)/((3*(d/2:ℕ)/4:ℝ)-(j:ℝ)-(k:ℝ))) := by unfold J; ring
_ ≤ 1+(3:ℝ)^j*(2:ℝ)^(3*(d/2:ℕ)/4:ℝ)*1 := by gcongr
_ ≤ 1+(3:ℝ)^d*(2:ℝ)^d := by rw [mul_one]; gcongr
_ = 1+(6:ℝ)^d := by rw [← mul_pow]; norm_num
_ ≤ (7:ℝ)^d := by
simpa only [one_pow, show (1:ℝ)+6=7 by norm_num] using
pow_add_pow_le (show (0:ℝ) ≤ 1 by norm_num) (show (0:ℝ) ≤ 6 by norm_num) (show d ≠ 0 by omega)
lemma cube_weighted_ambient {d : ℕ} (hd : 2 ≤ d) {f : Cube d → ℝ≥0∞}
{L : ℝ} (hf : CubeLocal f L) :
AmbientBallBound (((CubeLaw d).withDensity f).map (CubeEmbedding d)) L (3*(d/2:ℕ)/4:ℝ) := by
intro x r hr
rw [weighted_image_ball _ (cubeEmbedding_measurable d)]
apply (lintegral_mono_set (show {u | dist (CubeEmbedding d u) x ≤ r} ⊆
CubeBall d (Real.sqrt (d/2:ℕ) • cubeProject d x) r from ?_)).trans
(hf _ _ hr)
intro u hu
exact (cubeProject_parameter_dist hd u x).trans
(mul_le_mul_of_nonneg_left hu (Real.sqrt_nonneg _))
lemma cubeBall_embedding_subset {d : ℕ} (hd : 2 ≤ d) (z : Space (d/2)) {r : ℝ}
(u₀ : Cube d) (hu₀ : u₀ ∈ CubeBall d z r) :
CubeBall d z r ⊆ CubeEmbedding d ⁻¹' Metric.closedBall (CubeEmbedding d u₀) (4*r) := by
have hs : 0 < Real.sqrt (d/2:ℕ) := Real.sqrt_pos.2 (by exact_mod_cast (show 0 < d/2 by omega))
intro u hu
have hu' : dist u.val z ≤ Real.sqrt (d/2:ℕ)*r := hu
have hu₀' : dist u₀.val z ≤ Real.sqrt (d/2:ℕ)*r := hu₀
have hab : dist u.val u₀.val ≤ 2*Real.sqrt (d/2:ℕ)*r := by
have := dist_triangle u.val z u₀.val
rw [dist_comm z u₀.val] at this
linarith
apply (cubeEmbedding_dist hd u u₀).trans
calc
_ ≤ 2/Real.sqrt (d/2:ℕ)*(2*Real.sqrt (d/2:ℕ)*r) :=
mul_le_mul_of_nonneg_left hab (by positivity)
_ = 4*r := by field_simp [ne_of_gt hs]; ring
lemma weightedBlock_le_sum {X : Type*} {d m N : ℕ}
(ι : X → Space d) (f : Fin N → X → ℝ≥0∞)
{g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞}
(hg : ∀ x, ∑ w, g w x ≤ 1) (x : X) : WeightedBlock ι f g x ≤ ∑ w, f w x := by
unfold WeightedBlock
calc
_ ≤ ∫⁻ _A : Fin m → Space d, (∑ w, f w x) ∂Measure.pi (fun _ => stdGaussian (Space d)) := by
apply lintegral_mono
intro A
apply Finset.sum_le_sum
intro w _
exact mul_le_of_le_one_left' ((Finset.single_le_sum (fun _ _ => zero_le) (Finset.mem_univ w)).trans (hg _))
_ = _ := by simp
def CubeBlockConstant : ℝ := 4*1344^3
lemma cubeBlockConstant_one : 1 ≤ CubeBlockConstant := by norm_num [CubeBlockConstant]
lemma cube_constant_scale {d : ℕ} (hd : 2 ≤ d) {r : ℝ} (hr : 0 < r) :
(1344:ℝ)^d * (4*r)^(3*(d/2:ℕ)/4:ℝ) ≤ CubeBlockConstant^(d/2) * r^(3*(d/2:ℕ)/4:ℝ) := by
rw [Real.mul_rpow (by norm_num) hr.le]
have hp : (1344:ℝ)^d ≤ (1344^3:ℝ)^(d/2) := by
rw [← pow_mul]
apply pow_le_pow_right₀ (by norm_num)
omega
have h4 : (4:ℝ)^(3*(d/2:ℕ)/4:ℝ) ≤ (4:ℝ)^(d/2) := by
rw [← Real.rpow_natCast]
apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
have := Nat.cast_nonneg (α := ℝ) (d/2)
linarith
calc
_ = ((1344:ℝ)^d*(4:ℝ)^(3*(d/2:ℕ)/4:ℝ))*r^(3*(d/2:ℕ)/4:ℝ) := by ring
_ ≤ ((1344^3:ℝ)^(d/2)*(4:ℝ)^(d/2))*r^(3*(d/2:ℕ)/4:ℝ) := by gcongr
_ = _ := by rw [← mul_pow]; congr 2; norm_num [CubeBlockConstant]
def CubeBlockStatement : Prop :=
∃ C : ℝ, 1 ≤ C ∧ ∀ d k N : ℕ, 16 ≤ d → 1 ≤ k → k ≤ (d/2)/4 → 1 ≤ N →
∀ L : ℝ, 0 ≤ L → ∀ f : Fin N → Cube d → ℝ≥0∞,
(∀ v, Measurable (f v)) → (∀ v u, f v u ≤ 1) → (∀ v, CubeLocal (f v) L) →
∀ g : Fin N → ((Fin k → Space d) × (Fin k → ℝ)) → ℝ≥0∞,
(∀ v, Measurable (g v)) → (∀ x, ∑ v, g v x ≤ 1) →
CubeLocal (WeightedBlock (CubeEmbedding d) f g)
(C^(d/2) * (N:ℝ)^(1/(((d/2)/4:ℕ):ℝ)) * L)
theorem cube_block : CubeBlockStatement := by
refine ⟨CubeBlockConstant, cubeBlockConstant_one, ?_⟩
intro d k N hd hk hkp hN L hL f hf hf1 hloc g hg hgs z r hr
by_cases hL0 : L = 0
· simp only [hL0, mul_zero, zero_mul, ENNReal.ofReal_zero]
calc
_ ≤ ∫⁻ u in CubeBall d z r, ∑ w, f w u ∂CubeLaw d :=
lintegral_mono (weightedBlock_le_sum _ _ hgs)
_ = ∑ w, ∫⁻ u in CubeBall d z r, f w u ∂CubeLaw d :=
lintegral_finsetSum _ (fun w _ => hf w)
_ ≤ ∑ _w : Fin N, ENNReal.ofReal (L*r^(3*(d/2:ℕ)/4:ℝ)) :=
Finset.sum_le_sum (fun w _ => hloc w z r hr)
_ = 0 := by simp [hL0]
have hLpos : 0 < L := lt_of_le_of_ne hL (Ne.symm hL0)
by_cases hempty : CubeBall d z r = ∅
· simp [hempty]
obtain ⟨u₀,hu₀⟩ := Set.nonempty_iff_ne_empty.mpr hempty
let ν (w : Fin N) := ((CubeLaw d).withDensity (f w)).map (CubeEmbedding d)
let (w : Fin N) : IsFiniteMeasure (ν w) := weighted_image_finite (CubeLaw d) (CubeEmbedding d) (f w) (hf1 w)
have hν w : AmbientBallBound (ν w) L (3*(d/2:ℕ)/4:ℝ) := cube_weighted_ambient (by omega) (hloc w)
have hp : 2 ≤ (d/2)/4 := by omega
have hkd : k ≤ d := by omega
have hgap : (k:ℝ)+(((d/2)/4:ℕ):ℝ)-2 < (3*(d/2:ℕ)/4:ℝ) := by
have hp' : (((d/2)/4:ℕ):ℝ)*4 ≤ (d/2:ℕ) := by exact_mod_cast Nat.div_mul_le_self (d/2) 4
have hk' : (k:ℝ) ≤ ((d/2)/4:ℕ) := by exact_mod_cast hkp
have hn' : (8:ℝ) ≤ (d/2:ℕ) := by exact_mod_cast (show 8 ≤ d/2 by omega)
linarith
have hsel := localized_weighted_selection ν hp hk hkd (by norm_num : (1:ℝ) ≤ 7)
hLpos hgap (cube_J_bound hd hkp) hν g hg hgs (CubeEmbedding d u₀) (by positivity : 0 < 4*r)
have he := weightedBlock_mass_identity (CubeLaw d) (cubeEmbedding_measurable d) hf hf1 hg
(show MeasurableSet (Metric.closedBall (CubeEmbedding d u₀) (4*r)) from Metric.isClosed_closedBall.measurableSet)
apply (lintegral_mono_set (cubeBall_embedding_subset (by omega) z u₀ hu₀)).trans
rw [he]
apply hsel.trans (ENNReal.ofReal_le_ofReal ?_)
have hscale := cube_constant_scale (d := d) (by omega) hr
norm_num only [show (7:ℝ)*192=1344 by norm_num]
calc
_ = ((1344:ℝ)^d*(4*r)^(3*(d/2:ℕ)/4:ℝ))*((N:ℝ)^(1/(((d/2)/4:ℕ):ℝ))*L) := by ring
_ ≤ (CubeBlockConstant^(d/2)*r^(3*(d/2:ℕ)/4:ℝ))*((N:ℝ)^(1/(((d/2)/4:ℕ):ℝ))*L) :=
mul_le_mul_of_nonneg_right hscale (by positivity)
_ = _ := by ring
end MemoryPrecision
end
end OAI
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