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d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Estimates.AllDimensions
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
namespace OAI
/-!
# Geometry of the retained graph in arbitrary dimension
The graph tests use orthogonal distance to a line and one scalar longitudinal
constraint. Each longitudinal constraint is in the unknown endpoint's own
cap direction. In particular these bounds do not divide by the angle between
the two endpoint directions.
-/
namespace Falconer
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
/-- Perpendicular displacement from the line in a unit direction. -/
def perpendicularPart (w v : E) : E := v - inner ℝ w v • w
theorem perpendicularPart_sub (w u v : E) :
perpendicularPart w (u-v) = perpendicularPart w u - perpendicularPart w v := by
simp only [perpendicularPart, inner_sub_right, sub_smul]
abel
/-- Orthogonal projection is a contraction. -/
theorem norm_perpendicularPart_le {w : E} (hw : ‖w‖ = 1) (v : E) :
‖perpendicularPart w v‖ ≤ ‖v‖ := by
have hn := norm_sub_sq_real v (inner ℝ w v • w)
rw [real_inner_smul_right, real_inner_comm v w, norm_smul, hw,
mul_one, Real.norm_eq_abs, sq_abs] at hn
change ‖v - inner ℝ w v • w‖ ≤ ‖v‖
rw [real_inner_comm v w]
nlinarith [norm_nonneg (v - inner ℝ v w • w), norm_nonneg v, sq_nonneg (inner ℝ v w)]
/-- Moving a unit direction changes the perpendicular test by at most twice
its angular displacement times the size of the tested vector. -/
theorem perpendicularPart_direction_error {w z : E} (hw : ‖w‖ = 1) (hz : ‖z‖ = 1)
(v : E) :
‖perpendicularPart w v - perpendicularPart z v‖ ≤ 2 * ‖w-z‖ * ‖v‖ := by
have he : perpendicularPart w v - perpendicularPart z v =
inner ℝ z v • (z-w) + inner ℝ (z-w) v • w := by
simp only [perpendicularPart, inner_sub_left, sub_smul, smul_sub]
abel
rw [he]
calc
_ ≤ ‖inner ℝ z v • (z-w)‖ + ‖inner ℝ (z-w) v • w‖ := norm_add_le _ _
_ = |inner ℝ z v| * ‖z-w‖ + |inner ℝ (z-w) v| := by
rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs, hw, mul_one]
_ ≤ (‖z‖ * ‖v‖) * ‖z-w‖ + (‖z-w‖ * ‖v‖) := by
exact add_le_add (mul_le_mul_of_nonneg_right (abs_real_inner_le_norm z v)
(norm_nonneg _)) (abs_real_inner_le_norm (z-w) v)
_ = _ := by rw [hz, one_mul, norm_sub_rev z w]; ring
/-- The error in a graph perpendicular test from both point and cap
replacement is controlled uniformly in the dimension. -/
theorem perpendicularPart_test_error {w z : E} (hw : ‖w‖ = 1) (hz : ‖z‖ = 1)
(u v : E) :
‖perpendicularPart w u - perpendicularPart z v‖ ≤
‖u-v‖ + 2 * ‖w-z‖ * ‖v‖ := by
calc
_ ≤ ‖perpendicularPart w u - perpendicularPart w v‖ +
‖perpendicularPart w v - perpendicularPart z v‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
_ ≤ ‖u-v‖ + 2 * ‖w-z‖ * ‖v‖ := by
rw [← perpendicularPart_sub]
exact add_le_add (norm_perpendicularPart_le hw _) (perpendicularPart_direction_error hw hz v)
/-- Scalar phase error under point and cap replacement. -/
theorem longitudinal_test_error {w z : E} (hw : ‖w‖ = 1) (u v : E) :
|inner ℝ w u - inner ℝ z v| ≤ ‖u-v‖ + ‖w-z‖ * ‖v‖ := by
have he : inner ℝ w u - inner ℝ z v =
inner ℝ w (u-v) + inner ℝ (w-z) v := by
simp only [inner_sub_left, inner_sub_right]
ring
rw [he]
calc
_ ≤ |inner ℝ w (u-v)| + |inner ℝ (w-z) v| := abs_add_le _ _
_ ≤ ‖w‖ * ‖u-v‖ + ‖w-z‖ * ‖v‖ :=
add_le_add (abs_real_inner_le_norm _ _) (abs_real_inner_le_norm _ _)
_ = _ := by rw [hw, one_mul]
/-- A perpendicular tube and a longitudinal interval together confine the
unknown point to a ball, regardless of any other direction in the graph. -/
theorem norm_le_longitudinal_add_perpendicular {w : E} (hw : ‖w‖ = 1) (v : E) :
‖v‖ ≤ |inner ℝ w v| + ‖perpendicularPart w v‖ := by
have he : v = inner ℝ w v • w + perpendicularPart w v := by
dsimp [perpendicularPart]
abel
calc
‖v‖ = ‖inner ℝ w v • w + perpendicularPart w v‖ := congrArg norm he
_ ≤ ‖inner ℝ w v • w‖ + ‖perpendicularPart w v‖ := norm_add_le _ _
_ = _ := by rw [norm_smul, Real.norm_eq_abs, hw, mul_one]
/-- Difference of two points satisfying the same scalar interval and the same
perpendicular tube belongs to a ball of the expected radius. -/
theorem two_test_diameter {w : E} (hw : ‖w‖ = 1) {u v : E} {a b : ℝ}
(hlong : |inner ℝ w u - inner ℝ w v| ≤ a)
(hperp : ‖perpendicularPart w u - perpendicularPart w v‖ ≤ b) :
‖u-v‖ ≤ a+b := by
calc
‖u-v‖ ≤ |inner ℝ w (u-v)| + ‖perpendicularPart w (u-v)‖ :=
norm_le_longitudinal_add_perpendicular hw _
_ = |inner ℝ w u - inner ℝ w v| +
‖perpendicularPart w u - perpendicularPart w v‖ := by
rw [inner_sub_right, perpendicularPart_sub]
_ ≤ a+b := add_le_add hlong hperp
/-- When each point lies in a tube of radius τ and its scalar coordinate in
an interval of half-width τ, the full candidate diameter is at most 4τ. -/
theorem retained_candidates_diameter {w : E} (hw : ‖w‖ = 1) {u v : E} {c τ : ℝ}
(hu : |inner ℝ w u - c| ≤ τ) (hv : |inner ℝ w v - c| ≤ τ)
(hpu : ‖perpendicularPart w u‖ ≤ τ) (hpv : ‖perpendicularPart w v‖ ≤ τ) :
‖u-v‖ ≤ 4*τ := by
have hlong : |inner ℝ w u - inner ℝ w v| ≤ 2*τ := by
calc
_ = |(inner ℝ w u-c) - (inner ℝ w v-c)| := by congr 1; ring
_ ≤ |inner ℝ w u-c| + |inner ℝ w v-c| := abs_sub _ _
_ ≤ 2*τ := by linarith
have hperp : ‖perpendicularPart w u - perpendicularPart w v‖ ≤ 2*τ :=
(norm_sub_le _ _).trans (by linarith)
exact (two_test_diameter hw hlong hperp).trans_eq (by ring)
/-- Two centers retained in one cap lie in a slightly wider tube in any
nearby witness direction. This is the geometric input to the parallel test. -/
theorem witness_tube_bound {w w₀ : E} (hw : ‖w‖ = 1) (hw₀ : ‖w₀‖ = 1)
{u v z : E} {τ δ L : ℝ} (hδ : 0 ≤ δ)
(hu : ‖perpendicularPart w₀ (u-z)‖ ≤ τ)
(hv : ‖perpendicularPart w₀ (v-z)‖ ≤ τ)
(hclose : ‖w-w₀‖ ≤ δ) (hsize : ‖u-v‖ ≤ L) :
‖perpendicularPart w (u-v)‖ ≤ 2*τ + 2*δ*L := by
have he : (u-z) - (v-z) = u-v := by abel
have hbase : ‖perpendicularPart w₀ (u-v)‖ ≤ 2*τ := by
rw [← he, perpendicularPart_sub]
exact (norm_sub_le _ _).trans (by linarith)
have herror := perpendicularPart_direction_error hw hw₀ (u-v)
have hg : 2 * ‖w-w₀‖ * ‖u-v‖ ≤ 2*δ*L := by
exact mul_le_mul (mul_le_mul_of_nonneg_left hclose (by norm_num)) hsize
(norm_nonneg _) (mul_nonneg (by norm_num) hδ)
have ht := norm_sub_le_norm_sub_add_norm_sub
(perpendicularPart w (u-v)) (perpendicularPart w₀ (u-v)) 0
simp only [sub_zero] at ht
linarith
end Falconer
end OAI
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