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9655094 | 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 | import OAI.Analysis.C0Absorption.Finite
namespace OAI
open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty
open NormedSpace
namespace C0Absorption
noncomputable section
open Set Filter Topology NormedSpace
open scoped NNReal BigOperators ZeroAtInfty
variable {N : ℕ}
def finiteRecomputed (N : ℕ) (x : FInput N) : FInput N :=
finiteFrozen N (stateWeights (normalizedBall (finiteEmbed N x))) x
@[simp] theorem finiteRecomputed_zero (N : ℕ) : finiteRecomputed N 0=0 := map_zero _
theorem finiteFrozen_weight_error {W V : FrozenWeights} {x : FInput N} {q : ℝ}
(hq : 0≤q) (h : WeightError W V (finiteEmbed N x) q) :
‖finiteFrozen N W x-finiteFrozen N V x‖≤6*q := by
apply (pi_norm_le_iff_of_nonneg (mul_nonneg (by norm_num) hq)).mpr
rintro (i|γ)
· change |finiteRow W (i,Fin.last N) x-finiteRow V (i,Fin.last N) x|≤_
rw [finiteRow_eq,finiteRow_eq]
exact (rowScalar_weight_error W V _ _ (h.row _)).trans (by linarith)
· change |finiteFrozenFunction W x (.inr γ)-finiteFrozenFunction V x (.inr γ)|≤_
rw [finiteFrozen_block,finiteFrozen_block]
exact (cfun_norm_apply_le (frozenShift W (finiteEmbed N x)-frozenShift V (finiteEmbed N x)) (fLabel γ)).trans
(shift_weight_error hq h)
theorem finiteRecomputed_mixed_left (x y : FInput N) :
‖finiteRecomputed N x-finiteFrozen N (mixedWeights (normalizedBall (finiteEmbed N x)) (normalizedBall (finiteEmbed N y))) x‖≤12*eta*dist x y := by
have hh := finiteFrozen_weight_error (mul_nonneg (mul_nonneg (by norm_num) eta_pos.le) dist_nonneg)
(normalized_error_left (finiteEmbed N x) (finiteEmbed N y))
rw [LinearIsometry.dist_map] at hh
exact hh.trans (by apply le_of_eq; ring)
theorem finiteRecomputed_mixed_right (x y : FInput N) :
‖finiteRecomputed N y-finiteFrozen N (mixedWeights (normalizedBall (finiteEmbed N x)) (normalizedBall (finiteEmbed N y))) y‖≤12*eta*dist x y := by
have hh := finiteFrozen_weight_error (mul_nonneg (mul_nonneg (by norm_num) eta_pos.le) dist_nonneg)
(normalized_error_right (finiteEmbed N x) (finiteEmbed N y))
rw [LinearIsometry.dist_map] at hh
exact hh.trans (by apply le_of_eq; ring)
theorem finiteRecomputed_mixed_approx (x y : FInput N) :
‖finiteRecomputed N x-finiteRecomputed N y-
finiteFrozen N (mixedWeights (normalizedBall (finiteEmbed N x)) (normalizedBall (finiteEmbed N y))) (x-y)‖≤24*eta*dist x y := by
let W := mixedWeights (normalizedBall (finiteEmbed N x)) (normalizedBall (finiteEmbed N y))
have he : finiteRecomputed N x-finiteRecomputed N y-finiteFrozen N W (x-y)=
(finiteRecomputed N x-finiteFrozen N W x)-(finiteRecomputed N y-finiteFrozen N W y) := by rw [map_sub]; abel
rw [he]
exact (norm_sub_le _ _).trans (by linarith [finiteRecomputed_mixed_left x y,finiteRecomputed_mixed_right x y])
theorem fiveBounds_of_error {E : Type*} [NormedAddCommGroup E] {a b : E} {d e : ℝ}
(he : ‖a-b‖≤e*d) (hU : ‖b‖≤5*d) (hL : d≤5*‖b‖) :
(1/5-e)*d≤‖a‖ ∧ ‖a‖≤(5+e)*d := by
have hu : ‖a‖≤‖a-b‖+‖b‖ := by simpa only [sub_add_cancel] using norm_add_le (a-b) b
have hl : ‖b‖≤‖a-b‖+‖a‖ := by simpa only [sub_add_cancel,norm_sub_rev] using norm_add_le (b-a) a
constructor <;> nlinarith
theorem finiteRecomputed_global_bounds (x y : FInput N) :
(1/5-24*eta)*dist x y≤dist (finiteRecomputed N x) (finiteRecomputed N y) ∧
dist (finiteRecomputed N x) (finiteRecomputed N y)≤(5+24*eta)*dist x y := by
let W := mixedWeights (normalizedBall (finiteEmbed N x)) (normalizedBall (finiteEmbed N y))
let a := finiteRecomputed N x-finiteRecomputed N y
let b := finiteFrozen N W (x-y)
have he : ‖a-b‖≤24*eta*dist x y := finiteRecomputed_mixed_approx x y
have hU : ‖b‖≤5*dist x y := by
change ‖finiteFrozenFunction W (x-y)‖≤5*dist x y
rw [dist_eq_norm]
exact finiteFrozen_bound W (x-y)
have hL : dist x y≤5*‖b‖ := by simpa only [dist_eq_norm] using finiteFrozen_lower W (x-y)
rw [dist_eq_norm (finiteRecomputed N x) (finiteRecomputed N y)]
exact fiveBounds_of_error he hU hL
theorem finiteRecomputed_lipschitz (N : ℕ) : LipschitzWith (628/125) (finiteRecomputed N) := by
apply LipschitzWith.of_dist_le_mul
intro x y
have hh := (finiteRecomputed_global_bounds x y).2
norm_num [eta] at hh ⊢
exact hh
theorem finiteRecomputed_antilipschitz (N : ℕ) : AntilipschitzWith (125/22) (finiteRecomputed N) := by
apply AntilipschitzWith.of_le_mul_dist
intro x y
have h := (finiteRecomputed_global_bounds x y).1
norm_num [eta] at h ⊢
nlinarith
end
end C0Absorption
namespace C0Absorption
noncomputable section
open Set Filter Topology NormedSpace
open scoped NNReal BigOperators
section FiniteOperators
variable {A : Type*} [Fintype A] [DecidableEq A] {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
theorem finite_operator_norm_le_columns (T : (A → ℝ) →L[ℝ] E) :
‖T‖≤∑ a,‖T (Pi.single a 1)‖ := by
classical
apply ContinuousLinearMap.opNorm_le_bound _ (Finset.sum_nonneg (fun a _ => norm_nonneg _))
intro x
have he : (∑ a, x a • Pi.single a (1 : ℝ))=x := by
ext b
simp [Pi.single_apply]
have hEq : T x=∑ a,x a • T (Pi.single a 1) := by
simpa only [map_sum,map_smul] using congrArg T he.symm
rw [hEq]
calc
_ ≤ ∑ a, ‖x a • T (Pi.single a 1)‖ := norm_sum_le _ _
_ = ∑ a, ‖T (Pi.single a 1)‖*‖x a‖ := by simp only [norm_smul,mul_comm]
_ ≤ ∑ a, ‖T (Pi.single a 1)‖*‖x‖ := Finset.sum_le_sum (fun a _ =>
mul_le_mul_of_nonneg_left (norm_le_pi_norm x a) (norm_nonneg _))
_ = _ := by rw [Finset.sum_mul]
theorem tendsto_finiteOperator_of_columns {X : Type*} {l : Filter X}
(T : X → (A → ℝ) →L[ℝ] E) (S : (A → ℝ) →L[ℝ] E)
(h : ∀ a, Tendsto (fun z => T z (Pi.single a 1)) l (nhds (S (Pi.single a 1)))) :
Tendsto T l (nhds S) := by
classical
rw [tendsto_iff_norm_sub_tendsto_zero]
apply squeeze_zero (fun z => norm_nonneg _) (fun z => finite_operator_norm_le_columns (T z-S))
have hh : Tendsto (fun z => ∑ a, ‖T z (Pi.single a 1)-S (Pi.single a 1)‖) l (nhds (∑ a : A, (0 : ℝ))) :=
tendsto_finsetSum _ (fun a _ => tendsto_iff_norm_sub_tendsto_zero.mp (h a))
simpa only [sub_apply,Finset.sum_const_zero] using hh
end FiniteOperators
theorem diagonal_neighborhood {X Y G : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[NormedAddCommGroup G] (q : X → Y) {x : X} (hq : ContinuousAt q x)
(A : Y × Y → G) (g : G) (hA : Tendsto A (nhds (q x,q x)) (nhds g))
{ε : ℝ} (hε : 0<ε) :
∃ s : Set X, s∈nhds x ∧ ∀ y∈s, ∀ z∈s, ‖A (q y,q z)-g‖<ε := by
have hp : Tendsto (fun p : X × X => (q p.1,q p.2)) (nhds (x,x)) (nhds (q x,q x)) :=
(hq.comp continuous_fst.continuousAt).prodMk (hq.comp continuous_snd.continuousAt)
have ht := hA.comp hp
have he : {p : X × X | ‖A (q p.1,q p.2)-g‖<ε}∈nhds (x,x) := by
have hh := Metric.tendsto_nhds.mp ht ε hε
change ∀ᶠ p : X × X in nhds (x,x), ‖A (q p.1,q p.2)-g‖<ε
filter_upwards [hh] with p h
simpa only [Function.comp_def,dist_eq_norm] using h
rw [nhds_prod_eq] at he
obtain ⟨s,hs,t,ht,hst⟩ := Filter.mem_prod_iff.mp he
exact ⟨s∩t,inter_mem hs ht,fun y hy z hz => @hst (y,z) ⟨hy.1,hz.2⟩⟩
end
end C0Absorption
end OAI
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