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e5a97c6 | 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 | import OAI.Combinatorics.MatchingPSD.PolynomialInterpolation
namespace OAI
universe u_1 u_2
namespace MatchingInterpolation
open MatchingSmoothing Matrix PerfectMatchingPSD
open scoped BigOperators MatrixOrder
attribute [local instance] Classical.propDecidable
noncomputable def specialPoint : Fin 5 → Point :=
![origin, origin+dirP, origin+dirStar, origin+dirA, origin+dirB]
def specialCoeff : Fin 5 → ℝ := ![1,-2,-1,1,1]
noncomputable def scaledNode {d : ℕ} (L : ℝ) (g : GridIndex d) : Point :=
fun a => origin a+L*(g a:ℝ)
noncomputable def weightMass (d : ℕ) (L : ℝ) : ℝ := ∑ g, |weight d L g|
lemma weightMass_nonneg (d : ℕ) (L : ℝ) : 0 ≤ weightMass d L :=
Finset.sum_nonneg (fun _ _ => abs_nonneg _)
lemma sum_abs_specialCoeff : (∑ j, |specialCoeff j|) = 6 := by
norm_num [specialCoeff, Fin.sum_univ_succ]
lemma five_interpolation {d : ℕ} (P : Point → ℝ) (hP : P ∈ lowDegree coord d)
{L : ℝ} (hL : L ≠ 0) :
(∑ j, specialCoeff j * P (specialPoint j)) = ∑ g, weight d L g * P (scaledNode L g) := by
have h := weighted_interpolation d P hP L hL
simp only [specialCoeff, specialPoint, Fin.sum_univ_succ, Matrix.cons_val_zero,
Matrix.cons_val_succ, one_mul, neg_mul, Fin.sum_univ_zero, add_zero] at *
change _ = ∑ g, weight d L g * P (fun a => origin a+L*(g a:ℝ))
linarith
lemma finite_weighted_error {X : Type u_1} [Fintype X]
(c e : X → ℝ) {η : ℝ} (he : ∀ x, |e x| ≤ η) :
|∑ x, c x*e x| ≤ (∑ x, |c x|)*η := by
calc
_ ≤ ∑ x, |c x*e x| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ x, |c x| *η := by
apply Finset.sum_le_sum
intro x _
rw [abs_mul]
exact mul_le_mul_of_nonneg_left (he x) (abs_nonneg _)
_ = _ := (Finset.sum_mul _ _ _).symm
lemma interpolation_error {d : ℕ} (F P : Point → ℝ)
(hP : P ∈ lowDegree coord d) {L η : ℝ} (hL : L ≠ 0)
(hs : ∀ j, |F (specialPoint j)-P (specialPoint j)| ≤ η)
(hg : ∀ g : GridIndex d, |F (scaledNode L g)-P (scaledNode L g)| ≤ η) :
|(∑ j, specialCoeff j*F (specialPoint j)) -
∑ g, weight d L g*F (scaledNode L g)| ≤ (6+weightMass d L)*η := by
have hidentity : (∑ j, specialCoeff j*F (specialPoint j)) -
∑ g, weight d L g*F (scaledNode L g) =
(∑ j, specialCoeff j*(F (specialPoint j)-P (specialPoint j))) -
∑ g, weight d L g*(F (scaledNode L g)-P (scaledNode L g)) := by
simp only [mul_sub, Finset.sum_sub_distrib, five_interpolation P hP hL]
ring
rw [hidentity]
have h1 := finite_weighted_error specialCoeff (fun j => F (specialPoint j)-P (specialPoint j)) hs
have h2 := finite_weighted_error (weight d L) (fun g => F (scaledNode L g)-P (scaledNode L g)) hg
rw [sum_abs_specialCoeff] at h1
calc
_ ≤ |∑ j, specialCoeff j*(F (specialPoint j)-P (specialPoint j))| +
|∑ g, weight d L g*(F (scaledNode L g)-P (scaledNode L g))| := abs_sub _ _
_ ≤ 6*η + weightMass d L*η := add_le_add h1 h2
_ = _ := by ring
section Matrices
variable {R : Type u_2} [Fintype R] [DecidableEq R]
lemma interpolation_matrix_domination (d : ℕ)
(B : Point → Matrix R R ℝ) (hB : ∀ x, (B x).PosSemidef)
(P : R → R → Point → ℝ) (hP : ∀ u v, P u v ∈ lowDegree coord d)
{L η : ℝ} (hL : L ≠ 0) (hη : 0 ≤ η)
(hs : ∀ j u v, |B (specialPoint j) u v-P u v (specialPoint j)| ≤ η)
(hg : ∀ g : GridIndex d, ∀ u v, |B (scaledNode L g) u v-P u v (scaledNode L g)| ≤ η) :
∃ E : Matrix R R ℝ, E.PosSemidef ∧
B (origin+dirA)+B (origin+dirB) ≤ B (origin+dirStar)+(2:ℝ) • B (origin+dirP)+E ∧
∀ (Q : Matrix R R ℝ) (m k : ℝ), Q.trace ≤ m →
(∀ g : GridIndex d, (Q*B (scaledNode L g)).trace ≤ k) →
(Q*E).trace ≤ weightMass d L*k +
(Fintype.card R : ℝ)*(6+weightMass d L)*η*m := by
let H := (∑ j, specialCoeff j • B (specialPoint j)) -
∑ g, weight d L g • B (scaledNode L g)
let b := (6+weightMass d L)*η
let c := (Fintype.card R : ℝ)*b
let W := ∑ g, |weight d L g| • B (scaledNode L g)
have hb : 0 ≤ b := mul_nonneg (by linarith [weightMass_nonneg d L]) hη
have hc : 0 ≤ c := mul_nonneg (Nat.cast_nonneg _) hb
have hH : H.IsHermitian := by
have hh (x : Point) := (hB x).isHermitian.eq
simp only [H, Matrix.IsHermitian, Matrix.conjTranspose_sub, Matrix.conjTranspose_sum,
Matrix.conjTranspose_smul, star_trivial, hh]
have hentry (u v : R) : |H u v| ≤ b := by
simpa only [H, Matrix.sub_apply, Matrix.sum_apply, Matrix.smul_apply, smul_eq_mul] using
interpolation_error (fun x => B x u v) (P u v) (hP u v) hL (fun j => hs j u v) (fun g => hg g u v)
have hbound : H ≤ c • (1 : Matrix R R ℝ) := le_card_entry_bound hH hb hentry
have hW : W.PosSemidef := matrix_sum_psd _ _ (fun g _ => (hB _).smul (abs_nonneg _))
let E := W+c • (1 : Matrix R R ℝ)
have hE : E.PosSemidef := hW.add (Matrix.PosSemidef.one.smul hc)
refine ⟨E,hE,?_,?_⟩
· apply sub_nonneg.mp
apply Matrix.PosSemidef.nonneg
have hdiff : (∑ g : GridIndex d, (|weight d L g|-weight d L g) • B (scaledNode L g)).PosSemidef :=
matrix_sum_psd _ _ (fun g _ => (hB _).smul (sub_nonneg.mpr (le_abs_self _)))
convert ((sub_nonneg.mpr hbound).posSemidef.add (hB origin)).add hdiff using 1
dsimp [E,W,H]
simp only [sub_smul, Finset.sum_sub_distrib, specialPoint, specialCoeff, Fin.sum_univ_succ,
Matrix.cons_val_zero, Matrix.cons_val_succ, one_smul, neg_smul,
Fin.sum_univ_zero, add_zero]
module
· intro Q m k hm hk
change (Q*(W+c • (1 : Matrix R R ℝ))).trace ≤ _
rw [mul_add, Matrix.trace_add, mul_smul_comm, Matrix.trace_smul, mul_one]
have hWt : (Q*W).trace ≤ weightMass d L*k := by
dsimp [W,weightMass]
rw [Finset.mul_sum, Matrix.trace_sum, Finset.sum_mul]
apply Finset.sum_le_sum
intro g _
rw [mul_smul_comm, Matrix.trace_smul]
exact mul_le_mul_of_nonneg_left (hk g) (abs_nonneg _)
have he := add_le_add hWt (mul_le_mul_of_nonneg_left hm hc)
calc
_ ≤ weightMass d L*k + c*m := he
_ = _ := by dsimp [c,b]; ring
end Matrices
end MatchingInterpolation
end OAI
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