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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Localization.DiagonalHessianJoined
namespace OAI
section
noncomputable section
namespace SKGap
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
lemma real_opNorm_add (M N : Matrix ι ι ℝ) : opNorm (M+N) ≤ opNorm M+opNorm N := by
unfold opNorm
rw [map_add,map_add]
exact norm_add_le _ _
lemma real_opNorm_smul (c : ℝ) (M : Matrix ι ι ℝ) : opNorm (c • M)=|c| * opNorm M := by
unfold opNorm
rw [map_smul,map_smul,norm_smul,Real.norm_eq_abs]
lemma operator_vecMulVec_le (u v : EuclideanSpace ℝ ι) :
opNorm (vecMulVec u.ofLp v.ofLp) ≤ ‖u‖*‖v‖ := by
apply (operator_le_frobenius _).trans
apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _))).mp
rw [Matrix.frobenius_norm_def]
simp only [Real.rpow_two,← Real.sqrt_eq_rpow]
rw [Real.sq_sqrt (Finset.sum_nonneg (fun i _=>Finset.sum_nonneg (fun k _=>sq_nonneg _))),
mul_pow,EuclideanSpace.real_norm_sq_eq,EuclideanSpace.real_norm_sq_eq]
simp only [vecMulVec_apply,Real.norm_eq_abs,abs_mul,mul_pow,sq_abs]
rw [Finset.sum_mul]
apply le_of_eq
apply Finset.sum_congr rfl
intro i _
rw [Finset.mul_sum]
lemma operator_mulVec_norm_le (W : Matrix ι ι ℝ) (u : EuclideanSpace ℝ ι) :
‖WithLp.toLp 2 (W*ᵥu.ofLp)‖ ≤ opNorm W*‖u‖ := by
exact W.toEuclideanLin.toContinuousLinearMap.le_opNorm u
lemma goeBilinear_norm_bound (r : ℝ) (u : EuclideanSpace ℝ ι)
(g : MatrixCoordinates ι → ℝ) :
|goeBilinear r u.ofLp u.ofLp g| ≤ opNorm (goeMatrix r g)*‖u‖^2 := by
have hh := matrix_quadratic_op_bound (goeMatrix r g) u.ofLp
change |goeBilinear r u.ofLp u.ofLp g| ≤ opNorm (goeMatrix r g)*(u.ofLp⬝ᵥu.ofLp) at hh
convert hh using 1
rw [EuclideanSpace.real_norm_sq_eq]
simp only [dotProduct,← sq]
lemma conditionalBlockMatrix_norm {r j q s S a B κ ell : ℝ}
(u Z : EuclideanSpace ℝ ι) (g : MatrixCoordinates ι → ℝ) (hu : ‖u‖=1) :
opNorm (conditionalBlockMatrix r j q s S a B κ ell u.ofLp Z.ofLp g) ≤
(1+2*|κ-1|+|ell-2*κ+1|)*opNorm (goeMatrix r g)+
2*(|j*sqrt (q/s)| * ‖Z‖+|j*a/s|)+|2*j*(a+B*q)/S| := by
let W := goeMatrix r g
let Wu : EuclideanSpace ℝ ι := WithLp.toLp 2 (W*ᵥu.ofLp)
let w : EuclideanSpace ℝ ι := (j*sqrt (q/s)) • Z-(j*a/s) • u
have hW : 0 ≤ opNorm W := norm_nonneg _
have hwu : ‖Wu‖ ≤ opNorm W := by simpa only [hu,mul_one] using operator_mulVec_norm_le W u
have huu : opNorm (vecMulVec u.ofLp u.ofLp) ≤ 1 := by simpa only [hu,mul_one] using operator_vecMulVec_le u u
have h1 : opNorm (vecMulVec u.ofLp Wu.ofLp+vecMulVec Wu.ofLp u.ofLp) ≤ 2*opNorm W := by
apply (real_opNorm_add _ _).trans
have ha := operator_vecMulVec_le u Wu
have hb := operator_vecMulVec_le Wu u
rw [hu,one_mul] at ha
rw [hu,mul_one] at hb
linarith
have hα : |goeBilinear r u.ofLp u.ofLp g| ≤ opNorm W := by
simpa only [hu,one_pow,mul_one] using goeBilinear_norm_bound r u g
have hw : ‖w‖ ≤ |j*sqrt (q/s)| * ‖Z‖+|j*a/s| := by
apply (norm_sub_le _ _).trans
simp only [norm_smul,Real.norm_eq_abs,hu,mul_one]
rfl
have h2 : opNorm (vecMulVec u.ofLp w.ofLp+vecMulVec w.ofLp u.ofLp) ≤
2*(|j*sqrt (q/s)| * ‖Z‖+|j*a/s|) := by
apply (real_opNorm_add _ _).trans
have ha := operator_vecMulVec_le u w
have hb := operator_vecMulVec_le w u
rw [hu,one_mul] at ha
rw [hu,mul_one] at hb
linarith
have he : conditionalBlockMatrix r j q s S a B κ ell u.ofLp Z.ofLp g=
(W+(κ-1) • (vecMulVec u.ofLp Wu.ofLp+vecMulVec Wu.ofLp u.ofLp)+
((ell-2*κ+1)*goeBilinear r u.ofLp u.ofLp g) • vecMulVec u.ofLp u.ofLp)+
(vecMulVec u.ofLp w.ofLp+vecMulVec w.ofLp u.ofLp)+
(2*j*(a+B*q)/S) • vecMulVec u.ofLp u.ofLp := by
unfold conditionalBlockMatrix
rw [goeBlockResidual_expand]
dsimp [W,Wu,w]
abel
rw [he]
have hn := real_opNorm_add ((W+(κ-1) • (vecMulVec u.ofLp Wu.ofLp+vecMulVec Wu.ofLp u.ofLp)+
((ell-2*κ+1)*goeBilinear r u.ofLp u.ofLp g) • vecMulVec u.ofLp u.ofLp)+
(vecMulVec u.ofLp w.ofLp+vecMulVec w.ofLp u.ofLp)) ((2*j*(a+B*q)/S) • vecMulVec u.ofLp u.ofLp)
apply hn.trans
have hd := real_opNorm_add (W+(κ-1) • (vecMulVec u.ofLp Wu.ofLp+vecMulVec Wu.ofLp u.ofLp)+
((ell-2*κ+1)*goeBilinear r u.ofLp u.ofLp g) • vecMulVec u.ofLp u.ofLp)
(vecMulVec u.ofLp w.ofLp+vecMulVec w.ofLp u.ofLp)
have hc := real_opNorm_add (W+(κ-1) • (vecMulVec u.ofLp Wu.ofLp+vecMulVec Wu.ofLp u.ofLp))
(((ell-2*κ+1)*goeBilinear r u.ofLp u.ofLp g) • vecMulVec u.ofLp u.ofLp)
have hb := real_opNorm_add W ((κ-1) • (vecMulVec u.ofLp Wu.ofLp+vecMulVec Wu.ofLp u.ofLp))
rw [real_opNorm_smul] at hb
rw [real_opNorm_smul,abs_mul] at hc
rw [real_opNorm_smul]
have hp1 := mul_le_mul_of_nonneg_left h1 (abs_nonneg (κ-1))
have hp2 : |ell-2*κ+1| * |goeBilinear r u.ofLp u.ofLp g| * opNorm (vecMulVec u.ofLp u.ofLp) ≤
|ell-2*κ+1| * opNorm W := by
apply (mul_le_mul_of_nonneg_left huu (by positivity)).trans
rw [mul_one]
exact mul_le_mul_of_nonneg_left hα (abs_nonneg _)
have hp3 := mul_le_mul_of_nonneg_left huu (abs_nonneg (2*j*(a+B*q)/S))
nlinarith only [hd,hc,hb,hp1,hp2,hp3,h2]
end SKGap
end
end
end OAI
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