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135052a | 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 | import Mathlib
import OAI.Probability.SKGap.Localization.SparseQuadratic
namespace OAI
section
noncomputable section
namespace SKGap
open Matrix Real
open scoped BigOperators RealInnerProductSpace Matrix.Norms.L2Operator
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
lemma weighted_square_le {v z : ℝ} (hv : 0 ≤ v) (hv1 : v ≤ 1) :
(v*z)^2 ≤ v*z^2 := by
have hh := mul_le_mul_of_nonneg_left hv1 hv
have h := mul_le_mul_of_nonneg_right hh (sq_nonneg z)
nlinarith only [h]
lemma weighted_square_le_small {v z η : ℝ} (hv : 0 ≤ v) (hvη : v ≤ η) :
(v*z)^2 ≤ η*(v*z^2) := by
have hh := mul_le_mul_of_nonneg_left hvη hv
have h := mul_le_mul_of_nonneg_right hh (sq_nonneg z)
nlinarith only [h]
lemma weighted_sparse_form (B : Matrix ι ι ℝ) (S : Finset ι) (v z : ι→ℝ)
{η ε M : ℝ} (hη : 0 ≤ η) (hε : 0 ≤ ε) (hM : 0 ≤ M)
(hv : ∀ i,0 ≤ v i) (hv1 : ∀ i,v i ≤ 1) (hsmall : ∀ i,i ∉ S → v i ≤ η)
(hB : ‖B‖ ≤ M)
(hs : ∀ w : EuclideanSpace ℝ ι,(∀ i,i ∉ S → w i=0) →
|⟪w,Matrix.toEuclideanCLM (𝕜 := ℝ) (n := ι) B w⟫| ≤ ε*‖w‖^2) :
|∑ i,∑ j,B i j*(v i*z i)*(v j*z j)| ≤
(ε+M*(2*sqrt η+η))*(∑ i,v i*z i^2) := by
let b : EuclideanSpace ℝ ι := WithLp.toLp 2 (fun i=>if i ∈ S then v i*z i else 0)
let c : EuclideanSpace ℝ ι := WithLp.toLp 2 (fun i=>if i ∈ S then 0 else v i*z i)
let q := ∑ i,v i*z i^2
have hq : 0 ≤ q := Finset.sum_nonneg (fun i _=>mul_nonneg (hv i) (sq_nonneg _))
have hb : ‖b‖^2 ≤ q := by
rw [EuclideanSpace.real_norm_sq_eq]
apply Finset.sum_le_sum
intro i _
change (if i ∈ S then v i*z i else 0)^2 ≤ v i*z i^2
split_ifs
· exact weighted_square_le (hv i) (hv1 i)
· exact (by simpa only [zero_pow (by omega : (2:ℕ) ≠ 0)] using mul_nonneg (hv i) (sq_nonneg (z i)))
have hc : ‖c‖^2 ≤ η*q := by
rw [EuclideanSpace.real_norm_sq_eq,Finset.mul_sum]
apply Finset.sum_le_sum
intro i _
change (if i ∈ S then 0 else v i*z i)^2 ≤ η*(v i*z i^2)
split_ifs with hi
· exact (by simpa only [zero_pow (by omega : (2:ℕ) ≠ 0)] using mul_nonneg hη (mul_nonneg (hv i) (sq_nonneg (z i))))
· exact weighted_square_le_small (hv i) (hsmall i hi)
have hsb : |⟪b,Matrix.toEuclideanCLM (𝕜 := ℝ) (n := ι) B b⟫| ≤ ε*q := by
apply (hs b (by intro i hi;change (if i ∈ S then v i*z i else 0)=0;simp [hi])).trans
exact mul_le_mul_of_nonneg_left hb hε
have hh := sparse_quadratic_split (Matrix.toEuclideanCLM (𝕜 := ℝ) (n := ι) B) b c hq hη hM
(by simpa only [Matrix.l2_opNorm_toEuclideanCLM] using hB) hb hc hsb
have hbc : b+c=WithLp.toLp 2 (fun i=>v i*z i) := by
ext i
change (if i ∈ S then v i*z i else 0)+(if i ∈ S then 0 else v i*z i)=v i*z i
split_ifs <;> simp
rw [hbc,Matrix.inner_toEuclideanCLM] at hh
have he : (WithLp.toLp 2 (fun i=>v i*z i) : EuclideanSpace ℝ ι) ⬝ᵥ
B *ᵥ (WithLp.toLp 2 (fun i=>v i*z i) : EuclideanSpace ℝ ι)=
∑ i,∑ j,B i j*(v i*z i)*(v j*z j) := by
simp only [dotProduct,Matrix.mulVec]
simp_rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _;ring
rwa [he] at hh
end SKGap
end
end
end OAI
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