| 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 |
|
|
