| import Mathlib |
| import OAI.Probability.BinarySweep.MatrixBounds.SchattenBasic |
| |
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical Matrix.Norms.L2Operator |
| |
| namespace BinaryCoordinateSweeps.TraceHolder |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| lemma matrixMoment_sum_le {α : Type*} [Fintype α] {q : ℕ} (hq : 0<q) (A : α → M ι) : |
| matrixMoment q (∑a, A a) ≤ (∑a, schatten q (A a))^(2*q) := by |
| let B : Fin (2*q) → α → M ι := fun j a => if j.val%2=0 then star (A a) else A a |
| have he : (fun j : Fin (2*q) => if j.val%2=0 then star (∑a, A a) else ∑a, A a)= |
| fun j => ∑a, B j a := by |
| funext j |
| dsimp [B] |
| split_ifs < |
| calc |
| _ ≤ ‖Matrix.trace ((star (∑a, A a)*(∑a, A a))^q)‖ := Complex.re_le_norm _ |
| _ = ‖∑x : Fin (2*q) → α, Matrix.trace (List.ofFn (fun j => B j (x j))).prod‖ := by |
| rw [← alternating_gram,he,prod_ofFn_sum,Matrix.trace_sum] |
| _ ≤ ∑x : Fin (2*q) → α, ‖Matrix.trace (List.ofFn (fun j => B j (x j))).prod‖ := norm_sum_le _ _ |
| _ ≤ ∑x : Fin (2*q) → α, ∏j, schatten q (A (x j)) := by |
| apply Finset.sum_le_sum |
| intro x _ |
| convert matrix_trace_holder hq (fun j => B j (x j)) using 1 |
| apply Finset.prod_congr rfl |
| intro coordinate _ |
| dsimp [B,schatten] |
| split_ifs < |
| _ = (∑a, schatten q (A a))^(2*q) := by |
| rw [← Fintype.prod_sum (fun (_ : Fin (2*q)) a => schatten q (A a))] |
| simp |
|
|
| lemma schatten_mul_left {q : ℕ} (hq : 0<q) (C A : M ι) : |
| schatten q (C*A) ≤ ‖C‖*schatten q A := by |
| apply (pow_le_pow_iff_left₀ (schatten_nonneg _ _) |
| (mul_nonneg (norm_nonneg _) (schatten_nonneg _ _)) (by omega : 2*q≠0)).mp |
| rw [mul_pow,schatten_pow hq,schatten_pow hq] |
| exact matrixMoment_mul_left hq C A |
|
|
| lemma schatten_mul_right {q : ℕ} (hq : 0<q) (A C : M ι) : |
| schatten q (A*C) ≤ schatten q A*‖C‖ := by |
| rw [← schatten_star q (A*C),star_mul] |
| simpa only [norm_star,schatten_star,mul_comm] using schatten_mul_left hq (star C) (star A) |
|
|
| lemma norm_le_schatten {q : ℕ} (hq : 0<q) (A : M ι) : ‖A‖≤ schatten q A := by |
| by_cases hz : matrixMoment q A=0 |
| · rw [(matrixMoment_eq_zero_iff hq A).mp hz,norm_zero] |
| exact schatten_nonneg _ _ |
| have ht := lt_of_le_of_ne (matrixMoment_nonneg q A) (Ne.symm hz) |
| obtain ⟨U,V,d,hU,hV,hd,hs,he⟩ := normalized_svd hq A ht |
| have hd1 : ∀i, d i≤ 1 := by |
| intro i |
| rw [← hs] |
| exact Finset.single_le_sum (fun j _ => hd j) (Finset.mem_univ i) |
| have hD : ‖diagPower d ((((2*q:ℕ):ℝ)⁻¹:ℝ):ℂ)‖≤ 1 := |
| diagPower_norm_le_one d hd hd1 (by simp) |
| conv_lhs => rw [he,norm_smul,Real.norm_eq_abs] |
| change |schatten q A| * _ ≤ schatten q A |
| rw [abs_of_nonneg (schatten_nonneg q A)] |
| apply mul_le_of_le_one_right (schatten_nonneg _ _) |
| refine (norm_mul_le _ _).trans ((mul_le_of_le_one_left (norm_nonneg _) ?_).trans hV) |
| exact (norm_mul_le _ _).trans ((mul_le_of_le_one_left (norm_nonneg _) hU).trans hD) |
|
|
| lemma schatten_sandwich {q : ℕ} (hq : 0<q) (A B C : M ι) : |
| schatten q (A*B*C)≤ schatten q A*‖B‖*schatten q C := by |
| exact (schatten_mul_right hq (A*B) C).trans |
| ((mul_le_mul_of_nonneg_right (schatten_mul_right hq A B) (norm_nonneg _)).trans |
| (mul_le_mul_of_nonneg_left (norm_le_schatten hq C) |
| (mul_nonneg (schatten_nonneg _ _) (norm_nonneg _)))) |
|
|
| end BinaryCoordinateSweeps.TraceHolder |
| |
| end |
|
|
| end OAI |
|
|
