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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 | import Mathlib
import OAI.Combinatorics.MatchingPSD.PSDEvaluation
namespace OAI
namespace AffinePSD
open Matrix Finset
open scoped BigOperators
variable {I R : Type*} [Fintype I] [Fintype R] [DecidableEq R]
lemma zero_col_of_diag_zero {A : Matrix R R ℝ} (hA : A.PosSemidef)
(i : R) (hi : A i i = 0) (j : R) : A j i = 0 := by
have h := (hA.dotProduct_mulVec_zero_iff (x := Pi.single i 1)).mp (by simpa using hi)
simpa using congrFun h j
lemma zero_row_of_diag_zero {A : Matrix R R ℝ} (hA : A.PosSemidef)
(i : R) (hi : A i i = 0) (j : R) : A i j = 0 := by
have hs : A i j = A j i := by
simpa using congrArg (fun M : Matrix R R ℝ => M j i) hA.isHermitian.eq
rw [hs]
exact zero_col_of_diag_zero hA i hi j
lemma diagonal_compression (B : I → Matrix R R ℝ) (hB : ∀ i, (B i).PosSemidef)
(d : R → ℝ) (hs : ∑ i, B i = diagonal d) (A : Matrix R R ℝ)
(hA : A.PosSemidef) :
∃ A' : Matrix R R ℝ, A'.PosSemidef ∧ evaluation B A' = evaluation B A ∧
(∀ j, d j = 0 → A' j j = 0) := by
classical
let p : R → ℝ := fun j => if d j = 0 then 0 else 1
let P := diagonal p
have hdiag (i : I) (j : R) (hj : d j = 0) : B i j j = 0 := by
have hle : B i j j ≤ ∑ k, B k j j :=
Finset.single_le_sum (fun k _ => (hB k).diag_nonneg) (Finset.mem_univ i)
have he : ∑ k, B k j j = d j := by
simpa only [Matrix.sum_apply, Matrix.diagonal_apply_eq] using
congrArg (fun M : Matrix R R ℝ => M j j) hs
have hn := (hB i).diag_nonneg (i := j)
rw [he, hj] at hle
exact le_antisymm hle hn
have hpB (i : I) : P * B i = B i := by
ext j k
simp only [P, Matrix.diagonal_mul, p]
split_ifs with hj
· simp [zero_row_of_diag_zero (hB i) j (hdiag i j hj) k]
· simp
have hBp (i : I) : B i * P = B i := by
ext j k
simp only [P, Matrix.mul_diagonal, p]
split_ifs with hk
· simp [zero_col_of_diag_zero (hB i) k (hdiag i k hk) j]
· simp
refine ⟨P * A * P, ?_, ?_, ?_⟩
· simpa only [Matrix.conjTranspose_eq_transpose_of_trivial, P, Matrix.diagonal_transpose]
using hA.conjTranspose_mul_mul_same P
· ext i
change (P * A * P * B i).trace = (A * B i).trace
rw [Matrix.mul_assoc P A P, Matrix.mul_assoc P (A * P) (B i),
Matrix.trace_mul_comm P, Matrix.mul_assoc A P (B i), hpB,
Matrix.mul_assoc A (B i) P, hBp]
· intro j hj
simp [P, Matrix.mul_diagonal, Matrix.diagonal_mul, p, hj]
omit [Fintype R] in
lemma sum_diagonal_nonneg (B : I → Matrix R R ℝ) (hB : ∀ i, (B i).PosSemidef)
(d : R → ℝ) (hs : ∑ i, B i = diagonal d) : ∀ j, 0 ≤ d j := by
intro j
have he : ∑ i, B i j j = d j := by
simpa only [Matrix.sum_apply, Matrix.diagonal_apply_eq] using
congrArg (fun M : Matrix R R ℝ => M j j) hs
rw [← he]
exact Finset.sum_nonneg (fun i _ => (hB i).diag_nonneg)
lemma evaluation_sum_eq (B : I → Matrix R R ℝ) (d : R → ℝ)
(hs : ∑ i, B i = diagonal d) (A : Matrix R R ℝ) :
∑ i, evaluation B A i = ∑ j, A j j * d j := by
simp only [evaluation_apply]
rw [← Matrix.trace_sum, ← Matrix.mul_sum, hs]
simp [Matrix.trace, Matrix.diag, Matrix.mul_diagonal]
lemma diagonal_trace_bound (B : I → Matrix R R ℝ) (hB : ∀ i, (B i).PosSemidef)
(d : R → ℝ) (hs : ∑ i, B i = diagonal d) (A : Matrix R R ℝ)
(hA : A.PosSemidef) (hz : ∀ j, d j = 0 → A j j = 0) :
A.trace ≤ (∑ j, (d j)⁻¹) * ∑ i, evaluation B A i := by
have hd := sum_diagonal_nonneg B hB d hs
have he := evaluation_sum_eq B d hs A
rw [he]
change (∑ j, A j j) ≤ _
rw [Finset.sum_mul]
apply Finset.sum_le_sum
intro j _
by_cases hj : d j = 0
· simp [hj, hz j hj]
· have hp : 0 < d j := lt_of_le_of_ne (hd j) (Ne.symm hj)
have hle : A j j * d j ≤ ∑ k, A k k * d k :=
Finset.single_le_sum (fun k _ => mul_nonneg (hA.diag_nonneg (i := k)) (hd k)) (Finset.mem_univ j)
calc
A j j = (d j)⁻¹ * (A j j * d j) := by field_simp
_ ≤ (d j)⁻¹ * ∑ k, A k k * d k := mul_le_mul_of_nonneg_left hle (inv_nonneg.mpr hp.le)
end AffinePSD
end OAI
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