| import OAI.Combinatorics.MatchingPSD.BitTests |
| |
| namespace OAI |
|
|
| universe u_1 u_2 |
|
|
| namespace PerfectMatchingPSD |
| open Matrix |
| open scoped MatrixOrder |
| variable {R : Type u_1} [Fintype R] [DecidableEq R] |
|
|
| lemma entry_product_bound {a x y b : ℝ} (hb : 0 ≤ b) (ha : |a| ≤ b) : |
| x*a*y ≤ b*(x^2+y^2)/2 := by |
| have hxy : 2 * |x| * |y| ≤ x^2+y^2 := by |
| nlinarith [sq_nonneg (|x|-|y|), sq_abs x, sq_abs y] |
| have he : x*a*y ≤ b*(|x| * |y|) := by |
| calc |
| _ ≤ |x*a*y| := le_abs_self _ |
| _ = |a| *(|x| * |y|) := by rw [abs_mul, abs_mul] |
| _ ≤ _ := mul_le_mul_of_nonneg_right ha (mul_nonneg (abs_nonneg _) (abs_nonneg _)) |
| nlinarith [mul_le_mul_of_nonneg_left hxy hb] |
|
|
| lemma le_card_entry_bound {E : Matrix R R ℝ} (hE : E.IsHermitian) |
| {b : ℝ} (hb : 0 ≤ b) (hbound : ∀ i j, |E i j| ≤ b) : |
| E ≤ ((Fintype.card R : ℝ)*b) • (1 : Matrix R R ℝ) := by |
| apply sub_nonneg.mp |
| apply Matrix.PosSemidef.nonneg |
| apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg |
| · exact (isHermitian_one.smul (.of_nonneg (mul_nonneg (Nat.cast_nonneg _) hb))).sub hE |
| · intro x |
| have hsum : (∑ i, ∑ j, x i*E i j*x j) ≤ |
| (Fintype.card R : ℝ)*b*(∑ i, (x i)^2) := by |
| calc |
| _ ≤ ∑ i, ∑ j, (b/2*(x i)^2 + b/2*(x j)^2) := by |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j _ |
| have he := entry_product_bound (x := x i) (y := x j) hb (hbound i j) |
| linarith |
| _ = _ := by |
| simp only [Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ, |
| nsmul_eq_mul, ← Finset.mul_sum] |
| ring |
| simp only [Matrix.sub_mulVec, Matrix.smul_mulVec, Matrix.one_mulVec, |
| dotProduct_sub, dotProduct_smul, star_trivial] |
| rw [sub_nonneg] |
| simpa only [dotProduct, Matrix.mulVec, Finset.mul_sum, pow_two, smul_eq_mul, mul_assoc] using hsum |
|
|
| lemma posSemidef_entry_abs_le_trace {A : Matrix R R ℝ} (hA : A.PosSemidef) (i j : R) : |
| |A i j| ≤ A.trace := by |
| have hi : 0 ≤ A i i := by |
| have he := hA.dotProduct_mulVec_nonneg (Pi.single i 1) |
| simpa using he |
| have hj : 0 ≤ A j j := by |
| have he := hA.dotProduct_mulVec_nonneg (Pi.single j 1) |
| simpa using he |
| have hsym : A j i = A i j := by |
| simpa using congrArg (fun M : Matrix R R ℝ => M i j) hA.isHermitian.eq |
| have hplus : -(A i i + A j j)/2 ≤ A i j := by |
| have he := hA.dotProduct_mulVec_nonneg (Pi.single i 1 + Pi.single j 1) |
| simp only [star_trivial, Matrix.mulVec_add, dotProduct_add, |
| add_dotProduct, Matrix.mulVec_single_one, single_dotProduct, one_mul, Matrix.col_apply] at he |
| rw [hsym] at he |
| linarith |
| have hminus : A i j ≤ (A i i + A j j)/2 := by |
| have he := hA.dotProduct_mulVec_nonneg (Pi.single i 1 - Pi.single j 1) |
| simp only [star_trivial, Matrix.mulVec_sub, dotProduct_sub, |
| sub_dotProduct, Matrix.mulVec_single_one, single_dotProduct, one_mul, Matrix.col_apply] at he |
| rw [hsym] at he |
| linarith |
| have hdiag (k : R) : A k k ≤ A.trace := by |
| change A k k ≤ ∑ l, A l l |
| apply Finset.single_le_sum (f := fun l => A l l) |
| · intro l _ |
| have he := hA.dotProduct_mulVec_nonneg (Pi.single l 1) |
| simpa using he |
| · exact Finset.mem_univ k |
| exact (abs_le.mpr ⟨by linarith, hminus⟩).trans (by linarith [hdiag i, hdiag j]) |
| |
| omit [Fintype R] [DecidableEq R] in |
| lemma matrix_sum_psd {X : Type u_2} (s : Finset X) (f : X → Matrix R R ℝ) |
| (hf : ∀ x ∈ s, (f x).PosSemidef) : (∑ x ∈ s, f x).PosSemidef := |
| (Finset.sum_nonneg (fun x hx => (hf x hx).nonneg)).posSemidef |
|
|
| end PerfectMatchingPSD |
|
|
| end OAI |
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