| import OAI.Combinatorics.MatchingPSD.CellAverages |
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| namespace OAI |
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| universe u_1 u_2 u_3 |
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| namespace PerfectMatchingPSD |
| open Matrix Set |
| variable {R : Type u_1} {I : Type u_2} {J : Type u_3} [Fintype R] [DecidableEq R] |
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| omit [Fintype R] [DecidableEq R] in |
| lemma convex_posSemidef : Convex ℝ {A : Matrix R R ℝ | A.PosSemidef} := by |
| intro A hA B hB a b ha hb _ |
| exact (hA.smul ha).add (hB.smul hb) |
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| lemma hasDerivAt_det_one_add (A : Matrix R R ℝ) : |
| HasDerivAt (fun t : ℝ => (1 + t • A).det) A.trace 0 := by |
| let p : Polynomial ℝ := (1 + (Polynomial.X : Polynomial ℝ) • A.map Polynomial.C).det |
| have h := p.hasDerivAt 0 |
| rw [show p.derivative.eval 0 = A.trace from Matrix.derivative_det_one_add_X_smul A] at h |
| have he : (fun t : ℝ => (1 + t • A).det) = (fun t => p.eval t) := by |
| funext t |
| change _ = (Polynomial.evalRingHom t) p |
| rw [RingHom.map_det] |
| congr 1 |
| ext i j |
| change (if i = j then 1 else 0) + t * A i j = |
| Polynomial.eval t ((if i = j then 1 else 0) + |
| Polynomial.X * Polynomial.C (A i j)) |
| rw [Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_X, Polynomial.eval_C] |
| split_ifs <;> simp only [Polynomial.eval_one, Polynomial.eval_zero] |
| rwa [he] |
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| lemma hasDerivAt_det_add (D E : Matrix R R ℝ) (hD : IsUnit D) : |
| HasDerivAt (fun t : ℝ => (D + t • E).det) |
| (D.det * (D⁻¹ * E).trace) 0 := by |
| have h := (hasDerivAt_det_one_add (D⁻¹ * E)).const_mul D.det |
| have he : (fun t : ℝ => (D + t • E).det) = |
| (fun t => D.det * (1 + t • (D⁻¹ * E)).det) := by |
| funext t |
| rw [← Matrix.det_mul] |
| congr 1 |
| rw [mul_add, mul_one, mul_smul_comm, ← mul_assoc, |
| Matrix.mul_nonsing_inv D ((Matrix.isUnit_iff_isUnit_det D).mp hD), one_mul] |
| rwa [he] |
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| lemma derivative_nonpos_at_right_max {f : ℝ → ℝ} {c : ℝ} |
| (h : HasDerivAt f c 0) (hm : ∀ t ∈ Set.Icc (0 : ℝ) 1, f t ≤ f 0) : c ≤ 0 := by |
| have he : IsLocalMaxOn f (Set.Icc (0 : ℝ) 1) 0 := |
| (show IsMaxOn f (Set.Icc (0 : ℝ) 1) 0 from hm).isLocalMaxOn |
| have ht : (1 : ℝ) ∈ posTangentConeAt (Set.Icc (0 : ℝ) 1) 0 := by |
| apply mem_posTangentConeAt_of_segment_subset |
| simp [segment_eq_Icc] |
| simpa using he.hasFDerivWithinAt_nonpos h.hasFDerivAt.hasFDerivWithinAt ht |
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| lemma trace_inv_mul_le_of_det_max (D B : Matrix R R ℝ) (hD : D.PosDef) |
| (hm : ∀ t ∈ Set.Icc (0 : ℝ) 1, (D + t • (B - D)).det ≤ D.det) : |
| (D⁻¹ * B).trace ≤ (Fintype.card R : ℝ) := by |
| have hd : 0 < D.det := hD.det_pos |
| have he := derivative_nonpos_at_right_max (hasDerivAt_det_add D (B - D) hD.isUnit) |
| (by simpa using hm) |
| have ht : (D⁻¹ * (B - D)).trace ≤ 0 := by nlinarith |
| rw [mul_sub, Matrix.trace_sub, Matrix.nonsing_inv_mul D ((Matrix.isUnit_iff_isUnit_det D).mp hD.isUnit), Matrix.trace_one] at ht |
| linarith |
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| lemma exists_regularized_det_max [Fintype I] [Nonempty I] |
| (B : I → Matrix R R ℝ) (hB : ∀ i, (B i).PosSemidef) |
| (ε : ℝ) (hε : 0 < ε) : |
| ∃ D : Matrix R R ℝ, D.PosDef ∧ |
| D ∈ (fun A => ε • (1 : Matrix R R ℝ) + A) '' convexHull ℝ (Set.range B) ∧ |
| ∀ i, (D⁻¹ * B i).trace ≤ (Fintype.card R : ℝ) := by |
| let S := convexHull ℝ (Set.range B) |
| let f : Matrix R R ℝ → Matrix R R ℝ := fun A => ε • (1 : Matrix R R ℝ) + A |
| have hc : IsCompact S := (Set.finite_range B).isCompact_convexHull ℝ |
| have hn : S.Nonempty := (Set.range_nonempty B).mono (subset_convexHull ℝ _) |
| have hf : Continuous f := by fun_prop |
| have hp : ∀ A ∈ S, A.PosSemidef := |
| convexHull_min (Set.range_subset_iff.mpr hB) convex_posSemidef |
| have hpd : ∀ D ∈ f '' S, D.PosDef := by |
| rintro _ ⟨A, hA, rfl⟩ |
| exact (Matrix.PosDef.one.smul hε).add_posSemidef (hp A hA) |
| obtain ⟨D, hDs, hmax⟩ := (hc.image hf).exists_isMaxOn (hn.image f) |
| (by fun_prop : ContinuousOn Matrix.det (f '' S)) |
| refine ⟨D, hpd D hDs, hDs, ?_⟩ |
| intro i |
| have hi : f (B i) ∈ f '' S := ⟨B i, subset_convexHull ℝ _ (Set.mem_range_self i), rfl⟩ |
| have hs : Convex ℝ (f '' S) := by |
| exact (convex_convexHull ℝ (Set.range B)).translate (ε • (1 : Matrix R R ℝ)) |
| have hm := trace_inv_mul_le_of_det_max D (f (B i)) (hpd D hDs) (by |
| intro t ht |
| apply hmax |
| have he : D + t • (f (B i) - D) = (1 - t) • D + t • f (B i) := by module |
| rw [he] |
| exact hs hDs hi (sub_nonneg.mpr ht.2) ht.1 (by ring)) |
| have htr : 0 ≤ (D⁻¹).trace := (hpd D hDs).inv.posSemidef.trace_nonneg |
| have he : (D⁻¹ * f (B i)).trace = ε * (D⁻¹).trace + (D⁻¹ * B i).trace := by |
| simp [f, mul_add, Matrix.trace_add, Matrix.trace_smul] |
| rw [he] at hm |
| nlinarith |
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| open scoped MatrixOrder |
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| lemma trace_mul_nonneg {A B : Matrix R R ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : |
| 0 ≤ (A * B).trace := by |
| let G := CFC.sqrt B |
| have hG : G.PosSemidef := (CFC.sqrt_nonneg B).posSemidef |
| have hGG : G * G = B := CFC.sqrt_mul_sqrt_self B hB.nonneg |
| have h := (hA.conjTranspose_mul_mul_same G).trace_nonneg |
| rw [hG.isHermitian.eq, Matrix.trace_mul_cycle, hGG] at h |
| simpa only [Matrix.trace_mul_comm B A] using h |
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| lemma trace_mul_le_of_le {A B D : Matrix R R ℝ} (hA : A.PosSemidef) (hBD : B ≤ D) : |
| (A * B).trace ≤ (A * D).trace := by |
| have h := trace_mul_nonneg hA hBD |
| rwa [mul_sub, Matrix.trace_sub, sub_nonneg] at h |
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| lemma trace_mul_convexHull_le [Fintype I] (A : Matrix R R ℝ) |
| (B : I → Matrix R R ℝ) (m : ℝ) (hB : ∀ i, (A * B i).trace ≤ m) |
| {D : Matrix R R ℝ} (hD : D ∈ convexHull ℝ (Set.range B)) : |
| (A * D).trace ≤ m := by |
| have hc : Convex ℝ {D : Matrix R R ℝ | (A * D).trace ≤ m} := by |
| intro X hX Y hY a b ha hb hab |
| change (A * (a • X + b • Y)).trace ≤ m |
| rw [mul_add, mul_smul_comm, mul_smul_comm, Matrix.trace_add, |
| Matrix.trace_smul, Matrix.trace_smul] |
| change a * (A * X).trace + b * (A * Y).trace ≤ m |
| calc |
| _ ≤ a * m + b * m := add_le_add |
| (mul_le_mul_of_nonneg_left hX ha) (mul_le_mul_of_nonneg_left hY hb) |
| _ = m := by rw [← add_mul, hab, one_mul] |
| exact convexHull_min (Set.range_subset_iff.mpr hB) hc hD |
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| lemma exists_normalized_factors [Fintype I] [Nonempty I] [Fintype J] |
| (A : J → Matrix R R ℝ) (B : I → Matrix R R ℝ) |
| (hA : ∀ j, (A j).PosSemidef) (hB : ∀ i, (B i).PosSemidef) |
| (m : ℝ) (hpair : ∀ j i, (A j * B i).trace ≤ m) : |
| ∃ (A' : J → Matrix R R ℝ) (B' : I → Matrix R R ℝ), |
| (∀ j, (A' j).PosSemidef) ∧ (∀ i, (B' i).PosSemidef) ∧ |
| (∀ j i, (A' j * B' i).trace = (A j * B i).trace) ∧ |
| (∀ j, (A' j).trace ≤ m + 1) ∧ |
| (∀ i, (B' i).trace ≤ (Fintype.card R : ℝ)) := by |
| classical |
| let S : ℝ := ∑ j, (A j).trace |
| have hS : 0 ≤ S := Finset.sum_nonneg (fun j _ => (hA j).trace_nonneg) |
| let ε : ℝ := 1 / (1 + S) |
| have hε : 0 < ε := one_div_pos.mpr (by linarith) |
| have hεA (j : J) : ε * (A j).trace ≤ 1 := by |
| have hj : (A j).trace ≤ S := Finset.single_le_sum |
| (fun j _ => (hA j).trace_nonneg) (Finset.mem_univ j) |
| change 1 / (1 + S) * (A j).trace ≤ 1 |
| rw [one_div_mul_eq_div] |
| exact (div_le_one (by linarith : 0 < 1 + S)).mpr (by linarith) |
| obtain ⟨D, hD, ⟨D₀, hD₀, hDeq⟩, htr⟩ := exists_regularized_det_max B hB ε hε |
| let G := CFC.sqrt D |
| have hG : G.PosSemidef := (CFC.sqrt_nonneg D).posSemidef |
| have hGG : G * G = D := CFC.sqrt_mul_sqrt_self D hD.posSemidef.nonneg |
| have huG : IsUnit G := (IsUnit.mul_iff.mp (hGG ▸ hD.isUnit)).1 |
| have hunit := (Matrix.isUnit_iff_isUnit_det G).mp huG |
| have hGinv : G⁻¹.PosSemidef := hG.inv |
| have hleft : G⁻¹ * G = 1 := Matrix.nonsing_inv_mul G hunit |
| have hright : G * G⁻¹ = 1 := Matrix.mul_nonsing_inv G hunit |
| have hinv : G⁻¹ * G⁻¹ = D⁻¹ := by |
| rw [← hGG, Matrix.mul_inv_rev] |
| refine ⟨fun j => G * A j * G, fun i => G⁻¹ * B i * G⁻¹, |
| ?_, ?_, ?_, ?_, ?_⟩ |
| · intro j |
| simpa only [hG.isHermitian.eq] using (hA j).conjTranspose_mul_mul_same G |
| · intro i |
| simpa only [hGinv.isHermitian.eq] using (hB i).conjTranspose_mul_mul_same G⁻¹ |
| · intro j i |
| change (G * A j * G * (G⁻¹ * B i * G⁻¹)).trace = _ |
| have he : G * A j * G * (G⁻¹ * B i * G⁻¹) = G * (A j * B i) * G⁻¹ := by |
| calc |
| _ = G * A j * (G * G⁻¹) * B i * G⁻¹ := by simp only [mul_assoc] |
| _ = _ := by rw [hright, mul_one]; simp only [mul_assoc] |
| rw [he, Matrix.trace_mul_cycle, hleft, one_mul] |
| · intro j |
| rw [Matrix.trace_mul_cycle, hGG, Matrix.trace_mul_comm, ← hDeq, mul_add, Matrix.trace_add, |
| mul_smul_comm, Matrix.trace_smul, mul_one] |
| change ε * (A j).trace + (A j * D₀).trace ≤ m + 1 |
| have hl := trace_mul_convexHull_le (A j) B m (hpair j) hD₀ |
| linarith [hεA j] |
| · intro i |
| rw [Matrix.trace_mul_cycle, hinv] |
| exact htr i |
|
|
| end PerfectMatchingPSD |
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| end OAI |
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