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namespace OAI
noncomputable section
section
open scoped BigOperators ComplexOrder
open Matrix
namespace BinaryCoordinateSweeps.Density
variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq J]
def blockVec (f : I → J) (a : J) (x : I → ℂ) : I → ℂ :=
fun i => if f i = a then x i else 0
def pinch (f : I → J) (A : Matrix I I ℂ) : Matrix I I ℂ :=
fun i j => if f i = f j then A i j else 0
omit [Fintype I] in
lemma sum_blockVec (f : I → J) (x : I → ℂ) : ∑ a, blockVec f a x = x := by
ext i
simp [blockVec]
omit [Fintype I] [Fintype J] in
lemma pinch_hermitian (f : I → J) {A : Matrix I I ℂ} (hA : A.IsHermitian) :
(pinch f A).IsHermitian := by
ext i j
change star (if f j = f i then A j i else 0) = if f i = f j then A i j else 0
by_cases h : f i = f j
· rw [ite_eq_left h, ite_eq_left h.symm]
exact congrFun (congrFun hA i) j
· rw [ite_eq_right h, ite_eq_right (Ne.symm h), star_zero]
lemma pinch_form (f : I → J) (A : Matrix I I ℂ) (x : I → ℂ) :
star x ⬝ᵥ (pinch f A).mulVec x =
∑ a, star (blockVec f a x) ⬝ᵥ A.mulVec (blockVec f a x) := by
simp only [dotProduct, mulVec, blockVec, pinch, Pi.star_apply, apply_ite, star_zero,
Finset.mul_sum]
conv_rhs => rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro i hi
conv_rhs => rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro j hj
simp only [ite_mul, zero_mul, mul_ite, mul_zero]
by_cases h : f i = f j
· rw [h]
simp
· simp [h]
lemma pinch_posSemidef (f : I → J) {A : Matrix I I ℂ} (hA : A.PosSemidef) :
(pinch f A).PosSemidef := by
refine .of_dotProduct_mulVec_nonneg (pinch_hermitian f hA.1) (fun x => ?_)
rw [pinch_form]
exact Finset.sum_nonneg (fun a _ => hA.dotProduct_mulVec_nonneg _)
lemma form_cross_le {A : Matrix I I ℂ} (hA : A.PosSemidef) (x y : I → ℂ) :
star x ⬝ᵥ A.mulVec y + star y ⬝ᵥ A.mulVec x ≤
star x ⬝ᵥ A.mulVec x + star y ⬝ᵥ A.mulVec y := by
have h := hA.dotProduct_mulVec_nonneg (x-y)
rw [star_sub, mulVec_sub, sub_dotProduct, dotProduct_sub, dotProduct_sub] at h
exact sub_nonneg.mp (by convert h using 1; abel)
omit [Fintype J] in
lemma block_form_zero (f : I → J) (A : Matrix I I ℂ) (a b : J)
(h : ∀ i j, f i = a → f j = b → A i j = 0) (x : I → ℂ) :
star (blockVec f a x) ⬝ᵥ A.mulVec (blockVec f b x) = 0 := by
simp only [dotProduct, mulVec, blockVec, Pi.star_apply, apply_ite, star_zero,
Finset.mul_sum]
apply Finset.sum_eq_zero
intro i hi
apply Finset.sum_eq_zero
intro j hj
by_cases ha : f i = a
· by_cases hb : f j = b
· simp [ha, hb, h i j ha hb]
· simp [hb]
· simp [ha]
theorem sparse_pinching (f : I → J) (R : J → J → Prop) [DecidableRel R]
(hR : ∀ ⦃left right⦄, R left right → R right left)
(D : ℕ) (hD : ∀ a, (Finset.univ.filter (R a)).card ≤ D)
{A : Matrix I I ℂ} (hA : A.PosSemidef)
(hz : ∀ i j, ¬ R (f i) (f j) → A i j = 0) :
((D : ℂ) • pinch f A - A).PosSemidef := by
have hp := pinch_hermitian f hA.1
refine .of_dotProduct_mulVec_nonneg ((hp.smul (by simp)).sub hA.1) (fun x => ?_)
let t : J → J → ℂ := fun a b => star (blockVec f a x) ⬝ᵥ A.mulVec (blockVec f b x)
have hdiag (a : J) : 0 ≤ t a a := hA.dotProduct_mulVec_nonneg _
have ht (a b : J) (hab : ¬R a b) : t a b = 0 := by
apply block_form_zero
intro i j hi hj
exact hz i j (by simpa [hi, hj] using hab)
have hsum : ∑ a, ∑ b ∈ Finset.univ.filter (R a), t a b = star x ⬝ᵥ A.mulVec x := by
simp_rw [Finset.sum_filter]
have ht' : ∀ a b, (if R a b then t a b else 0) = t a b := by
intro a b
split_ifs with h
· rfl
· exact (ht a b h).symm
simp_rw [ht']
dsimp [t]
simp_rw [← dotProduct_sum, ← mulVec_sum, sum_blockVec]
rw [← sum_dotProduct, ← star_sum, sum_blockVec]
have hswap : ∑ a, ∑ b ∈ Finset.univ.filter (R a), t b a =
∑ a, ∑ b ∈ Finset.univ.filter (R a), t a b := by
simp only [Finset.sum_filter]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro a ha
apply Finset.sum_congr rfl
intro b hb
have he : R b a ↔ R a b := ⟨fun h => hR h, fun h => hR h⟩
simp only [he]
have hdeg : ∑ a, ∑ b ∈ Finset.univ.filter (R a), (t a a + t b b) =
2 * ∑ a, ((Finset.univ.filter (R a)).card : ℂ) * t a a := by
simp only [Finset.sum_add_distrib]
have hs : ∑ a, ∑ b ∈ Finset.univ.filter (R a), t b b =
∑ a, ∑ b ∈ Finset.univ.filter (R a), t a a := by
simp only [Finset.sum_filter]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro a ha
apply Finset.sum_congr rfl
intro b hb
simp only [show R b a ↔ R a b from ⟨fun h => hR h, fun h => hR h⟩]
rw [hs]
simp [two_mul]
have hle := Finset.sum_le_sum (fun a (_ : a ∈ (Finset.univ : Finset J)) =>
Finset.sum_le_sum (fun b (_ : b ∈ Finset.univ.filter (R a)) =>
form_cross_le hA (blockVec f a x) (blockVec f b x)))
change (∑ a, ∑ b ∈ Finset.univ.filter (R a), (t a b + t b a)) ≤ _ at hle
simp only [Finset.sum_add_distrib] at hle
rw [hswap, hsum] at hle
have hc : star x ⬝ᵥ A.mulVec x ≤ ∑ a, ((Finset.univ.filter (R a)).card : ℂ) * t a a := by
have hh := hdeg
simp only [Finset.sum_add_distrib] at hh
rw [hh] at hle
simp only [two_mul, Complex.le_def, Complex.add_re, Complex.add_im] at hle
exact Complex.le_def.mpr ⟨by linarith [hle.1], by linarith [hle.2]⟩
have hd : ∑ a, ((Finset.univ.filter (R a)).card : ℂ) * t a a ≤
(D : ℂ) * ∑ a, t a a := by
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro a ha
exact mul_le_mul_of_nonneg_right (by exact_mod_cast hD a) (hdiag a)
have hf := hc.trans hd
rw [sub_mulVec, dotProduct_sub, smul_mulVec, dotProduct_smul, smul_eq_mul]
apply sub_nonneg.mpr
simpa only [pinch_form, t] using hf
end BinaryCoordinateSweeps.Density
end
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.Density
variable {I : Type*} [Fintype I] [LinearOrder I]
def subsetCode (k : ℕ) (s : Finset I) : Fin k → Option I :=
fun i => if h : i.val < s.card then some (s.orderEmbOfFin rfl ⟨i.val,h⟩) else none
omit [Fintype I] in
lemma subsetCode_range (k : ℕ) (s : Finset I) (hs : s.card ≤ k) (j : I) :
some j ∈ Set.range (subsetCode k s) ↔ j ∈ s := by
constructor
· rintro ⟨i,hi⟩
dsimp only [subsetCode] at hi
split_ifs at hi with h
· have hj := Option.some.inj hi
rw [← hj]
exact Finset.orderEmbOfFin_mem s rfl _
· intro hj
have hr : j ∈ Set.range (s.orderEmbOfFin rfl) := by
rw [Finset.range_orderEmbOfFin]; exact hj
obtain ⟨i,hi⟩ := hr
refine ⟨⟨i.val, i.isLt.trans_le hs⟩, ?_⟩
simp only [subsetCode, dite_eq_left i.isLt]
exact congrArg some hi
lemma card_small_subsets_le (k : ℕ) :
Fintype.card {s : Finset I // s.card ≤ k} ≤ (Fintype.card I + 1)^k := by
let f : {s : Finset I // s.card ≤ k} → (Fin k → Option I) := fun s => subsetCode k s.val
have hf : Function.Injective f := by
intro s t h
apply Subtype.ext
ext j
rw [← subsetCode_range k s.val s.property j, ← subsetCode_range k t.val t.property j]
rw [show Set.range (subsetCode k s.val) = Set.range (subsetCode k t.val) from
congrArg Set.range h]
have hc := Fintype.card_le_of_injective f hf
simpa only [Fintype.card_fun, Fintype.card_fin, Fintype.card_option] using hc
def hamming (a b : I → Bool) : ℕ := (Finset.univ.filter (fun i => a i ≠ b i)).card
omit [LinearOrder I] in
lemma hamming_symm (a b : I → Bool) : hamming a b = hamming b a := by
unfold hamming
congr 1
ext i
simp only [Finset.mem_filter, Finset.mem_univ, true_and, ne_comm]
lemma hamming_degree (a : I → Bool) (k : ℕ) :
(Finset.univ.filter (fun b => hamming a b ≤ k)).card ≤ (Fintype.card I + 1)^k := by
let f : {b : I → Bool // hamming a b ≤ k} → {s : Finset I // s.card ≤ k} :=
fun b => ⟨Finset.univ.filter (fun i => a i ≠ b.val i), b.property⟩
have hf : Function.Injective f := by
intro b c h
apply Subtype.ext
funext i
have hi := Finset.ext_iff.mp (congrArg Subtype.val h) i
simp only [f, Finset.mem_filter, Finset.mem_univ, true_and] at hi
cases ha : a i <;> cases hb : b.val i <;> cases hc : c.val i <;>
simp_all only [Bool.false_eq_true, Bool.true_eq_false, not_false_eq_true, ne_eq,
not_true_eq_false, iff_false, iff_true]
have hh := (Fintype.card_le_of_injective f hf).trans (card_small_subsets_le (I := I) k)
simpa only [Fintype.card_subtype] using hh
end BinaryCoordinateSweeps.Density
end
end OAI
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