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9b6e014 | 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 | import Mathlib
import OAI.Probability.BinarySweep.MatrixBounds.Whitening
namespace OAI
noncomputable section
open scoped BigOperators Classical ComplexOrder MatrixOrder Matrix.Norms.L2Operator
open Matrix
namespace BinaryCoordinateSweeps.Density
variable {A I : Type*} [Fintype A] [DecidableEq A] [Nonempty A]
[Fintype I] [Nonempty I]
def meanDensity (r : I → Matrix A A ℂ) : Matrix A A ℂ :=
(Fintype.card I : ℂ)⁻¹ • ∑ i, r i
def regularizedMean (r : I → Matrix A A ℂ) (ε : ℝ) : Matrix A A ℂ :=
(1-ε : ℂ) • meanDensity r + (ε/(Fintype.card A) : ℂ) • 1
omit [Fintype A] [DecidableEq A] [Nonempty A] [Nonempty I] in
lemma meanDensity_psd (r : I → Matrix A A ℂ) (hr : ∀ i, (r i).PosSemidef) :
(meanDensity r).PosSemidef := by
exact (Matrix.posSemidef_sum _ (fun i _ => hr i)).smul (by positivity)
omit [DecidableEq A] [Nonempty A] in
lemma meanDensity_trace (r : I → Matrix A A ℂ) (ht : ∀ i, (r i).trace = 1) :
(meanDensity r).trace = 1 := by
simp [meanDensity,Matrix.trace_sum,Matrix.trace_smul,ht,
ne_of_gt (Fintype.card_pos : 0 < Fintype.card I)]
lemma regularizedMean_trace (r : I → Matrix A A ℂ) (ht : ∀ i, (r i).trace = 1) (ε : ℝ) :
(regularizedMean r ε).trace = 1 := by
simp [regularizedMean,Matrix.trace_smul,meanDensity_trace r ht,Matrix.trace_one,
ne_of_gt (Fintype.card_pos : 0 < Fintype.card A)]
omit [Nonempty I] in
lemma regularizedMean_posDef (r : I → Matrix A A ℂ) (hr : ∀ i, (r i).PosSemidef)
{ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) : (regularizedMean r ε).PosDef := by
rw [regularizedMean,add_comm]
apply Matrix.PosDef.add_posSemidef
· apply Matrix.PosDef.one.smul
exact_mod_cast div_pos hε (Nat.cast_pos.mpr (Fintype.card_pos : 0 < Fintype.card A))
· exact (meanDensity_psd r hr).smul (by exact_mod_cast sub_nonneg.mpr hε1)
omit [Nonempty A] [Nonempty I] in
lemma regularizedMean_even (r : I → Matrix A A ℂ) (p : A → Bool)
(he : ∀ i a b, p a ≠ p b → r i a b = 0) (ε : ℝ) :
∀ a b, p a ≠ p b → regularizedMean r ε a b = 0 := by
intro a b hab
have hn : a ≠ b := fun h => hab (congrArg p h)
simp [regularizedMean,meanDensity,Matrix.smul_apply,Matrix.sum_apply,he,hab,hn]
omit [Nonempty A] in
lemma whitened_column_budget (r : I → Matrix A A ℂ) (v C : Matrix A A ℂ)
(_hr : ∀ i, (r i).PosSemidef) (hv : v.PosSemidef) (htv : v.trace = 1)
{ε : ℝ} (hε : 0 ≤ ε) (hC : C.IsHermitian)
(hCMC : C * regularizedMean r ε * C = 1) :
(1-ε) * (∑ i, (C * r i * C * v).trace.re) ≤ Fintype.card I := by
have hd : (regularizedMean r ε - (1-ε:ℂ) • meanDensity r).PosSemidef := by
simp only [regularizedMean,add_sub_cancel_left]
exact Matrix.PosSemidef.one.smul (by exact_mod_cast div_nonneg hε (Nat.cast_nonneg (Fintype.card A)))
have hc := hd.mul_mul_conjTranspose_same C
rw [hC.eq,Matrix.mul_sub,Matrix.sub_mul,hCMC,Matrix.mul_smul,Matrix.smul_mul] at hc
have ht := trace_product_re_nonneg hc hv
rw [Matrix.sub_mul,Matrix.one_mul,Matrix.smul_mul,Matrix.trace_sub,Matrix.trace_smul,htv] at ht
simp only [Complex.sub_re,Complex.one_re,smul_eq_mul,Complex.mul_re,Complex.sub_im,Complex.one_im,Complex.ofReal_re,Complex.ofReal_im,sub_zero,zero_mul] at ht
have hn : (Fintype.card I : ℝ) > 0 := Nat.cast_pos.mpr Fintype.card_pos
have hs : (C * meanDensity r * C * v).trace.re =
(Fintype.card I : ℝ)⁻¹ * ∑ i, (C * r i * C * v).trace.re := by
simp [meanDensity,Matrix.mul_sum,Matrix.sum_mul,
Matrix.trace_sum,Matrix.trace_smul,Complex.mul_re]
rw [hs] at ht
have := mul_nonneg hn.le ht
field_simp at this
nlinarith
end BinaryCoordinateSweeps.Density
end
end OAI
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