File size: 2,987 Bytes
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 | import OAI.Combinatorics.MatchingEntropy.SeparableVariation
namespace OAI
noncomputable section
open scoped BigOperators Topology
open Matrix
universe uV uE uI uJ uA uB uC uW uF uD uι
namespace MatchingEntropyBounds
theorem spectral_diagonal {I : Type uI} [Fintype I] [DecidableEq I]
{C : Matrix I I ℝ} (hC : C.IsHermitian) (e : I) :
C e e = ∑ i, hC.eigenvalues i * (hC.eigenvectorBasis i e)^2 := by
have hh := congrArg (fun A : Matrix I I ℝ => A e e) hC.spectral_theorem
rw [Unitary.conjStarAlgAut_apply] at hh
simpa only [mul_apply, diagonal_apply, star_eq_conjTranspose, conjTranspose_apply,
star_trivial, Function.comp_apply, RCLike.ofReal_real_eq_id, id_eq,
mul_ite, ite_mul, mul_zero, zero_mul, Finset.sum_ite_eq', Finset.mem_univ, ite_true,
Matrix.IsHermitian.eigenvectorUnitary_apply, pow_two, mul_comm, mul_left_comm,
mul_assoc] using hh
theorem covariance_weighted_rank {I : Type uI} [Fintype I] [DecidableEq I]
{C : Matrix I I ℝ} (hC : C.PosSemidef) (w : I → ℝ)
(hsecond : ∀ u : I → ℝ,
u ⬝ᵥ (C *ᵥ u) ≤ ∑ e, w e * ((C *ᵥ u) e)^2) :
(C.rank : ℝ) ≤ ∑ e, w e * C e e := by
let eig := hC.isHermitian.eigenvalues
let b := hC.isHermitian.eigenvectorBasis
have heig : ∀ i, 0 ≤ eig i := hC.eigenvalues_nonneg
have hone : ∀ i, eig i ≠ 0 → 1 ≤ eig i * (∑ e, w e * (b i e)^2) := by
intro i hi
let v : I → ℝ := b i
have hv : v ⬝ᵥ v=1 := by
have hn := b.orthonormal.1 i
have hb : inner ℝ (b i) (b i)=1 := by
rw [real_inner_self_eq_norm_sq, hn]
norm_num
simpa only [EuclideanSpace.inner_eq_star_dotProduct, star_trivial] using hb
have he : C *ᵥ v=eig i • v := hC.isHermitian.mulVec_eigenvectorBasis i
have hs := hsecond v
rw [he] at hs
simp only [dotProduct_smul, smul_eq_mul, hv, mul_one, Pi.smul_apply] at hs
have heq : (∑ e, w e*(eig i * v e)^2) =
(eig i)^2 * (∑ e, w e*(v e)^2) := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro e _
ring
rw [heq] at hs
have hp : 0 < eig i := lt_of_le_of_ne (heig i) (Ne.symm hi)
nlinarith
rw [hC.isHermitian.rank_eq_card_non_zero_eigs]
calc
(Fintype.card {i // eig i ≠ 0} : ℝ) = ∑ i : {i // eig i ≠ 0}, (1:ℝ) := by simp
_ ≤ ∑ i : {i // eig i ≠ 0}, eig i * (∑ e, w e*(b i e)^2) :=
Finset.sum_le_sum fun i _ => hone i i.property
_ = ∑ i, eig i * (∑ e, w e*(b i e)^2) := by
exact sum_subtype_supported (fun i => eig i ≠ 0)
(fun i => eig i * (∑ e, w e*(b i e)^2))
(by intro i hi; rw [not_ne_iff.mp hi, zero_mul])
_ = ∑ e, w e * C e e := by
simp_rw [Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro e _
rw [spectral_diagonal hC.isHermitian e, Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
dsimp [eig, b]
ring
end MatchingEntropyBounds
end
end OAI
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