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a89d282 | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | import OAI.Probability.InvariantIsing.Haar.HaarPolynomialKernel
import OAI.Probability.InvariantIsing.Haar.HaarPlaneFrame
namespace OAI
/-! The averaging kernel peaks exactly at coincident orthogonal matrices. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory
open scoped BigOperators
namespace InvariantIsing
lemma specialOrthogonal_sum_sq {N : ℕ} (U : SpecialOrthogonal N) :
(∑ i, ∑ j, ((U : Matrix (Fin N) (Fin N) ℝ) i j)^2) = N := by
rw [← matrixFrobeniusPair_self]
unfold matrixFrobeniusPair
rw [(Matrix.mem_orthogonalGroup_iff' (Fin N) ℝ).mp
(Matrix.mem_specialOrthogonalGroup_iff.mp U.property).1]
simp
lemma haarKernelBase_dist {N : ℕ} (U V : SpecialOrthogonal N) :
haarKernelBase U V = 2*N-(frobeniusDistance U V)^2/2 := by
have hU := specialOrthogonal_sum_sq U
have hV := specialOrthogonal_sum_sq V
rw [frobeniusDistance,Real.sq_sqrt (Finset.sum_nonneg fun i _ =>
Finset.sum_nonneg fun j _ => sq_nonneg _)]
have hsum : (∑ i, ∑ j, ((U : Matrix (Fin N) (Fin N) ℝ) i j-
(V : Matrix (Fin N) (Fin N) ℝ) i j)^2) = 2*N-
2*(∑ i, ∑ j, (U : Matrix (Fin N) (Fin N) ℝ) i j*
(V : Matrix (Fin N) (Fin N) ℝ) i j) := by
simp only [sub_sq,Finset.sum_add_distrib,Finset.sum_sub_distrib]
simp_rw [mul_assoc,← Finset.mul_sum]
rw [hU,hV]
ring
rw [hsum]
unfold haarKernelBase
ring
lemma haarKernelBase_nonneg {N : ℕ} (U V : SpecialOrthogonal N) :
0 ≤ haarKernelBase U V := by
have h := Finset.sum_nonneg (s := Finset.univ) (fun i _ =>
Finset.sum_nonneg (s := Finset.univ) (fun j _ =>
sq_nonneg ((U : Matrix (Fin N) (Fin N) ℝ) i j+
(V : Matrix (Fin N) (Fin N) ℝ) i j)))
simp only [add_sq,Finset.sum_add_distrib] at h
simp_rw [mul_assoc,← Finset.mul_sum] at h
rw [specialOrthogonal_sum_sq U,specialOrthogonal_sum_sq V] at h
unfold haarKernelBase
linarith
lemma haarKernelBase_le {N : ℕ} (U V : SpecialOrthogonal N) :
haarKernelBase U V ≤ 2*N := by
rw [haarKernelBase_dist]
nlinarith [sq_nonneg (frobeniusDistance U V)]
lemma haarKernelBase_self {N : ℕ} (U : SpecialOrthogonal N) :
haarKernelBase U U = 2*N := by
unfold haarKernelBase
simp_rw [← pow_two]
rw [specialOrthogonal_sum_sq]
ring
lemma haarKernelBase_mul_left {N : ℕ} (W U V : SpecialOrthogonal N) :
haarKernelBase (W*U) (W*V) = haarKernelBase U V := by
have he : matrixFrobeniusPair
((W : Matrix (Fin N) (Fin N) ℝ)*(U : Matrix (Fin N) (Fin N) ℝ))
((W : Matrix (Fin N) (Fin N) ℝ)*(V : Matrix (Fin N) (Fin N) ℝ)) =
matrixFrobeniusPair (U : Matrix (Fin N) (Fin N) ℝ)
(V : Matrix (Fin N) (Fin N) ℝ) := by
unfold matrixFrobeniusPair
rw [Matrix.transpose_mul,mul_assoc,← mul_assoc (W : Matrix (Fin N) (Fin N) ℝ).transpose,
(Matrix.mem_orthogonalGroup_iff' (Fin N) ℝ).mp
(Matrix.mem_specialOrthogonalGroup_iff.mp W.property).1,one_mul]
unfold haarKernelBase
rw [← matrixFrobeniusPair_eq_sum,← matrixFrobeniusPair_eq_sum]
exact congrArg (fun x : ℝ => N+x) he
lemma haarKernelBase_mul_right {N : ℕ} (U V W : SpecialOrthogonal N) :
haarKernelBase (U*W) (V*W) = haarKernelBase U V := by
have he : matrixFrobeniusPair
((U : Matrix (Fin N) (Fin N) ℝ)*(W : Matrix (Fin N) (Fin N) ℝ))
((V : Matrix (Fin N) (Fin N) ℝ)*(W : Matrix (Fin N) (Fin N) ℝ)) =
matrixFrobeniusPair (U : Matrix (Fin N) (Fin N) ℝ)
(V : Matrix (Fin N) (Fin N) ℝ) := by
unfold matrixFrobeniusPair
rw [Matrix.transpose_mul,Matrix.trace_mul_cycle]
rw [mul_assoc (V : Matrix (Fin N) (Fin N) ℝ) W (W : Matrix (Fin N) (Fin N) ℝ).transpose,
(Matrix.mem_orthogonalGroup_iff (Fin N) ℝ).mp
(Matrix.mem_specialOrthogonalGroup_iff.mp W.property).1,mul_one,Matrix.trace_mul_comm]
unfold haarKernelBase
rw [← matrixFrobeniusPair_eq_sum,← matrixFrobeniusPair_eq_sum]
exact congrArg (fun x : ℝ => N+x) he
end InvariantIsing
end
end OAI
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