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c4cdbbd | 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 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Probability.InvariantIsing.Arrays.PerturbationFeatures
namespace OAI
/-! Mixed-rotation tensor covariances for Gaussian mean comparison. -/
noncomputable section
open scoped BigOperators
namespace InvariantIsing
def mixedProjectedOverlap {N : ℕ} (U V : Rotation N) (I : Finset (Fin N))
(σ τ : Spin N) : ℝ :=
(N : ℝ)⁻¹ * ∑ i ∈ I, U (spinVector σ) i * V (spinVector τ) i
lemma mixedProjectedOverlap_abs_le_one {N : ℕ} (U V : Rotation N)
(I : Finset (Fin N)) (σ τ : Spin N) : |mixedProjectedOverlap U V I σ τ| ≤ 1 := by
have hcs := Finset.sum_mul_sq_le_sq_mul_sq I
(fun i => U (spinVector σ) i) (fun i => V (spinVector τ) i)
have hs : (∑ i ∈ I, U (spinVector σ) i * V (spinVector τ) i) ^ 2 ≤ (N : ℝ) ^ 2 := by
apply hcs.trans
simpa only [pow_two] using
(mul_le_mul (projected_spin_sq_sum_le U I σ) (projected_spin_sq_sum_le V I τ)
(Finset.sum_nonneg fun _ _ => sq_nonneg _) (Nat.cast_nonneg _))
have ha : |∑ i ∈ I, U (spinVector σ) i * V (spinVector τ) i| ≤ N :=
(sq_le_sq₀ (abs_nonneg _) (Nat.cast_nonneg _)).mp (by simpa only [sq_abs] using hs)
unfold mixedProjectedOverlap
rw [abs_mul, abs_of_nonneg (show (0 : ℝ) ≤ (N : ℝ)⁻¹ by positivity)]
by_cases hN : N = 0
· simp [hN]
· have hn : (N : ℝ) ≠ 0 := by exact_mod_cast hN
exact (mul_le_mul_of_nonneg_left ha (by positivity)).trans_eq (inv_mul_cancel₀ hn)
lemma mixedProjectedOverlap_sub_first_le {N : ℕ} (hN : 0 < N)
(U V W : SpecialOrthogonal N) (I : Finset (Fin N)) (σ τ : Spin N) :
|mixedProjectedOverlap (specialRotation U) (specialRotation W) I σ τ -
mixedProjectedOverlap (specialRotation V) (specialRotation W) I σ τ| ≤
frobeniusDistance U V := by
have he : mixedProjectedOverlap (specialRotation U) (specialRotation W) I σ τ -
mixedProjectedOverlap (specialRotation V) (specialRotation W) I σ τ =
(N : ℝ)⁻¹ * ∑ i ∈ I,
(specialRotation U (spinVector σ) i - specialRotation V (spinVector σ) i) *
specialRotation W (spinVector τ) i := by
simp only [mixedProjectedOverlap, ← mul_sub, ← Finset.sum_sub_distrib, sub_mul]
rw [he, abs_mul, abs_of_nonneg (show (0 : ℝ) ≤ (N : ℝ)⁻¹ by positivity)]
have hn : (N : ℝ) ≠ 0 := by exact_mod_cast hN.ne'
calc
_ ≤ (N : ℝ)⁻¹ * (N * frobeniusDistance U V) :=
mul_le_mul_of_nonneg_left (abs_projected_cross_difference_sum_le U V W I σ τ) (by positivity)
_ = _ := by field_simp
lemma spectralSpinFeature_mixed_cross {N : ℕ} (U V : Rotation N)
(I : Finset (Fin N)) (σ τ : Spin N) :
(∑ i : I, spectralSpinFeature U I σ i * spectralSpinFeature V I τ i) =
mixedProjectedOverlap U V I σ τ := by
simp only [spectralSpinFeature, div_mul_div_comm, ← pow_two,
Real.sq_sqrt (Nat.cast_nonneg N), ← Finset.sum_div]
rw [div_eq_mul_inv, mul_comm]
unfold mixedProjectedOverlap
congr 1
exact Finset.sum_coe_sort I (fun i => U (spinVector σ) i * V (spinVector τ) i)
lemma spectralMonomialFeature_mixed_cross {N m : ℕ} (U V : Rotation N)
(I : Fin m → Finset (Fin N)) (d : Fin m → ℕ) (σ τ : Spin N) :
(∑ v : SpectralTensorIndex I d,
spectralMonomialFeature U I d σ v * spectralMonomialFeature V I d τ v) =
∏ a, mixedProjectedOverlap U V (I a) σ τ ^ d a := by
simp only [spectralMonomialFeature, ← Finset.prod_mul_distrib]
rw [← Fintype.prod_sum (fun a (v : Fin (d a) → I a) =>
IsingPerceptron.tensorFeature (d a) (spectralSpinFeature U (I a) σ) v *
IsingPerceptron.tensorFeature (d a) (spectralSpinFeature V (I a) τ) v)]
simp_rw [IsingPerceptron.tensorFeature_cross, spectralSpinFeature_mixed_cross]
/-- The cross covariance of a tensor increment with any rotated tensor
has a dimension-independent rotation modulus. -/
theorem spectralMonomialFeature_increment_cross_le {N m : ℕ} (hN : 0 < N)
(U V W : SpecialOrthogonal N) (I : Fin m → Finset (Fin N)) (d : Fin m → ℕ)
(σ τ : Spin N) :
|∑ v : SpectralTensorIndex I d,
(spectralMonomialFeature (specialRotation U) I d σ v -
spectralMonomialFeature (specialRotation V) I d σ v) *
spectralMonomialFeature (specialRotation W) I d τ v| ≤
(∑ a, (d a : ℝ)) * frobeniusDistance U V := by
simp_rw [sub_mul]
rw [Finset.sum_sub_distrib, spectralMonomialFeature_mixed_cross,
spectralMonomialFeature_mixed_cross]
have hpow (R : SpecialOrthogonal N) (a : Fin m) :
|mixedProjectedOverlap (specialRotation R) (specialRotation W) (I a) σ τ ^ d a| ≤ 1 := by
rw [abs_pow]
exact pow_le_one₀ (abs_nonneg _) (mixedProjectedOverlap_abs_le_one _ _ _ _ _)
calc
_ ≤ ∑ a, |mixedProjectedOverlap (specialRotation U) (specialRotation W) (I a) σ τ ^ d a -
mixedProjectedOverlap (specialRotation V) (specialRotation W) (I a) σ τ ^ d a| :=
abs_prod_sub_prod_le_sum Finset.univ _ _ (fun a _ => hpow U a) (fun a _ => hpow V a)
_ ≤ ∑ a, (d a : ℝ) * frobeniusDistance U V := by
apply Finset.sum_le_sum
intro a _
exact (abs_pow_sub_pow_le_degree (mixedProjectedOverlap_abs_le_one _ _ _ _ _)
(mixedProjectedOverlap_abs_le_one _ _ _ _ _) (d a)).trans
(mul_le_mul_of_nonneg_left (mixedProjectedOverlap_sub_first_le hN U V W (I a) σ τ)
(Nat.cast_nonneg _))
_ = _ := (Finset.sum_mul _ _ _).symm
end InvariantIsing
end
end OAI
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