| 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 |
|
|
