| import OAI.Probability.InvariantIsing.Haar.HaarKernelGeometry |
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| namespace OAI |
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| /-! Unnormalized Frobenius geometry for the averaging and tangent estimates. -/ |
| noncomputable section |
| open Matrix |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace InvariantIsing |
|
|
| lemma frobenius_norm_sq {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ) : |
| ‖A‖^2 = matrixFrobeniusPair A A := by |
| rw [Matrix.frobenius_norm_def] |
| simp only [Real.rpow_two,Real.norm_eq_abs,sq_abs] |
| have hs : 0 ≤ ∑ i : Fin N, ∑ j : Fin N, (A i j)^2 := |
| Finset.sum_nonneg fun i _ => Finset.sum_nonneg fun j _ => sq_nonneg _ |
| rw [← Real.sqrt_eq_rpow,Real.sq_sqrt hs,matrixFrobeniusPair_self] |
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| lemma frobeniusDistance_eq_norm {N : ℕ} (U V : SpecialOrthogonal N) : |
| frobeniusDistance U V = ‖(U : Matrix (Fin N) (Fin N) ℝ)-V‖ := by |
| rw [Matrix.frobenius_norm_def] |
| simp only [Real.rpow_two,Real.norm_eq_abs,sq_abs,Matrix.sub_apply] |
| exact Real.sqrt_eq_rpow _ |
|
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| lemma frobenius_norm_mul_orthogonal {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ) |
| (U : SpecialOrthogonal N) : ‖A*(U : Matrix (Fin N) (Fin N) ℝ)‖ = ‖A‖ := by |
| apply (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp |
| rw [frobenius_norm_sq,frobenius_norm_sq] |
| unfold matrixFrobeniusPair |
| rw [Matrix.transpose_mul,Matrix.trace_mul_cycle] |
| rw [mul_assoc A U (U : Matrix (Fin N) (Fin N) ℝ).transpose, |
| (Matrix.mem_orthogonalGroup_iff (Fin N) ℝ).mp |
| (Matrix.mem_specialOrthogonalGroup_iff.mp U.property).1,mul_one,Matrix.trace_mul_comm] |
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| lemma frobeniusDistance_mul_left {N : ℕ} (W U V : SpecialOrthogonal N) : |
| frobeniusDistance (W*U) (W*V) = frobeniusDistance U V := by |
| have h := congrArg (fun x : ℝ => 2*N-x) (haarKernelBase_mul_left W U V) |
| rw [haarKernelBase_dist,haarKernelBase_dist] at h |
| apply (sq_eq_sq₀ (Real.sqrt_nonneg _) (Real.sqrt_nonneg _)).mp |
| change (frobeniusDistance (W*U) (W*V))^2 = (frobeniusDistance U V)^2 |
| linarith |
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| lemma frobeniusDistance_comm {N : ℕ} (U V : SpecialOrthogonal N) : |
| frobeniusDistance U V = frobeniusDistance V U := by |
| rw [frobeniusDistance_eq_norm,frobeniusDistance_eq_norm,← norm_neg,neg_sub] |
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| lemma frobeniusDistance_mul_right {N : ℕ} (U V W : SpecialOrthogonal N) : |
| frobeniusDistance (U*W) (V*W) = frobeniusDistance U V := by |
| rw [frobeniusDistance_eq_norm,frobeniusDistance_eq_norm] |
| change ‖(U : Matrix (Fin N) (Fin N) ℝ)*W-(V : Matrix (Fin N) (Fin N) ℝ)*W‖ = _ |
| rw [← sub_mul,frobenius_norm_mul_orthogonal] |
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| end InvariantIsing |
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| end |
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| end OAI |
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