| import OAI.Probability.InvariantIsing.Gaussian.GaussianTranspose |
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|
| namespace OAI |
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|
| /-! The actual Gram eigenvalues dominate the squared transposed least singular value. -/ |
| noncomputable section |
| open Matrix |
| open scoped BigOperators RealInnerProductSpace |
| namespace InvariantIsing |
|
|
| lemma gaussianPattern_quadratic {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m)) |
| (x : EuclideanSpace ℝ (Fin N)) : |
| ⟪x,(gaussianPatternCoupling 1 z).toEuclideanLin x⟫ = |
| ‖gaussianPatternEuclideanOperator (gaussianPatternTranspose N m z) x‖^2/N := by |
| have hn : ‖gaussianPatternEuclideanOperator (gaussianPatternTranspose N m z) x‖^2 = |
| ((gaussianPatternArray z)ᵀ *ᵥ x.ofLp) ⬝ᵥ ((gaussianPatternArray z)ᵀ *ᵥ x.ofLp) := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| simp only [dotProduct,pow_two] |
| rfl |
| rw [hn] |
| rw [EuclideanSpace.inner_eq_star_dotProduct,dotProduct_comm] |
| simp only [star_trivial] |
| change x.ofLp ⬝ᵥ ((1/(N : ℝ)) • (gaussianPatternArray z*(gaussianPatternArray z)ᵀ)) *ᵥ x.ofLp = _ |
| rw [Matrix.smul_mulVec,dotProduct_smul,← Matrix.mulVec_mulVec,Matrix.dotProduct_transpose_mulVec] |
| simp only [smul_eq_mul] |
| ring |
|
|
| lemma gaussianPatternSingularMin_nonneg {N m : ℕ} (hm : 0 < m) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : 0 ≤ gaussianPatternSingularMin z := by |
| let : NeZero m := ⟨Nat.ne_of_gt hm⟩ |
| have : Nonempty (Metric.sphere (0 : EuclideanSpace ℝ (Fin m)) 1) := by |
| refine ⟨⟨EuclideanSpace.single (⟨0,hm⟩ : Fin m) (1 : ℝ),?_⟩⟩ |
| simp |
| exact le_ciInf (fun _ => norm_nonneg _) |
|
|
| lemma gaussianPatternEigenvalues_lower_singular {N m : ℕ} (hN : 0 < N) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) (i : Fin N) : |
| (gaussianPatternSingularMin (gaussianPatternTranspose N m z))^2/N ≤ |
| gaussianPatternEigenvalues z i := by |
| let hA := gaussianPatternCoupling_isHermitian 1 z |
| let u := hA.eigenvectorBasis i |
| have hu : ‖u‖ = 1 := hA.eigenvectorBasis.orthonormal.1 i |
| have hb : BddBelow (Set.range (fun x : Metric.sphere (0 : EuclideanSpace ℝ (Fin N)) 1 => |
| ‖gaussianPatternEuclideanOperator (gaussianPatternTranspose N m z) x‖)) := |
| ⟨0,by rintro _ ⟨x,rfl⟩; exact norm_nonneg _⟩ |
| have hl : gaussianPatternSingularMin (gaussianPatternTranspose N m z) ≤ |
| ‖gaussianPatternEuclideanOperator (gaussianPatternTranspose N m z) u‖ := |
| ciInf_le hb ⟨u,by simpa only [Metric.mem_sphere,dist_zero_right] using hu⟩ |
| have hs := sq_le_sq₀ (gaussianPatternSingularMin_nonneg hN _) (norm_nonneg _) |>.2 hl |
| have he : gaussianPatternEigenvalues z i = ⟪u,(gaussianPatternCoupling 1 z).toEuclideanLin u⟫ := by |
| unfold gaussianPatternEigenvalues |
| change hA.eigenvalues i = _ |
| rw [hA.eigenvalues_eq i,EuclideanSpace.inner_eq_star_dotProduct] |
| simp only [star_trivial,RCLike.re_to_real] |
| exact dotProduct_comm _ _ |
| rw [he,gaussianPattern_quadratic] |
| exact div_le_div_of_nonneg_right hs (Nat.cast_nonneg N) |
|
|
| end InvariantIsing |
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| end |
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| end OAI |
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