| import OAI.Analysis.HyperbolicCones.SpectralContinuity |
| import OAI.Analysis.HyperbolicCones.InverseOrder |
|
|
| namespace OAI |
|
|
| /-! The inverse-square-root kernel limit for every finite positive semidefinite matrix. -/ |
|
|
| noncomputable section |
| open scoped Matrix.Norms.L2Operator MatrixOrder |
| open Matrix Filter Topology |
| namespace Paper256 |
|
|
| theorem inverseSquareRoot_affine_eq_cfc {n : ℕ} (H : Sym n) (t : ℝ) : |
| inverseSquareRoot (1 + (t - 1) • H) = |
| cfc (fun x : ℝ => (Real.sqrt (1 + (t - 1) * x))⁻¹) (H : Mat n ℝ) := by |
| have haff : cfc (fun x : ℝ => 1 + (t - 1) * x) (H : Mat n ℝ) = |
| ((1 + (t - 1) • H : Sym n) : Mat n ℝ) := by |
| simpa using matrix_cfc_affine H 1 (t - 1) |
| have hcomp := cfc_comp (fun x : ℝ => (Real.sqrt x)⁻¹) |
| (fun x : ℝ => 1 + (t - 1) * x) (H : Mat n ℝ) H.property |
| ((H.val.finite_real_spectrum.image _).continuousOn _) |
| (H.val.finite_real_spectrum.continuousOn _) |
| rw [haff] at hcomp |
| exact (inverseSquareRoot_eq_cfc _).trans hcomp.symm |
|
|
| theorem inverseSquareRoot_affine_tendsto {n : ℕ} (H : Sym n) |
| (hH : (H : Mat n ℝ).PosSemidef) : |
| Tendsto (fun t : ℝ => inverseSquareRoot (1 + (t - 1) • H)) atTop |
| (𝓝 (kernelProjection (H : Mat n ℝ))) := by |
| have hc := matrix_cfc_tendsto (H : Mat n ℝ) H.property |
| (fun t x : ℝ => (Real.sqrt (1 + (t - 1) * x))⁻¹) |
| (fun x : ℝ => if x = 0 then 1 else 0) |
| (fun i => inverse_sqrt_affine_scalar_tendsto _ (hH.eigenvalues_nonneg i)) |
| simpa only [← inverseSquareRoot_affine_eq_cfc, ← kernelProjection_eq_cfc] using hc |
|
|
| theorem inverseSquareRoot_kernel_limit (n : ℕ) (H : Sym n) |
| (hH : (H : Mat n ℝ).PosSemidef) : |
| Tendsto (fun t : ℝ => operatorNorm |
| (inverseSquareRoot (1 + (t - 1) • H) - kernelProjection (H : Mat n ℝ))) |
| atTop (𝓝 0) := by |
| have h := ((inverseSquareRoot_affine_tendsto H hH).sub_const |
| (kernelProjection (H : Mat n ℝ))).norm |
| have hop (A : Mat n ℝ) : operatorNorm A = ‖A‖ := (Matrix.l2_opNorm_def A).symm |
| simpa only [hop, sub_self, norm_zero] using h |
|
|
| end Paper256 |
|
|
| end |
|
|
| end OAI |
|
|
