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9eeb9c7 | 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 | 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
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