File size: 2,806 Bytes
a89d282 | 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 52 53 54 55 56 57 58 59 60 61 62 63 64 | import OAI.Probability.InvariantIsing.Gaussian.GaussianGramLowerBound
import OAI.Probability.InvariantIsing.Gaussian.GaussianGramNormBound
namespace OAI
/-! Deterministic transport of the two singular-value edge errors to Gram spectral excess. -/
noncomputable section
open scoped Topology
namespace InvariantIsing
lemma gaussianGram_singular_bounds {n m : ℕ}
(z : EuclideanSpace ℝ (Fin (n+1) × Fin m)) (i : Fin (n+1)) :
(gaussianPatternSingularMin (gaussianPatternTranspose (n+1) m z)/Real.sqrt (n+1))^2 ≤
gaussianPatternEigenvalues z i ∧
gaussianPatternEigenvalues z i ≤ (gaussianPatternSingularMax z/Real.sqrt (n+1))^2 := by
have hs : (Real.sqrt (n+1))^2 = (n+1 : ℝ) := Real.sq_sqrt (by positivity)
constructor
· simpa only [div_pow,hs,Nat.cast_succ,Nat.cast_add,Nat.cast_one] using
gaussianPatternEigenvalues_lower_singular (Nat.succ_pos n) z i
· have h := (le_abs_self (gaussianPatternEigenvalues z i)).trans
(gaussianPatternEigenvalues_abs_le (Nat.succ_pos n) z i)
simpa only [div_pow,hs,Nat.cast_succ,Nat.cast_add,Nat.cast_one] using h
lemma squared_lower_edge_bound (L b : ℝ) (hL : 0 ≤ L) (hb : 0 ≤ b) :
L^2-2*L*max (L-b) 0 ≤ b^2 := by
by_cases h : L ≤ b
· rw [max_eq_right (sub_nonpos.mpr h)]
nlinarith
· rw [max_eq_left (by linarith : 0 ≤ L-b)]
nlinarith [sq_nonneg (L-b)]
lemma gaussianGram_spectralExcess_bound {n m : ℕ}
(z : EuclideanSpace ℝ (Fin (n+1) × Fin m)) (L U : ℝ) (hL : 0 ≤ L) (hU : 0 ≤ U) :
spectralExcess (gaussianPatternEigenvalues z) (L^2) (U^2) ≤
(U+max (gaussianPatternSingularMax z/Real.sqrt (n+1)-U) 0)^2-U^2 +
2*L*max (L-gaussianPatternSingularMin (gaussianPatternTranspose (n+1) m z)/Real.sqrt (n+1)) 0 := by
let a := gaussianPatternSingularMax z/Real.sqrt (n+1)
let b := gaussianPatternSingularMin (gaussianPatternTranspose (n+1) m z)/Real.sqrt (n+1)
let du := max (a-U) 0
let dl := max (L-b) 0
have ha : 0 ≤ a := div_nonneg (norm_nonneg _) (Real.sqrt_nonneg _)
have hb : 0 ≤ b := div_nonneg (gaussianPatternSingularMin_nonneg (Nat.succ_pos n) _)
(Real.sqrt_nonneg _)
have hdu : 0 ≤ du := le_max_right _ _
have hdl : 0 ≤ dl := le_max_right _ _
have hu : 0 ≤ (U+du)^2-U^2 := by nlinarith
have hl : 0 ≤ 2*L*dl := by positivity
apply spectralExcess_le _ _ _ _ (add_nonneg hu hl)
intro i
have hbounds := gaussianGram_singular_bounds z i
change b^2 ≤ gaussianPatternEigenvalues z i ∧ gaussianPatternEigenvalues z i ≤ a^2 at hbounds
have hblo := squared_lower_edge_bound L b hL hb
have hau : a ≤ U+du := by dsimp only [du]; linarith [le_max_left (a-U) 0]
constructor
· change L^2-2*L*dl ≤ b^2 at hblo
linarith [hbounds.1]
· nlinarith [hbounds.2]
end InvariantIsing
end
end OAI
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