| import OAI.MathematicalPhysics.CriticalSK.Main |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Topology NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter |
| namespace CriticalSK |
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| section ThresholdComparison |
| variable {ι : Type*} [Fintype ι] |
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| lemma kernel_centered_l1_bound {K : Matrix ι ι ℝ} {μ p : ι → ℝ} {c : ℝ} |
| (hmass : ∑ x, p x = ∑ x, μ x) |
| (hstat : ∀ y, ∑ x, μ x * K x y = μ y) |
| (hbound : ∀ x, ∑ y, |K x y - μ y| ≤ c) : |
| (∑ y, |(∑ x, p x * K x y) - μ y|) ≤ c * ∑ x, |p x - μ x| := by |
| have hid (y : ι) : (∑ x, p x * K x y) - μ y = |
| ∑ x, (p x - μ x) * (K x y - μ y) := by |
| simp only [sub_mul, mul_sub, Finset.sum_sub_distrib, ← Finset.sum_mul, |
| hstat, hmass] |
| ring |
| simp_rw [hid] |
| calc |
| _ ≤ ∑ y, ∑ x, |(p x - μ x) * (K x y - μ y)| := |
| Finset.sum_le_sum (fun _ _ => Finset.abs_sum_le_sum_abs _ _) |
| _ = ∑ x, |p x - μ x| * ∑ y, |K x y - μ y| := by |
| simp_rw [abs_mul] |
| rw [Finset.sum_comm] |
| simp_rw [Finset.mul_sum] |
| _ ≤ ∑ x, |p x - μ x| * c := |
| Finset.sum_le_sum (fun x _ => mul_le_mul_of_nonneg_left (hbound x) (abs_nonneg _)) |
| _ = _ := by rw [← Finset.sum_mul, mul_comm] |
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| variable [DecidableEq ι] |
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| lemma kernel_power_tv_bound {K : Matrix ι ι ℝ} {μ : ι → ℝ} {c : ℝ} |
| (hK : ∀ x y, 0 ≤ K x y) (hμ : ∀ y, 0 ≤ μ y) |
| (hrow : ∀ x, ∑ y, K x y = 1) (hmass : ∑ y, μ y = 1) |
| (hstat : ∀ y, ∑ x, μ x * K x y = μ y) |
| (hc : 0 ≤ c) (hbound : ∀ x, (1 / 2 : ℝ) * ∑ y, |K x y - μ y| ≤ c / 2) |
| (k : ℕ) (x : ι) : |
| (1 / 2 : ℝ) * ∑ y, |(K ^ k) x y - μ y| ≤ c ^ k := by |
| have hb : ∀ z, ∑ y, |K z y - μ y| ≤ c := fun z => by linarith [hbound z] |
| have hpow : ∀ k : ℕ, ∀ z, ∑ y, |(K ^ k) z y - μ y| ≤ 2 * c ^ k := by |
| intro k |
| induction k with |
| | zero => |
| intro z |
| simpa using kernel_row_l1_le_two (K := K ^ 0) |
| (kernel_pow_nonneg hK 0) hμ (kernel_pow_sum hrow 0) hmass z |
| | succ k ih => |
| intro z |
| rw [pow_succ] |
| simp only [Matrix.mul_apply] |
| calc |
| _ ≤ c * ∑ y, |(K ^ k) z y - μ y| := |
| kernel_centered_l1_bound ((kernel_pow_sum hrow k z).trans hmass.symm) hstat hb |
| _ ≤ c * (2 * c ^ k) := mul_le_mul_of_nonneg_left (ih z) hc |
| _ = 2 * c ^ (k + 1) := by rw [pow_succ]; ring |
| linarith [hpow k x] |
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| end ThresholdComparison |
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| end CriticalSK |
| end |
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| end OAI |
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