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dd14c26 | 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 | import Mathlib
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open scoped BigOperators
def polynomialBudget (P : Polynomial ℝ) : ℝ := ∑n ∈ P.support,|P.coeff n|
lemma polynomialBudget_nonneg (P : Polynomial ℝ) : 0≤polynomialBudget P :=
Finset.sum_nonneg (fun _ _ => abs_nonneg _)
lemma eval_abs_le_polynomialBudget (P : Polynomial ℝ) {ρ : ℝ} (h0 : 0≤ρ) (h1 : ρ≤1) :
|P.eval ρ|≤polynomialBudget P := by
rw [Polynomial.eval_eq_sum]
unfold Polynomial.sum polynomialBudget
apply (Finset.abs_sum_le_sum_abs _ _).trans
apply Finset.sum_le_sum
intro n _
rw [abs_mul,abs_pow,abs_of_nonneg h0]
exact mul_le_of_le_one_right (abs_nonneg _) (pow_le_one₀ h0 h1)
lemma scaledCoefficient_bound (P : Polynomial ℝ) (n : ℕ) {r L ρ : ℝ}
(hl0 : 0≤r*L) (hl1 : r*L≤1) (h0 : 0≤ρ) (h1 : ρ≤1) :
|(r*L)^n*P.eval ρ|≤polynomialBudget P := by
rw [abs_mul,abs_pow,abs_of_nonneg hl0]
exact (mul_le_mul_of_nonneg_right (pow_le_one₀ hl0 hl1) (abs_nonneg _)).trans
(by simpa using eval_abs_le_polynomialBudget P h0 h1)
end LogConcaveSampling
end
end
end
end OAI
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