| import Mathlib |
| import OAI.Probability.LogConcave.JetEstimates.Slice |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open scoped Classical BigOperators NNReal ENNReal |
|
|
| namespace TensorSum |
| variable {S : Type} [Fintype S] |
|
|
| def logDegree (A : TensorSum S) (j : ℕ) : ℕ := |
| ∑i,(A.term i).atom.expression.profileConstant*(j+1) |
|
|
| def logCoefficient (A : TensorSum S) (j : ℕ) (p : ℝ≥0∞) : ℝ := |
| ∑i,polynomialBudget (A.term i).coefficient* |
| (4*((A.term i).atom.expression.profileConstant:ℝ)*(p.toReal+j+1))^ |
| ((A.term i).atom.expression.profileConstant*(j+1)) |
|
|
| omit [Fintype S] in |
| lemma logCoefficient_nonneg (A : TensorSum S) (j : ℕ) (p : ℝ≥0∞) : |
| 0≤A.logCoefficient j p := by |
| apply Finset.sum_nonneg |
| intro i _ |
| exact mul_nonneg (polynomialBudget_nonneg _) (by positivity) |
|
|
| omit [Fintype S] in |
| lemma momentBudget_polylog (A : TensorSum S) (d j : ℕ) (p : ℝ≥0∞) : |
| (A.momentBudget d j p).toReal≤ |
| A.logCoefficient j p*(1+Real.log ((d:ℝ)+1))^(A.logDegree j) := by |
| have hl : 0≤Real.log ((d:ℝ)+1) := Real.log_nonneg (by linarith [Nat.cast_nonneg (α:=ℝ) d]) |
| have ht : 1≤1+Real.log ((d:ℝ)+1) := by linarith |
| have hf (i : A.index) : |
| ENNReal.ofReal (polynomialBudget (A.term i).coefficient)* |
| ((4*((A.term i).atom.expression.profileConstant:ℝ≥0∞))* |
| ENNReal.ofReal (p.toReal+j+Real.log ((d:ℝ)+1)+1))^ |
| ((A.term i).atom.expression.profileConstant*(j+1))≠⊤ := by |
| apply ne_of_lt |
| exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top |
| (ENNReal.pow_lt_top (ENNReal.mul_lt_top |
| (ENNReal.mul_lt_top (by norm_num) (ENNReal.natCast_lt_top _)) ENNReal.ofReal_lt_top)) |
| rw [momentBudget,ENNReal.toReal_sum (fun i _ => hf i),logCoefficient,Finset.sum_mul] |
| apply Finset.sum_le_sum |
| intro i _ |
| simp only [ENNReal.toReal_mul,ENNReal.toReal_pow,ENNReal.toReal_ofNat, |
| ENNReal.toReal_natCast,ENNReal.toReal_ofReal (polynomialBudget_nonneg _), |
| ENNReal.toReal_ofReal (by positivity : 0≤p.toReal+(j:ℝ)+Real.log ((d:ℝ)+1)+1)] |
| let c := (A.term i).atom.expression.profileConstant |
| let n := c*(j+1) |
| have hb : 4*(c:ℝ)*(p.toReal+j+Real.log ((d:ℝ)+1)+1)≤ |
| (4*(c:ℝ)*(p.toReal+j+1))*(1+Real.log ((d:ℝ)+1)) := by |
| have hh : 0≤(p.toReal+(j:ℝ))*Real.log ((d:ℝ)+1) := by positivity |
| have hh' := mul_nonneg (by positivity : 0≤4*(c:ℝ)) hh |
| nlinarith |
| have hn : n≤A.logDegree j := by |
| dsimp [n,c,logDegree] |
| exact Finset.single_le_sum (f:=fun i : A.index => (A.term i).atom.expression.profileConstant*(j+1)) (fun _ _ => Nat.zero_le _) (Finset.mem_univ i) |
| calc |
| _ ≤ polynomialBudget (A.term i).coefficient* |
| ((4*(c:ℝ)*(p.toReal+j+1))*(1+Real.log ((d:ℝ)+1)))^n := |
| mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) hb _) (polynomialBudget_nonneg _) |
| _ = polynomialBudget (A.term i).coefficient*(4*(c:ℝ)*(p.toReal+j+1))^n* |
| (1+Real.log ((d:ℝ)+1))^n := by rw [mul_pow]; ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left (pow_le_pow_right₀ ht hn) (by |
| exact mul_nonneg (polynomialBudget_nonneg _) (by positivity)) |
| end TensorSum |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
