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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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | 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
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