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67aee1d | 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 | import OAI.Analysis.FreeEntropy.GaussianBlockTail
namespace OAI
universe u1
noncomputable section
open MeasureTheory ProbabilityTheory Finset
namespace FiniteEntropySeparation
variable {ι : Type u1} [Fintype ι] [DecidableEq ι]
def coordinateMask (m : Finset ι) (x : EuclideanSpace ℝ ι) : EuclideanSpace ℝ ι :=
WithLp.toLp 2 (fun i => if i∈m then x i else 0)
theorem coordinateMask_sq (m : Finset ι) (x : EuclideanSpace ℝ ι) :
‖coordinateMask m x‖^2=∑i∈m,x i^2 := by
classical
simp [coordinateMask,EuclideanSpace.norm_sq_eq,Real.norm_eq_abs,sq_abs]
theorem coordinateMask_norm_le (m : Finset ι) (x : EuclideanSpace ℝ ι) :
‖coordinateMask m x‖≤‖x‖ := by
apply (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mp
rw [coordinateMask_sq,EuclideanSpace.norm_sq_eq]
simp only [Real.norm_eq_abs,sq_abs]
exact sum_le_sum_of_subset_of_nonneg (subset_univ m) (fun i _ _ => sq_nonneg _)
def coordinateMaskLinear (m : Finset ι) : EuclideanSpace ℝ ι →ₗ[ℝ] EuclideanSpace ℝ ι where
toFun := coordinateMask m
map_add' x y := by ext i; simp only [coordinateMask,PiLp.toLp_apply,PiLp.add_apply]; split_ifs <;> simp
map_smul' r x := by ext i; simp only [coordinateMask,PiLp.toLp_apply,PiLp.smul_apply,smul_eq_mul,RingHom.id_apply]; split_ifs <;> simp
theorem coordinateMask_continuous (m : Finset ι) : Continuous (coordinateMask m) :=
(coordinateMaskLinear m).continuous_of_finiteDimensional
theorem coordinateMask_norm_sub (m : Finset ι) (x y : EuclideanSpace ℝ ι) :
|‖coordinateMask m x‖-‖coordinateMask m y‖|≤‖x-y‖ := by
calc
_ ≤ ‖coordinateMask m x-coordinateMask m y‖ := abs_norm_sub_norm_le _ _
_ = ‖coordinateMask m (x-y)‖ := by exact congrArg norm (map_sub (coordinateMaskLinear m) x y).symm
_ ≤ _ := coordinateMask_norm_le _ _
theorem normalLaw_coordinateMask_tail (m : Finset ι) {s r : ℝ} (hs : 0<s) (hr : 0≤r) :
(normalLaw s) {x | r≤‖coordinateMask m x‖} ≤
ENNReal.ofReal (Real.exp (-r^2/(4*s))*Real.sqrt 2^m.card) := by
have he : {x : EuclideanSpace ℝ ι | r≤‖coordinateMask m x‖}={x | r^2≤∑i∈m,x i^2} := by
ext x
simp only [Set.mem_ofPred_eq,← coordinateMask_sq]
exact (sq_le_sq₀ hr (norm_nonneg _)).symm
rw [he]
exact normalLaw_mask_tail m hs _
end FiniteEntropySeparation
end
end OAI
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