File size: 2,703 Bytes
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 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 | import Mathlib
namespace OAI
open scoped BigOperators
open Finset
namespace ForestSpace
noncomputable section
attribute [local instance] Classical.propDecidable
section ModulusDefinitions
variable (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E]
/-- The closed finite-codimensional subspaces. -/
def Cofinite : Type :=
{F : Submodule ℝ E // IsClosed (F : Set E) ∧ FiniteDimensional ℝ (E ⧸ F)}
noncomputable def cofiniteTop : Cofinite E :=
⟨⊤, by simp, inferInstance⟩
/-- The inner one-sided excess values and their infimum. -/
def aucValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ :=
{a | ∃ y : E, y ∈ F.val ∧ N y = 1 ∧ a = N (x + t • y) - 1}
noncomputable def aucInner (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : ℝ :=
sInf (aucValues E N t x F)
noncomputable def aucCenter (N : E → ℝ) (t : ℝ) (x : E) : ℝ :=
sSup (Set.range (aucInner E N t x))
/-- The one-sided asymptotic uniform convexity modulus, with the order
of infimum, supremum and infimum. -/
noncomputable def aucModulus (N : E → ℝ) (t : ℝ) : ℝ :=
sInf {a | ∃ x : E, N x = 1 ∧ a = aucCenter E N t x}
/-- The inner averaged signed excess; it is not replaced by a maximum. -/
def averageValues (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : Set ℝ :=
{a | ∃ y : E, y ∈ F.val ∧ N y = 1 ∧
a = (N (x+t•y) + N (x-t•y))/((2 : ℕ) : ℝ) - 1}
noncomputable def averageInner (N : E → ℝ) (t : ℝ) (x : E) (F : Cofinite E) : ℝ :=
sInf (averageValues E N t x F)
noncomputable def averageCenter (N : E → ℝ) (t : ℝ) (x : E) : ℝ :=
sSup (Set.range (averageInner E N t x))
/-- The averaged asymptotic midpoint modulus. -/
noncomputable def averageModulus (N : E → ℝ) (t : ℝ) : ℝ :=
sInf {a | ∃ x : E, N x = 1 ∧ a = averageCenter E N t x}
end ModulusDefinitions
/-- An equivalent norm is kept as its exact real seminorm, with explicit positive
comparison constants. The lower inequality makes it a norm. -/
structure EquivalentNorm (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E]
(α β : ℝ) where
N : Seminorm ℝ E
alpha_pos : 0 < α
alpha_le_beta : α ≤ β
lower : ∀ x : E, α*‖x‖ ≤ N x
upper : ∀ x : E, N x ≤ β*‖x‖
/-- Canonical (not merely abstract) reflexivity. -/
def Reflexive (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] : Prop :=
Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E)
/-- AUC requires positivity at every strictly positive parameter. -/
def IsAUC {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] (N : E → ℝ) : Prop :=
∀ t : ℝ, 0 < t → 0 < aucModulus E N t
end
end ForestSpace
end OAI
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