The tau functional for multidistance

Definition of the tau functional and basic facts

Main definitions:

Main results

class MeasureableFinGroup (G : Type u_1) extends AddCommGroup G, Fintype G, MeasurableSpace G, MeasurableSingletonClass G : Type u_1

We will frequently be working with finite additive groups with the discrete sigma-algebra.

• add : G → G → G
• add_assoc (a b c : G) : a + b + c = a + (b + c)
• zero : G
• zero_add (a : G) : 0 + a = a
• add_zero (a : G) : a + 0 = a
• nsmul : ℕ → G → G
• nsmul_zero (x : G) : AddMonoid.nsmul 0 x = 0
• nsmul_succ (n : ℕ) (x : G) : AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
• neg : G → G
• sub : G → G → G
• sub_eq_add_neg (a b : G) : a - b = a + -b
• zsmul : ℤ → G → G
• zsmul_zero' (a : G) : SubNegMonoid.zsmul 0 a = 0
• zsmul_succ' (n : ℕ) (a : G) : SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a
• zsmul_neg' (n : ℕ) (a : G) : SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a
• neg_add_cancel (a : G) : -a + a = 0
• add_comm (a b : G) : a + b = b + a
• elems : Finset G
• complete (x : G) : x ∈ elems
• MeasurableSet' : Set G → Prop
• measurableSet_empty : MeasureableFinGroup.toMeasurableSpace.MeasurableSet' ∅
• measurableSet_compl (s : Set G) : MeasureableFinGroup.toMeasurableSpace.MeasurableSet' s → MeasureableFinGroup.toMeasurableSpace.MeasurableSet' sᶜ
• measurableSet_iUnion (f : ℕ → Set G) : (∀ (i : ℕ), MeasureableFinGroup.toMeasurableSpace.MeasurableSet' (f i)) → MeasureableFinGroup.toMeasurableSpace.MeasurableSet' (⋃ (i : ℕ), f i)
• measurableSet_singleton (x : G) : MeasurableSet {x}

structure multiRefPackage (G : Type u_1) (Ω₀ : Type u_2) [MeasureableFinGroup G] [MeasureTheory.MeasureSpace Ω₀] : Type (max u_1 u_2)

A structure that packages all the fixed information in the main argument. See https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Problem.20when.20instances.20are.20inside.20a.20structure for more discussion of the design choices here.

• m : ℕ

  The torsion index of the group we are considering.

• hm : self.m ≥ 2
• htorsion (x : G) : self.m • x = 0
• hprob : MeasureTheory.IsProbabilityMeasure MeasureTheory.volume
• X₀ : Ω₀ → G

  The random variable

• hmeas : Measurable self.X₀
• η : ℝ

  A small constant. The argument will only work for suitably small η.

• hη : 0 < self.η
• hη' : self.η ≤ 1

noncomputable def multiTau {G : Type u_1} {Ω₀ : Type u_2} [MeasureableFinGroup G] [MeasureTheory.MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u_3) (hΩ : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω i)) (X : (i : Fin p.m) → Ω i → G) : ℝ

If $(X_i{)}_{1\le i\le m}$ is a tuple, we define its $\tau$-functional $$\tau [(X_i{)}_{1\le i\le m}]:=D[(X_i{)}_{1\le i\le m}]+\eta \sum _{i=1}^{m}d[X_i;X^0].$$

Equations

• multiTau p Ω hΩ X = D[X ; hΩ] + p.η * ∑ i : Fin p.m, d[X i # p.X₀]

def multiTauMinimizes {G Ω₀ : Type u} [MeasureableFinGroup G] [MeasureTheory.MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω i)) (X : (i : Fin p.m) → Ω i → G) : Prop

A $\tau$-minimizer is a tuple $(X_i{)}_{1\le i\le m}$ that minimizes the $\tau$-functional among all tuples of $G$-valued random variables.

Equations

• multiTauMinimizes p Ω hΩ X = ∀ (Ω' : Fin p.m → Type ?u.48) (hΩ' : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω' i)) (X' : (i : Fin p.m) → Ω' i → G), multiTau p Ω hΩ X ≤ multiTau p Ω' hΩ' X'

theorem multiTau_continuous {G Ω₀ : Type u} [MeasureableFinGroup G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] [MeasureTheory.MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) : Continuous fun (μ : Fin p.m → MeasureTheory.ProbabilityMeasure G) => multiTau p (fun (x : Fin p.m) => G) (fun (i : Fin p.m) => { toMeasurableSpace := MeasureableFinGroup.toMeasurableSpace, volume := ↑(μ i) }) fun (x : Fin p.m) => id

If $G$ is finite, then a $\tau$ is continuous.

theorem multiTau_min_exists {G Ω₀ : Type u} [MeasureableFinGroup G] [MeasureTheory.MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) : ∃ (Ω : Fin p.m → Type u) (hΩ : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω i)) (X : (i : Fin p.m) → Ω i → G), multiTauMinimizes p Ω hΩ X

If $G$ is finite, then a $\tau$-minimizer exists.

theorem multiTau_min_sum_le {G Ω₀ : Type u} [hG : MeasureableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω i)) (hprobΩ : ∀ (i : Fin p.m), MeasureTheory.IsProbabilityMeasure MeasureTheory.volume) (X : (i : Fin p.m) → Ω i → G) (hX : ∀ (i : Fin p.m), Measurable (X i)) (h_min : multiTauMinimizes p Ω hΩ X) : ∑ i : Fin p.m, d[X i # p.X₀] ≤ 2 * ↑p.m * p.η⁻¹ * d[p.X₀ # p.X₀]

If $(X_i{)}_{1\le i\le m}$ is a $\tau$-minimizer, then $\sum _{i=1}^{m}d[X_i;X^0]\le \frac{2m}{\eta }d[X^0;X^0]$.

theorem sub_multiDistance_le {G Ω₀ : Type u} [MeasureableFinGroup G] [hΩ₀ : MeasureTheory.MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω i)) (hΩprob : ∀ (i : Fin p.m), MeasureTheory.IsProbabilityMeasure MeasureTheory.volume) (X : (i : Fin p.m) → Ω i → G) (hmeasX : ∀ (i : Fin p.m), Measurable (X i)) (h_min : multiTauMinimizes p Ω hΩ X) (Ω' : Fin p.m → Type u) (hΩ' : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω' i)) (hΩprob' : ∀ (i : Fin p.m), MeasureTheory.IsProbabilityMeasure MeasureTheory.volume) (X' : (i : Fin p.m) → Ω' i → G) (hmeasX' : ∀ (i : Fin p.m), Measurable (X' i)) : D[X ; hΩ] - D[X' ; hΩ'] ≤ p.η * ∑ i : Fin p.m, d[X i # X' i]

If $(X_i{)}_{1\le i\le m}$ is a $\tau$-minimizer, and $k:=D[(X_i{)}_{1\le i\le m}]$, then for any other tuple $(X_i^{\prime }{)}_{1\le i\le m}$, one has $$k-D[(X_i^{\prime }{)}_{1\le i\le m}]\le \eta \sum _{i=1}^{m}d[X_i;X_i^{\prime }].$$

theorem sub_condMultiDistance_le {G Ω₀ : Type u} [MeasureableFinGroup G] [MeasureTheory.MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω i)) (hΩprob : ∀ (i : Fin p.m), MeasureTheory.IsProbabilityMeasure MeasureTheory.volume) (X : (i : Fin p.m) → Ω i → G) (hmeasX : ∀ (i : Fin p.m), Measurable (X i)) (h_min : multiTauMinimizes p Ω hΩ X) (Ω' : Fin p.m → Type u) (hΩ' : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω' i)) (hΩ'prob : ∀ (i : Fin p.m), MeasureTheory.IsProbabilityMeasure MeasureTheory.volume) (X' : (i : Fin p.m) → Ω' i → G) (hmeasX' : ∀ (i : Fin p.m), Measurable (X' i)) {S : Type u} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] (Y : (i : Fin p.m) → Ω' i → S) (hY : ∀ (i : Fin p.m), Measurable (Y i)) : D[X ; hΩ] - D[X' | Y ; hΩ'] ≤ p.η * ∑ i : Fin p.m, d[X i # X' i | Y i]

If $(X_i{)}_{1\le i\le m}$ is a $\tau$-minimizer, and $k:=D[(X_i{)}_{1\le i\le m}]$, then for any other tuples $(X_i^{\prime }{)}_{1\le i\le m}$ and $(Y_i{)}_{1\le i\le m}$ with the $X_i^{\prime }$ $G$-valued, one has $$k-D[(X_i^{\prime }{)}_{1\le i\le m}|(Y_i{)}_{1\le i\le m}]\le \eta \sum _{i=1}^{m}d[X_i;X_i^{\prime }|Y_i].$$

theorem sub_condMultiDistance_le' {G Ω₀ : Type u} [MeasureableFinGroup G] [MeasureTheory.MeasureSpace Ω₀] (p : multiRefPackage G Ω₀) (Ω : Fin p.m → Type u) (hΩ : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω i)) (hΩprob : ∀ (i : Fin p.m), MeasureTheory.IsProbabilityMeasure MeasureTheory.volume) (X : (i : Fin p.m) → Ω i → G) (hmeasX : ∀ (i : Fin p.m), Measurable (X i)) (h_min : multiTauMinimizes p Ω hΩ X) (Ω' : Fin p.m → Type u) (hΩ' : (i : Fin p.m) → MeasureTheory.MeasureSpace (Ω' i)) (hΩ'prob : ∀ (i : Fin p.m), MeasureTheory.IsProbabilityMeasure MeasureTheory.volume) (X' : (i : Fin p.m) → Ω' i → G) (hmeasX' : ∀ (i : Fin p.m), Measurable (X' i)) {S : Type u} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] (Y : (i : Fin p.m) → Ω' i → S) (hY : ∀ (i : Fin p.m), Measurable (Y i)) (φ : Equiv.Perm (Fin p.m)) : D[X ; hΩ] - D[X' | Y ; hΩ'] ≤ p.η * ∑ i : Fin p.m, d[X (φ i) # X' i | Y i]

With the notation of the previous lemma, we have \begin{equation}\label{5.3-conv} k - D[ X'{[m]} | Y{[m]} ] \leq \eta \sum_{i=1}^m d[X_{\sigma(i)};X'_i|Y_i] \end{equation} for any permutation $\sigma :\{1,\dots ,m\}\to \{1,\dots ,m\}$.