Datasets:

roseking
/

openAI-math-dataset

import OAI.Combinatorics.MatchingEntropy.Calculus

namespace OAI

noncomputable section

open scoped BigOperators Topology
open Filter

universe uV uE uI uJ uA uB uC uW uF uD uι

namespace MatchingEntropyBounds

variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]

open Matrix

variable {A : Type uA} {I : Type uI} [Fintype A] [Fintype I] [DecidableEq A] [DecidableEq I]

def weightedColumns (p : A → ℝ) (b : A → I → ℝ) : Matrix I A ℝ :=
  fun i a => Real.sqrt (p a) * b a i

def covariance (p : A → ℝ) (b : A → I → ℝ) : Matrix I I ℝ :=
  weightedColumns p b * (weightedColumns p b)ᵀ

omit [DecidableEq A] [DecidableEq I] in
theorem covariance_posSemidef (p : A → ℝ) (b : A → I → ℝ) :
    (covariance p b).PosSemidef := by
  simpa only [covariance, conjTranspose_eq_transpose_of_trivial] using
    posSemidef_self_mul_conjTranspose (weightedColumns p b)

omit [Fintype I] [DecidableEq A] [DecidableEq I] in
theorem covariance_entry {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ) (i j : I) :
    covariance p b i j = ∑ a, p a*b a i*b a j := by
  unfold covariance
  rw [Matrix.mul_apply]
  apply Finset.sum_congr rfl
  intro a _
  simp only [weightedColumns, Matrix.transpose_apply]
  calc
    _ = (Real.sqrt (p a))^2 * b a i * b a j := by ring
    _ = _ := by rw [Real.sq_sqrt (hp a)]

omit [DecidableEq A] [DecidableEq I] in
theorem covariance_mulVec {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ)
    (u : I → ℝ) (i : I) :
    (covariance p b *ᵥ u) i = ∑ a, p a*b a i*(b a ⬝ᵥ u) := by
  simp only [Matrix.mulVec, dotProduct, covariance_entry hp, Finset.sum_mul]
  rw [Finset.sum_comm]
  apply Finset.sum_congr rfl
  intro a _
  rw [Finset.mul_sum]
  apply Finset.sum_congr rfl
  intro j _
  ring

omit [DecidableEq A] [DecidableEq I] in
theorem covariance_quadratic {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ)
    (u : I → ℝ) : u ⬝ᵥ (covariance p b *ᵥ u) = ∑ a, p a*(b a ⬝ᵥ u)^2 := by
  rw [dotProduct]
  simp_rw [covariance_mulVec hp, Finset.mul_sum]
  rw [Finset.sum_comm]
  apply Finset.sum_congr rfl
  intro a _
  calc
    (∑ i, u i*(p a*b a i*(b a ⬝ᵥ u))) =
      (∑ i, b a i*u i)*(p a*(b a ⬝ᵥ u)) := by
      rw [Finset.sum_mul]
      apply Finset.sum_congr rfl
      intro i _
      ring
    _ = p a*(b a ⬝ᵥ u)^2 := by rw [← dotProduct]; ring

def weightedDirectionSpace (p : A → ℝ) (b : A → I → ℝ) : Submodule ℝ (I → ℝ) :=
  Submodule.span ℝ (Set.range fun a : {a // 0 < p a} => b a.val)

omit [Fintype A] [Fintype I] [DecidableEq A] [DecidableEq I] in
theorem weightedColumns_span {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ) :
    Submodule.span ℝ (Set.range (weightedColumns p b).col) = weightedDirectionSpace p b := by
  apply le_antisymm
  · apply Submodule.span_le.mpr
    rintro _ ⟨a,rfl⟩
    by_cases ha : 0 < p a
    · have hb : b a ∈ weightedDirectionSpace p b := Submodule.subset_span ⟨⟨a,ha⟩,rfl⟩
      exact Submodule.smul_mem _ (Real.sqrt (p a)) hb
    · have hz := le_antisymm (le_of_not_gt ha) (hp a)
      have he : (weightedColumns p b).col a = 0 := by
        ext i
        simp [weightedColumns, Matrix.col, hz]
      rw [he]
      exact (weightedDirectionSpace p b).zero_mem
  · apply Submodule.span_le.mpr
    rintro _ ⟨a,rfl⟩
    have hb : (weightedColumns p b).col a.val ∈
        Submodule.span ℝ (Set.range (weightedColumns p b).col) := Submodule.subset_span ⟨a.val,rfl⟩
    have hs : Real.sqrt (p a.val) ≠ 0 := ne_of_gt (Real.sqrt_pos.2 a.property)
    have hh := Submodule.smul_mem _ (Real.sqrt (p a.val))⁻¹ hb
    have he : (Real.sqrt (p a.val))⁻¹ • (weightedColumns p b).col a.val = b a.val := by
      ext i
      change (Real.sqrt (p a.val))⁻¹ * (Real.sqrt (p a.val) * b a.val i) = _
      rw [← mul_assoc, inv_mul_cancel₀ hs, one_mul]
    rw [he] at hh
    exact hh

omit [DecidableEq A] [DecidableEq I] in
theorem covariance_rank {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ) :
    (covariance p b).rank = Module.finrank ℝ (weightedDirectionSpace p b) := by
  rw [covariance, rank_self_mul_transpose, rank_eq_finrank_span_cols, weightedColumns_span hp]

def normalizedCentered (v : A → I → ℝ) (x : I → ℝ) (a : A) (i : I) : ℝ :=
  (v a i-x i)/Real.sqrt (x i)

omit [Fintype I] [DecidableEq A] [DecidableEq I] in
theorem normalizedCentered_mean {p : A → ℝ} (hp : ∑ a, p a = 1)
    (v : A → I → ℝ) (x : I → ℝ) (hx : ∀ i, ∑ a, p a*v a i = x i) (i : I) :
    ∑ a, p a*normalizedCentered v x a i = 0 := by
  simp only [normalizedCentered, ← mul_div_assoc, mul_sub, ← Finset.sum_div,
    Finset.sum_sub_distrib, ← Finset.sum_mul, hp, one_mul, hx, sub_self, zero_div]

omit [Fintype I] [DecidableEq A] [DecidableEq I] in
theorem normalizedCentered_variance {p : A → ℝ} (hp : ∑ a, p a = 1)
    (v : A → I → ℝ) (hv : ∀ a i, (v a i)^2=v a i) (x : I → ℝ)
    (hx : ∀ i, ∑ a, p a*v a i = x i) (hxp : ∀ i, 0 < x i) (i : I) :
    ∑ a, p a*(normalizedCentered v x a i)^2 = 1-x i := by
  have heq : ∀ a, p a*(normalizedCentered v x a i)^2 =
      (p a*v a i-2*x i*(p a*v a i)+(x i)^2*p a)/x i := by
    intro a
    rw [normalizedCentered, div_pow, Real.sq_sqrt (hxp i).le, ← mul_div_assoc]
    congr 1
    calc
      _ = p a*(v a i)^2-2*x i*(p a*v a i)+(x i)^2*p a := by ring
      _ = _ := by rw [hv]
  simp_rw [heq]
  rw [← Finset.sum_div]
  simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum,
    hx, hp, mul_one]
  field_simp [ne_of_gt (hxp i)]
  ring

omit [DecidableEq A] [DecidableEq I] in
theorem normalized_covariance_trace {p : A → ℝ} (hp : p ∈ probabilitySimplex A)
    (v : A → I → ℝ) (hv : ∀ a i, (v a i)^2=v a i) (x : I → ℝ)
    (hx : ∀ i, ∑ a, p a*v a i = x i) (hxp : ∀ i, 0 < x i) :
    (covariance p (normalizedCentered v x)).trace = Fintype.card I - ∑ i, x i := by
  unfold Matrix.trace Matrix.diag
  simp_rw [covariance_entry hp.1]
  have heq : ∀ i, (∑ a, p a*normalizedCentered v x a i*normalizedCentered v x a i) = 1-x i := by
    intro i
    simpa only [pow_two, mul_assoc] using
      normalizedCentered_variance hp.2 v hv x hx hxp i
  simp_rw [heq]
  simp

end MatchingEntropyBounds

end

end OAI

	import OAI.Combinatorics.MatchingEntropy.Calculus

	namespace OAI

	noncomputable section

	open scoped BigOperators Topology
	open Filter

	universe uV uE uI uJ uA uB uC uW uF uD uι

	namespace MatchingEntropyBounds

	variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]

	open Matrix

	variable {A : Type uA} {I : Type uI} [Fintype A] [Fintype I] [DecidableEq A] [DecidableEq I]

	def weightedColumns (p : A → ℝ) (b : A → I → ℝ) : Matrix I A ℝ :=
	fun i a => Real.sqrt (p a) * b a i

	def covariance (p : A → ℝ) (b : A → I → ℝ) : Matrix I I ℝ :=
	weightedColumns p b * (weightedColumns p b)ᵀ

	omit [DecidableEq A] [DecidableEq I] in
	theorem covariance_posSemidef (p : A → ℝ) (b : A → I → ℝ) :
	(covariance p b).PosSemidef := by
	simpa only [covariance, conjTranspose_eq_transpose_of_trivial] using
	posSemidef_self_mul_conjTranspose (weightedColumns p b)

	omit [Fintype I] [DecidableEq A] [DecidableEq I] in
	theorem covariance_entry {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ) (i j : I) :
	covariance p b i j = ∑ a, p ab a ib a j := by
	unfold covariance
	rw [Matrix.mul_apply]
	apply Finset.sum_congr rfl
	intro a _
	simp only [weightedColumns, Matrix.transpose_apply]
	calc
	_ = (Real.sqrt (p a))^2 * b a i * b a j := by ring
	_ = _ := by rw [Real.sq_sqrt (hp a)]

	omit [DecidableEq A] [DecidableEq I] in
	theorem covariance_mulVec {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ)
	(u : I → ℝ) (i : I) :
	(covariance p b ᵥ u) i = ∑ a, p ab a i*(b a ⬝ᵥ u) := by
	simp only [Matrix.mulVec, dotProduct, covariance_entry hp, Finset.sum_mul]
	rw [Finset.sum_comm]
	apply Finset.sum_congr rfl
	intro a _
	rw [Finset.mul_sum]
	apply Finset.sum_congr rfl
	intro j _
	ring

	omit [DecidableEq A] [DecidableEq I] in
	theorem covariance_quadratic {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ)
	(u : I → ℝ) : u ⬝ᵥ (covariance p b ᵥ u) = ∑ a, p a(b a ⬝ᵥ u)^2 := by
	rw [dotProduct]
	simp_rw [covariance_mulVec hp, Finset.mul_sum]
	rw [Finset.sum_comm]
	apply Finset.sum_congr rfl
	intro a _
	calc
	(∑ i, u i(p ab a i*(b a ⬝ᵥ u))) =
	(∑ i, b a iu i)(p a*(b a ⬝ᵥ u)) := by
	rw [Finset.sum_mul]
	apply Finset.sum_congr rfl
	intro i _
	ring
	_ = p a*(b a ⬝ᵥ u)^2 := by rw [← dotProduct]; ring

	def weightedDirectionSpace (p : A → ℝ) (b : A → I → ℝ) : Submodule ℝ (I → ℝ) :=
	Submodule.span ℝ (Set.range fun a : {a // 0 < p a} => b a.val)

	omit [Fintype A] [Fintype I] [DecidableEq A] [DecidableEq I] in
	theorem weightedColumns_span {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ) :
	Submodule.span ℝ (Set.range (weightedColumns p b).col) = weightedDirectionSpace p b := by
	apply le_antisymm
	· apply Submodule.span_le.mpr
	rintro _ ⟨a,rfl⟩
	by_cases ha : 0 < p a
	· have hb : b a ∈ weightedDirectionSpace p b := Submodule.subset_span ⟨⟨a,ha⟩,rfl⟩
	exact Submodule.smul_mem _ (Real.sqrt (p a)) hb
	· have hz := le_antisymm (le_of_not_gt ha) (hp a)
	have he : (weightedColumns p b).col a = 0 := by
	ext i
	simp [weightedColumns, Matrix.col, hz]
	rw [he]
	exact (weightedDirectionSpace p b).zero_mem
	· apply Submodule.span_le.mpr
	rintro _ ⟨a,rfl⟩
	have hb : (weightedColumns p b).col a.val ∈
	Submodule.span ℝ (Set.range (weightedColumns p b).col) := Submodule.subset_span ⟨a.val,rfl⟩
	have hs : Real.sqrt (p a.val) ≠ 0 := ne_of_gt (Real.sqrt_pos.2 a.property)
	have hh := Submodule.smul_mem _ (Real.sqrt (p a.val))⁻¹ hb
	have he : (Real.sqrt (p a.val))⁻¹ • (weightedColumns p b).col a.val = b a.val := by
	ext i
	change (Real.sqrt (p a.val))⁻¹ * (Real.sqrt (p a.val) * b a.val i) = _
	rw [← mul_assoc, inv_mul_cancel₀ hs, one_mul]
	rw [he] at hh
	exact hh

	omit [DecidableEq A] [DecidableEq I] in
	theorem covariance_rank {p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (b : A → I → ℝ) :
	(covariance p b).rank = Module.finrank ℝ (weightedDirectionSpace p b) := by
	rw [covariance, rank_self_mul_transpose, rank_eq_finrank_span_cols, weightedColumns_span hp]

	def normalizedCentered (v : A → I → ℝ) (x : I → ℝ) (a : A) (i : I) : ℝ :=
	(v a i-x i)/Real.sqrt (x i)

	omit [Fintype I] [DecidableEq A] [DecidableEq I] in
	theorem normalizedCentered_mean {p : A → ℝ} (hp : ∑ a, p a = 1)
	(v : A → I → ℝ) (x : I → ℝ) (hx : ∀ i, ∑ a, p a*v a i = x i) (i : I) :
	∑ a, p a*normalizedCentered v x a i = 0 := by
	simp only [normalizedCentered, ← mul_div_assoc, mul_sub, ← Finset.sum_div,
	Finset.sum_sub_distrib, ← Finset.sum_mul, hp, one_mul, hx, sub_self, zero_div]

	omit [Fintype I] [DecidableEq A] [DecidableEq I] in
	theorem normalizedCentered_variance {p : A → ℝ} (hp : ∑ a, p a = 1)
	(v : A → I → ℝ) (hv : ∀ a i, (v a i)^2=v a i) (x : I → ℝ)
	(hx : ∀ i, ∑ a, p a*v a i = x i) (hxp : ∀ i, 0 < x i) (i : I) :
	∑ a, p a*(normalizedCentered v x a i)^2 = 1-x i := by
	have heq : ∀ a, p a*(normalizedCentered v x a i)^2 =
	(p av a i-2x i(p av a i)+(x i)^2*p a)/x i := by
	intro a
	rw [normalizedCentered, div_pow, Real.sq_sqrt (hxp i).le, ← mul_div_assoc]
	congr 1
	calc
	_ = p a(v a i)^2-2x i(p av a i)+(x i)^2*p a := by ring
	_ = _ := by rw [hv]
	simp_rw [heq]
	rw [← Finset.sum_div]
	simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum,
	hx, hp, mul_one]
	field_simp [ne_of_gt (hxp i)]
	ring

	omit [DecidableEq A] [DecidableEq I] in
	theorem normalized_covariance_trace {p : A → ℝ} (hp : p ∈ probabilitySimplex A)
	(v : A → I → ℝ) (hv : ∀ a i, (v a i)^2=v a i) (x : I → ℝ)
	(hx : ∀ i, ∑ a, p a*v a i = x i) (hxp : ∀ i, 0 < x i) :
	(covariance p (normalizedCentered v x)).trace = Fintype.card I - ∑ i, x i := by
	unfold Matrix.trace Matrix.diag
	simp_rw [covariance_entry hp.1]
	have heq : ∀ i, (∑ a, p anormalizedCentered v x a inormalizedCentered v x a i) = 1-x i := by
	intro i
	simpa only [pow_two, mul_assoc] using
	normalizedCentered_variance hp.2 v hv x hx hxp i
	simp_rw [heq]
	simp

	end MatchingEntropyBounds

	end

	end OAI