| import Mathlib |
| import OAI.Probability.SKRatio.Quantization.BinProjected |
| import OAI.Probability.SKRatio.Quantization.BinDirections |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Matrix |
| open MeasureTheory ProbabilityTheory Filter Real |
| namespace SKRatio.Bins |
| open Planted Scalar SKRatioClock.Regression |
| variable {ι α : Type*} [Fintype ι] [DecidableEq ι] [Fintype α] [DecidableEq α] |
| attribute [local instance] Classical.propDecidable |
|
|
| def binT (σ : ι → α) (a : α) : ℝ := sqrt (count σ a:ℝ)/sqrt (Fintype.card ι:ℝ) |
|
|
| omit [DecidableEq ι] in |
| lemma sum_count (σ : ι → α) : (∑ a, (count σ a:ℝ))=(Fintype.card ι:ℝ) := by |
| have hh := sum_fibers σ (fun _ => (1:ℝ)) |
| simpa only [sum,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one,count] using hh |
|
|
| omit [DecidableEq ι] [Fintype α] in |
| lemma binT_nonneg (σ : ι → α) (a : α) : 0≤binT σ a := by unfold binT; positivity |
|
|
| omit [DecidableEq ι] [Fintype α] in |
| lemma binT_sq (σ : ι → α) (a : α) : binT σ a^2=(count σ a:ℝ)/(Fintype.card ι:ℝ) := by |
| simp only [binT,div_pow,sq_sqrt (Nat.cast_nonneg _)] |
|
|
| omit [DecidableEq ι] in |
| lemma binT_unit [Nonempty ι] (σ : ι → α) : (∑ a, binT σ a^2)=1 := by |
| simp only [binT_sq,←Finset.sum_div,sum_count] |
| exact div_self (Nat.cast_ne_zero.mpr Fintype.card_ne_zero) |
|
|
| omit [DecidableEq ι] [Fintype α] in |
| lemma sqrt_count_mul_div {σ : ι → α} (hcount : ∀ a, 0<count σ a) (B : α → ℝ) (a : α) : |
| (count σ a:ℝ)*(B a/sqrt (count σ a:ℝ))=sqrt (count σ a:ℝ)*B a := by |
| have hc : sqrt (count σ a:ℝ)≠0 := (sqrt_pos.mpr (Nat.cast_pos.mpr (hcount a))).ne' |
| field_simp |
| rw [sq_sqrt (Nat.cast_nonneg _)]; ring |
|
|
| omit [DecidableEq ι] in |
| lemma vectorOf_sum {σ : ι → α} (hcount : ∀ a, 0<count σ a) |
| (z : Parameters α) (u : Directions σ) (a : α) : |
| sum σ (vectorOf z u) a=sqrt (count σ a:ℝ)*paramB z a := by |
| unfold vectorOf |
| rw [sum_add_bin,sum_const_bin σ (fun a => paramB z a/sqrt (count σ a:ℝ)), |
| zeroSum_scale_bin (paramR z) (directions_zeroSum u) a,add_zero] |
| exact sqrt_count_mul_div hcount (paramB z) a |
|
|
| omit [DecidableEq ι] in |
| lemma vectorOf_sq_sum {σ : ι → α} (hcount : ∀ a, 0<count σ a) |
| (z : Parameters α) (u : Directions σ) (a : α) : |
| sum σ (fun i => vectorOf z u i^2) a=(paramB z a)^2+(paramR z a)^2 := by |
| simp only [vectorOf] |
| rw [bin_orthogonal_decomposition (zeroSum_scale_bin (paramR z) (directions_zeroSum u)) |
| (fun a => paramB z a/sqrt (count σ a:ℝ)) a, |
| overlap_scale_bin,directions_overlap,mul_one,div_pow,sq_sqrt (Nat.cast_nonneg _)] |
| congr 1 |
| exact mul_div_cancel₀ _ (Nat.cast_ne_zero.mpr (hcount a).ne') |
|
|
| def moment (t B f : α → ℝ) : ℝ := ∑ a, t a*B a*f a |
|
|
| omit [DecidableEq ι] in |
| lemma vectorOf_moment {σ : ι → α} (hcount : ∀ a, 0<count σ a) |
| (z : Parameters α) (u : Directions σ) (f : α → ℝ) : |
| (∑ i, vectorOf z u i*f (σ i))/sqrt (Fintype.card ι:ℝ)= |
| moment (binT σ) (paramB z) f := by |
| rw [sum_weighted,Finset.sum_div] |
| simp only [vectorOf_sum hcount, moment,binT] |
| apply Finset.sum_congr rfl |
| intro a _ |
| ring |
|
|
| def finiteRadicalSq (t v : α → ℝ) (z : Parameters α) (a : α) : ℝ := |
| (∑ d, (v a+v d)^2*((paramB z d)^2+(paramR z d)^2))- |
| (moment t (paramB z) (fun d => v a+v d))^2 |
|
|
| def finiteGaussian (β : ℝ) (t v : α → ℝ) (z : Parameters α) : ℝ := |
| β*∑ a, t a*paramR z a*sqrt (finiteRadicalSq t v z a) |
|
|
| omit [DecidableEq ι] in |
| lemma binVariance_normalized [Nonempty ι] {σ : ι → α} (hcount : ∀ a, 0<count σ a) |
| (v B : α → ℝ) (a : α) : |
| binVariance σ (crossCoeff σ v (fun d => B d/sqrt (count σ d:ℝ))) a= |
| (∑ d, (v a+v d)^2*B d^2)-(moment (binT σ) B (fun d => v a+v d))^2 := by |
| let f : ι → ℝ := fun i => (v a+v (σ i))*(B (σ i)/sqrt (count σ (σ i):ℝ)) |
| have he : binVariance σ (crossCoeff σ v (fun d => B d/sqrt (count σ d:ℝ))) a = |
| ∑ i, centered f i^2 := rfl |
| rw [he,centered_sq] |
| have hsum : (∑ i, f i)=sqrt (Fintype.card ι:ℝ)*moment (binT σ) B (fun d => v a+v d) := by |
| rw [←sum_fibers σ] |
| have hf (d : α) : sum σ f d=sqrt (count σ d:ℝ)*B d*(v a+v d) := by |
| change sum σ (fun i => (fun d => (v a+v d)*(B d/sqrt (count σ d:ℝ))) (σ i)) d=_ |
| rw [sum_const_bin σ (fun d => (v a+v d)*(B d/sqrt (count σ d:ℝ)))] |
| rw [show (count σ d:ℝ)*((v a+v d)*(B d/sqrt (count σ d:ℝ)))= |
| ((count σ d:ℝ)*(B d/sqrt (count σ d:ℝ)))*(v a+v d) by ring, |
| sqrt_count_mul_div hcount] |
| simp only [hf,moment,Finset.mul_sum,binT] |
| apply Finset.sum_congr rfl |
| intro d _ |
| have hn : sqrt (Fintype.card ι:ℝ)≠0 := (sqrt_pos.mpr (Nat.cast_pos.mpr Fintype.card_pos)).ne' |
| field_simp |
| have hsq : (∑ i, f i^2)=∑ d, (v a+v d)^2*B d^2 := by |
| rw [←sum_fibers σ] |
| apply Finset.sum_congr rfl |
| intro d _ |
| change sum σ (fun i => (fun d => ((v a+v d)*(B d/sqrt (count σ d:ℝ)))^2) (σ i)) d=_ |
| rw [sum_const_bin σ (fun d => ((v a+v d)*(B d/sqrt (count σ d:ℝ)))^2),mul_pow,div_pow,sq_sqrt (Nat.cast_nonneg _)] |
| have hc : (count σ d:ℝ)≠0 := Nat.cast_ne_zero.mpr (hcount d).ne' |
| field_simp |
| rw [hsq,average,hsum] |
| have hn : (Fintype.card ι:ℝ)≠0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| rw [div_pow,mul_pow,sq_sqrt (Nat.cast_nonneg _)] |
| field_simp |
|
|
| omit [DecidableEq ι] in |
| lemma radicalSq_normalized [Nonempty ι] {σ : ι → α} (hcount : ∀ a, 0<count σ a) |
| (v : α → ℝ) (z : Parameters α) (a : α) : |
| radicalSq σ (crossCoeff σ v (fun d => paramB z d/sqrt (count σ d:ℝ))) |
| (fun a d => v a+v d) (fun d => (paramR z d)^2) a= |
| finiteRadicalSq (binT σ) v z a := by |
| rw [radicalSq,binVariance_normalized hcount] |
| simp only [finiteRadicalSq,mul_add,Finset.sum_add_distrib] |
| ring |
|
|
| theorem normalized_projected_tail {n : ℕ} (hn : 0<n) {σ : Fin n → α} |
| (hcount : ∀ a, 2≤count σ a) (v : α → ℝ) (hv : ∀ a, |v a|≤1) |
| {β : ℝ} (hβ : 0<β) (z : Parameters α) {a : ℝ} (ha : 0≤a) : |
| (standardArrayLaw (Fin n × Fin n)).real {g | |
| finiteGaussian β (binT σ) v z+a≤ |
| projectedSup β (fun i => v (σ i)) (vectorOf z : Directions σ → Fin n → ℝ) g}≤ |
| exp (-a^2*(n:ℝ)/(π^2*β^2)) := by |
| have : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn |
| have : Nonempty (Directions σ) := directions_nonempty hcount |
| have hc : ∀ a, 0<count σ a := fun a => lt_of_lt_of_le (by norm_num) (hcount a) |
| have hh := projectedSup_tail hn σ v (fun a => paramB z a/sqrt (count σ a:ℝ)) hv hβ |
| (scaled_directions_continuous z) |
| (fun u => zeroSum_scale_bin (paramR z) (directions_zeroSum u)) |
| (show ∀ u : Directions σ, overlap σ (fun i => paramR z (σ i)*u.val i) |
| (fun i => paramR z (σ i)*u.val i)=(fun a => (paramR z a)^2) from by |
| intro u |
| ext a |
| rw [overlap_scale_bin,directions_overlap,mul_one]) |
| (fun u => vectorOf_unit hc z u) ha |
| have he : (∑ d, β/sqrt n*sqrt (radicalSq σ |
| (crossCoeff σ v (fun a => paramB z a/sqrt (count σ a:ℝ))) |
| (fun a d => v a+v d) (fun a => (paramR z a)^2) d)* |
| sqrt ((paramR z d)^2)*sqrt ((Finset.univ.filter (fun i => σ i=d)).card : ℝ))= |
| finiteGaussian β (binT σ) v z := by |
| simp_rw [radicalSq_normalized hc,sqrt_sq (paramR_nonneg z _)] |
| unfold finiteGaussian binT |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro d _ |
| simp only [count,Fintype.card_fin] |
| ring |
| rw [he] at hh |
| exact hh |
|
|
| def finiteA (β : ℝ) (t h : α → ℝ) (z : Parameters α) : ℝ := |
| moment t (paramB z) h*moment t (paramB z) (fun a => v (h a))+ |
| moment t (paramB z) (fun _ => 1)*moment t (paramB z) (fun a => h a*v (h a))- |
| β^2*moment t (paramB z) (fun _ => 1)*moment t (paramB z) (fun a => v (h a))+ |
| 2*β^2*moment t (paramB z) (fun _ => 1)*moment t (paramB z) (fun a => m (h a)*v (h a))+ |
| β^2*(∑ a, ∑ d, t d^2*(v (h d)^2/(w (h a)+w (h d)))* |
| ((paramB z a)^2+(paramR z a)^2))+ |
| β^2*(∑ a, ∑ d, t a*paramB z a*kernel (h a) (h d)*(t d*paramB z d)) |
|
|
| def finiteV (β : ℝ) (t h : α → ℝ) (z : Parameters α) : ℝ := |
| finiteA β t h z+finiteGaussian β t (fun a => v (h a)) z |
|
|
| omit [DecidableEq α] |
|
|
| lemma continuous_paramB (a : α) : Continuous (fun z : Parameters α => paramB z a) := |
| (EuclideanSpace.proj (Sum.inl a : α ⊕ α)).continuous.comp continuous_subtype_val |
| lemma continuous_paramR (a : α) : Continuous (fun z : Parameters α => paramR z a) := |
| (EuclideanSpace.proj (Sum.inr a : α ⊕ α)).continuous.comp continuous_subtype_val |
|
|
| lemma continuous_moment : Continuous (fun q : (α → ℝ) × (α → ℝ) × (α → ℝ) => |
| moment q.1 q.2.1 q.2.2) := by |
| unfold moment |
| fun_prop |
|
|
| lemma continuous_finiteRadicalSq (v : α → ℝ) (a : α) : |
| Continuous (fun q : (α → ℝ) × Parameters α => finiteRadicalSq q.1 v q.2 a) := by |
| unfold finiteRadicalSq moment |
| have hb (d : α) := (continuous_paramB d).comp (continuous_snd (X := α → ℝ)) |
| have hr (d : α) := (continuous_paramR d).comp (continuous_snd (X := α → ℝ)) |
| fun_prop |
|
|
| lemma continuous_finiteGaussian (β : ℝ) (v : α → ℝ) : |
| Continuous (fun q : (α → ℝ) × Parameters α => finiteGaussian β q.1 v q.2) := by |
| unfold finiteGaussian |
| apply Continuous.const_mul |
| apply continuous_finsetSum |
| intro a _ |
| exact (((continuous_apply a).comp continuous_fst).mul |
| ((continuous_paramR a).comp continuous_snd)).mul |
| (continuous_sqrt.comp (continuous_finiteRadicalSq v a)) |
|
|
| lemma continuous_finiteA (β : ℝ) (h : α → ℝ) : |
| Continuous (fun q : (α → ℝ) × Parameters α => finiteA β q.1 h q.2) := by |
| unfold finiteA moment |
| have hb (a : α) := (continuous_paramB a).comp (continuous_snd (X := α → ℝ)) |
| have hr (a : α) := (continuous_paramR a).comp (continuous_snd (X := α → ℝ)) |
| fun_prop |
|
|
| lemma continuous_finiteV (β : ℝ) (h : α → ℝ) : |
| Continuous (fun q : (α → ℝ) × Parameters α => finiteV β q.1 h q.2) := |
| (continuous_finiteA β h).add (continuous_finiteGaussian β (fun a => v (h a))) |
|
|
| end SKRatio.Bins |
|
|
| end |
|
|
| end OAI |
|
|
