| import Mathlib |
| import OAI.Probability.SKRatio.Matrices.GaussianAffineGaussianSup |
| import OAI.Probability.SKRatio.Matrices.GaussianProductUpperTailLipschitz |
| import OAI.Probability.SKRatio.Quantization.BinGaussian |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Matrix NNReal |
| open MeasureTheory ProbabilityTheory Filter Real |
| namespace SKRatio.Bins |
| variable {ι α : Type*} [Fintype ι] [Nonempty ι] [Fintype α] [DecidableEq α] |
| attribute [local instance] Classical.propDecidable |
|
|
| def average (p : ι → ℝ) : ℝ := (∑ i, p i)/(Fintype.card ι:ℝ) |
| def centered (p : ι → ℝ) : ι → ℝ := fun i => p i-average p |
|
|
| omit [Nonempty ι] in |
| lemma average_add (p q : ι → ℝ) : average (fun i => p i+q i)=average p+average q := by |
| simp only [average,Finset.sum_add_distrib,add_div] |
| omit [Nonempty ι] in |
| lemma average_mul (p : ι → ℝ) (c : ℝ) : average (fun i => c*p i)=c*average p := by |
| simp only [average,←Finset.mul_sum,mul_div_assoc] |
| lemma average_const (c : ℝ) : average (fun _ : ι => c)=c := by |
| have hn : (Fintype.card ι:ℝ)≠0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| simp [average,mul_div_cancel_left₀ _ hn] |
| omit [Nonempty ι] in |
| lemma average_zeroSum {σ : ι → α} {u : ι → ℝ} (hu : ZeroSum σ u) : average u=0 := by |
| have h := zero_sum_weighted hu (fun _ => 1) |
| simpa only [mul_one, average, zero_div] using congrArg (fun z : ℝ => z/(Fintype.card ι:ℝ)) h |
|
|
| lemma centered_sq (p : ι → ℝ) : (∑ i, centered p i^2)=(∑ i, p i^2)- |
| (Fintype.card ι:ℝ)*average p^2 := by |
| have hn : (Fintype.card ι:ℝ)≠0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero |
| have hsum : (∑ i, p i)=(Fintype.card ι:ℝ)*average p := by |
| dsimp [average] |
| field_simp |
| simp only [centered,sub_sq,Finset.sum_add_distrib,Finset.sum_sub_distrib, |
| ←Finset.sum_mul,←Finset.mul_sum,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| rw [hsum] |
| ring |
| lemma centered_sq_le (p : ι → ℝ) : (∑ i, centered p i^2)≤∑ i, p i^2 := by |
| rw [centered_sq] |
| exact sub_le_self _ (mul_nonneg (Nat.cast_nonneg _) (sq_nonneg _)) |
|
|
| def fullCoeff (v p : ι → ℝ) (δ : ℝ) : Coordinates ι → ℝ |
| | .inl (i,j) => δ/sqrt 2*(centered p i*centered (fun k => v k*p k) j+ |
| centered p j*centered (fun k => v k*p k) i) |
| | .inr _ => 0 |
|
|
| omit [Nonempty ι] in |
| lemma continuous_centered : Continuous (centered (ι := ι)) := by |
| apply continuous_pi |
| intro i |
| exact (continuous_apply i).sub ((continuous_finsetSum _ (fun j _ => continuous_apply j)).div_const _) |
|
|
| omit [Nonempty ι] in |
| lemma continuous_fullCoeff (v : ι → ℝ) (δ : ℝ) : Continuous (fun p => fullCoeff v p δ) := by |
| apply continuous_pi |
| intro k |
| cases k with |
| | inr i => exact continuous_const |
| | inl ij => |
| change Continuous (fun p => δ/sqrt 2*(centered p ij.1*centered (fun k => v k*p k) ij.2+ |
| centered p ij.2*centered (fun k => v k*p k) ij.1)) |
| have hm : Continuous (fun p : ι → ℝ => fun k => v k*p k) := by fun_prop |
| exact (((continuous_apply ij.1).comp continuous_centered |>.mul |
| ((continuous_apply ij.2).comp (continuous_centered.comp hm))).add |
| (((continuous_apply ij.2).comp continuous_centered).mul |
| ((continuous_apply ij.1).comp (continuous_centered.comp hm)))).const_mul _ |
|
|
| def crossCoeff (σ : ι → α) (v b : α → ℝ) (a d : α) : ℝ := |
| (v a+v d)*b d-average (fun i => (v a+v (σ i))*b (σ i)) |
|
|
| omit [Nonempty ι] [Fintype α] [DecidableEq α] in |
| lemma crossCoeff_eq (σ : ι → α) (v b : α → ℝ) (a : α) (j : ι) : |
| crossCoeff σ v b a (σ j)=centered (fun i => v (σ i)*b (σ i)) j+ |
| v a*centered (fun i => b (σ i)) j := by |
| have he : (fun i => (v a+v (σ i))*b (σ i))= |
| (fun i => v a*b (σ i)+v (σ i)*b (σ i)) := by ext i; ring |
| rw [crossCoeff,he,average_add,average_mul] |
| dsimp only [centered] |
| ring |
|
|
| omit [Nonempty ι] in |
| lemma centered_add_zeroSum {σ : ι → α} (b : α → ℝ) {u : ι → ℝ} (hu : ZeroSum σ u) : |
| centered (fun i => b (σ i)+u i)=fun i => centered (fun k => b (σ k)) i+u i := by |
| ext i |
| simp only [centered,average_add,average_zeroSum hu,add_zero] |
| ring |
|
|
| omit [Nonempty ι] [Fintype α] in |
| lemma zeroSum_mul {σ : ι → α} (v : α → ℝ) {u : ι → ℝ} (hu : ZeroSum σ u) : |
| ZeroSum σ (fun i => v (σ i)*u i) := by |
| intro a |
| have he : sum σ (fun i => v (σ i)*u i) a = v a*sum σ u a := by |
| unfold sum |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i hi |
| change v (σ i)*u i=v a*u i |
| rw [(Finset.mem_filter.mp hi).2] |
| rw [he,hu a,mul_zero] |
|
|
| omit [Nonempty ι] in |
| lemma fullCoeff_decomposition {σ : ι → α} (v b : α → ℝ) (δ : ℝ) {u : ι → ℝ} |
| (hu : ZeroSum σ u) : |
| fullCoeff (fun i => v (σ i)) (fun i => b (σ i)+u i) δ = |
| fun k => fullCoeff (fun i => v (σ i)) (fun i => b (σ i)) δ k+ |
| residualCoeff σ (crossCoeff σ v b) (fun a d => v a+v d) δ u k := by |
| have he : (fun i => v (σ i)*(b (σ i)+u i))= |
| fun i => v (σ i)*b (σ i)+v (σ i)*u i := by ext i; ring |
| have hcen := centered_add_zeroSum (fun a => v a*b a) (zeroSum_mul v hu) |
| ext k |
| cases k with |
| | inr i => simp [fullCoeff,residualCoeff] |
| | inl ij => |
| rcases ij with ⟨i,j⟩ |
| simp only [fullCoeff,residualCoeff,centered_add_zeroSum b hu,he,hcen,residualMatrix, |
| Matrix.add_apply,Matrix.transpose_apply,linearMatrix,quadraticMatrix,crossCoeff_eq] |
| ring |
|
|
| lemma fullCoeff_sq_le {v p : ι → ℝ} (hv : ∀ i, |v i|≤1) (δ : ℝ) : |
| (∑ k, fullCoeff v p δ k^2) ≤ 2*δ^2*(∑ i, p i^2)^2 := by |
| let x := centered p |
| let y := centered (fun i => v i*p i) |
| have hx : (∑ i, x i^2)≤∑ i, p i^2 := centered_sq_le p |
| have hy : (∑ i, y i^2)≤∑ i, p i^2 := (centered_sq_le _).trans (by |
| apply Finset.sum_le_sum |
| intro i _ |
| have hvi : v i^2≤1 := (sq_le_one_iff_abs_le_one (v i)).mpr (hv i) |
| simpa only [mul_pow,one_mul] using mul_le_mul_of_nonneg_right hvi (sq_nonneg (p i))) |
| have hprod : (∑ i, x i^2)*(∑ j, y j^2)≤(∑ i, p i^2)^2 := by |
| rw [pow_two] |
| exact mul_le_mul hx hy (Finset.sum_nonneg (fun _ _ => sq_nonneg _)) |
| (Finset.sum_nonneg (fun _ _ => sq_nonneg _)) |
| have hsq : (∑ i, ∑ j, (x i*y j+x j*y i)^2)≤4*((∑ i, x i^2)*(∑ j, y j^2)) := by |
| calc |
| _ ≤ ∑ i, ∑ j, 2*((x i*y j)^2+(x j*y i)^2) := by |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j _ |
| nlinarith only [sq_nonneg (x i*y j-x j*y i)] |
| _ = _ := by |
| simp only [mul_add,mul_pow,Finset.sum_add_distrib,←Finset.mul_sum,←Finset.sum_mul] |
| ring |
| simp only [Fintype.sum_sum_type,fullCoeff,zero_pow (by norm_num : (2:ℕ)≠0), |
| Finset.sum_const_zero,add_zero,Fintype.sum_prod_type,mul_pow,div_pow, |
| sq_sqrt (by norm_num : (0:ℝ)≤2),←Finset.mul_sum] |
| change δ^2/2*(∑ i, ∑ j, (x i*y j+x j*y i)^2)≤_ |
| have h := mul_le_mul_of_nonneg_left (hsq.trans (mul_le_mul_of_nonneg_left hprod (by norm_num))) |
| (show 0≤δ^2/2 by positivity) |
| convert h using 1 |
| first | rfl | ring |
|
|
| lemma fullCoeff_norm_le {v p : ι → ℝ} (hv : ∀ i, |v i|≤1) {δ : ℝ} (hδ : 0≤δ) |
| (hp : ∑ i, p i^2=1) : ‖WithLp.toLp 2 (fullCoeff v p δ)‖≤ sqrt 2*δ := by |
| have h := fullCoeff_sq_le (p := p) hv δ |
| rw [hp] at h |
| have he : ‖WithLp.toLp 2 (fullCoeff v p δ)‖^2=∑ k, fullCoeff v p δ k^2 := by |
| simp only [EuclideanSpace.norm_sq_eq,Real.norm_eq_abs,sq_abs] |
| have hs := sq_sqrt (show (0:ℝ)≤2 by norm_num) |
| have hpos : 0≤ sqrt 2*δ := mul_nonneg (sqrt_nonneg _) hδ |
| apply (sq_le_sq₀ (norm_nonneg _) hpos).mp |
| rw [mul_pow,hs,he] |
| simpa only [one_pow,mul_one] using h |
|
|
| section Translation |
| variable {κ I : Type*} [Fintype κ] [TopologicalSpace I] [CompactSpace I] [Nonempty I] |
|
|
| lemma gaussianSup_add_common {a : I → κ → ℝ} (ha : Continuous a) (c : κ → ℝ) |
| (g : κ → ℝ) : |
| SKRatioGaussian.gaussianSup (fun _ : I => 0) (fun t k => c k+a t k) g = |
| (∑ k, c k*g k)+SKRatioGaussian.gaussianSup (fun _ : I => 0) a g := by |
| have hac : Continuous (fun t k => c k+a t k) := continuous_const.add ha |
| have he (t : I) : SKRatioGaussian.gaussianAffine (fun _ : I => 0) (fun t k => c k+a t k) g t = |
| (∑ k, c k*g k)+SKRatioGaussian.gaussianAffine (fun _ : I => 0) a g t := by |
| simp only [SKRatioGaussian.gaussianAffine,add_mul,Finset.sum_add_distrib,zero_add] |
| apply le_antisymm |
| · apply SKRatioGaussian.gaussianSup_le continuous_const hac |
| intro t |
| rw [he] |
| exact add_le_add le_rfl (SKRatioGaussian.gaussianAffine_le_gaussianSup continuous_const ha g t) |
| · obtain ⟨t, ht, _⟩ := SKRatioGaussian.gaussianSup_attained continuous_const ha g |
| rw [ht,←he] |
| exact SKRatioGaussian.gaussianAffine_le_gaussianSup continuous_const hac g t |
|
|
| lemma integral_gaussianSup_add_common {a : I → κ → ℝ} (ha : Continuous a) (c : κ → ℝ) : |
| (∫ g, SKRatioGaussian.gaussianSup (fun _ : I => 0) (fun t k => c k+a t k) g |
| ∂SKRatioGaussian.gaussianCoordinates κ) = |
| ∫ g, SKRatioGaussian.gaussianSup (fun _ : I => 0) a g ∂SKRatioGaussian.gaussianCoordinates κ := by |
| have hi (k : κ) : Integrable (fun g : κ → ℝ => c k*g k) (SKRatioGaussian.gaussianCoordinates κ) := |
| ((SKRatioClock.Regression.coordinate_hasLaw k).hasGaussianLaw.integrable).const_mul _ |
| simp_rw [gaussianSup_add_common ha c] |
| rw [integral_add (integrable_finsetSum _ (fun k _ => hi k)) |
| (SKRatioGaussian.integrable_gaussianSup continuous_const ha), |
| integral_finsetSum _ (fun k _ => hi k)] |
| have hm (k : κ) : (∫ g : κ → ℝ, c k*g k ∂SKRatioGaussian.gaussianCoordinates κ)=0 := by |
| rw [integral_const_mul] |
| change c k*(∫ g : κ → ℝ, g k ∂SKRatioClock.Regression.standardArrayLaw κ)=0 |
| rw [SKRatioClock.Regression.coordinate_mean,mul_zero] |
| simp only [hm,Finset.sum_const_zero,zero_add] |
| end Translation |
|
|
| variable {I : Type*} [TopologicalSpace I] [CompactSpace I] [Nonempty I] |
|
|
| theorem full_sup_expectation_le (σ : ι → α) (v b : α → ℝ) |
| {s : α → ℝ} {δ : ℝ} (hδ : 0≤δ) {u : I → ι → ℝ} |
| (huc : Continuous u) (hu0 : ∀ t, ZeroSum σ (u t)) |
| (hus : ∀ t, overlap σ (u t) (u t)=s) : |
| (∫ g, SKRatioGaussian.gaussianSup (fun _ : I => 0) |
| (fun t => fullCoeff (fun i => v (σ i)) (fun i => b (σ i)+u t i) δ) g |
| ∂SKRatioGaussian.gaussianCoordinates (Coordinates ι)) ≤ |
| ∑ a, δ*sqrt (radicalSq σ (crossCoeff σ v b) (fun a d => v a+v d) s a)* |
| sqrt (s a)*sqrt ((Finset.univ.filter (fun i => σ i=a)).card : ℝ) := by |
| have hd (t : I) := fullCoeff_decomposition v b δ (hu0 t) |
| simp_rw [hd] |
| have hi := integral_gaussianSup_add_common |
| ((continuous_residualCoeff σ (crossCoeff σ v b) (fun a d => v a+v d) δ).comp huc) |
| (fullCoeff (fun i => v (σ i)) (fun i => b (σ i)) δ) |
| exact hi.trans_le (residual_sup_expectation_le σ _ _ (fun a d => add_comm _ _) hδ huc hu0 hus) |
|
|
| theorem full_sup_lipschitz (v : ι → ℝ) (hv : ∀ i, |v i|≤1) |
| {δ : ℝ} (hδ : 0≤δ) {p : I → ι → ℝ} (hpc : Continuous p) |
| (hp : ∀ t, ∑ i, p t i^2=1) : |
| LipschitzWith ⟨sqrt 2*δ,by positivity⟩ |
| (fun g : EuclideanSpace ℝ (Coordinates ι) => |
| SKRatioGaussian.gaussianSup (fun _ : I => 0) (fun t => fullCoeff v (p t) δ) g.ofLp) := by |
| apply SKRatioGaussian.gaussianSup_lipschitz continuous_const ((continuous_fullCoeff v δ).comp hpc) |
| intro t |
| exact fullCoeff_norm_le hv hδ (hp t) |
|
|
| theorem full_sup_tail (σ : ι → α) (v b : α → ℝ) (hv : ∀ a, |v a|≤1) |
| {s : α → ℝ} {δ : ℝ} (hδ : 0<δ) {u : I → ι → ℝ} |
| (huc : Continuous u) (hu0 : ∀ t, ZeroSum σ (u t)) |
| (hus : ∀ t, overlap σ (u t) (u t)=s) |
| (hp : ∀ t, ∑ i, (b (σ i)+u t i)^2=1) {a : ℝ} (ha : 0≤a) : |
| (SKRatioGaussian.gaussianCoordinates (Coordinates ι)).real {g | |
| (∑ d, δ*sqrt (radicalSq σ (crossCoeff σ v b) (fun a d => v a+v d) s d)* |
| sqrt (s d)*sqrt ((Finset.univ.filter (fun i => σ i=d)).card : ℝ))+a ≤ |
| SKRatioGaussian.gaussianSup (fun _ : I => 0) |
| (fun t => fullCoeff (fun i => v (σ i)) (fun i => b (σ i)+u t i) δ) g} ≤ |
| exp (-a^2/(π^2*δ^2)) := by |
| let p : I → ι → ℝ := fun t i => b (σ i)+u t i |
| have hpc : Continuous p := continuous_const.add huc |
| let f := SKRatioGaussian.gaussianSup (fun _ : I => 0) |
| (fun t => fullCoeff (fun i => v (σ i)) (p t) δ) |
| have hL : (0:ℝ≥0) < ⟨sqrt 2*δ,by positivity⟩ := by |
| change (0:ℝ)<sqrt 2*δ |
| positivity |
| have hc := SKRatioGaussian.gaussianProduct_upper_tail_lipschitz |
| (full_sup_lipschitz _ (fun i => hv (σ i)) hδ.le hpc hp) hL ha |
| have he := full_sup_expectation_le σ v b hδ.le huc hu0 hus |
| have hsub : {g | (∑ d, δ*sqrt (radicalSq σ (crossCoeff σ v b) (fun a d => v a+v d) s d)* |
| sqrt (s d)*sqrt ((Finset.univ.filter (fun i => σ i=d)).card : ℝ))+a ≤ f g} ⊆ |
| {g | a ≤ f g-∫ y, f y ∂SKRatioGaussian.gaussianCoordinates (Coordinates ι)} := by |
| intro g hg |
| change _+a≤f g at hg |
| change (∫ y, f y ∂SKRatioGaussian.gaussianCoordinates (Coordinates ι))≤_ at he |
| change a≤f g-(∫ y, f y ∂SKRatioGaussian.gaussianCoordinates (Coordinates ι)) |
| linarith only [hg,he] |
| have hcont : Continuous f := SKRatioGaussian.continuous_gaussianSup continuous_const |
| ((continuous_fullCoeff _ _).comp hpc) |
| apply (measureReal_mono hsub).trans |
| convert hc using 1 <;> try rfl |
| apply congrArg Real.exp |
| change -a^2/(π^2*δ^2) = -2*a^2/(π^2*(sqrt 2*δ)^2) |
| rw [mul_pow,sq_sqrt (by norm_num : (0:ℝ)≤2)] |
| ring |
|
|
| end SKRatio.Bins |
|
|
| end |
|
|
| end OAI |
|
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