| import OAI.Probability.GaussianPropeller.SoftMinimum |
|
|
| namespace OAI |
|
|
| universe uX uι uκ |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
| open scoped RealInnerProductSpace |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal RealInnerProductSpace |
| open Filter |
| open scoped Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
| open scoped RealInnerProductSpace |
| open Set Filter |
| open scoped Topology RealInnerProductSpace |
| open scoped NNReal |
| open Set Filter |
| open scoped Topology RealInnerProductSpace NNReal |
|
|
| namespace GaussianPropeller.EhrhardCone |
| open GaussianPropeller.Quantile GaussianPropeller.EhrhardSmooth |
| variable {ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ] [Nonempty κ] |
| local notation "E" => EuclideanSpace ℝ ι |
| local notation "γ" => stdGaussian E |
|
|
| def cone (v : κ → E) : Set E := {y | ∀ i, 0 ≤ ⟪v i,y⟫} |
| def shifted (v : κ → E) (x : E) : Set E := {y | x+y ∈ cone v} |
| noncomputable def prob (v : κ → E) (x : E) : ℝ := (γ).real (shifted v x) |
|
|
| omit [Nonempty κ] in |
| lemma measurableSet_shifted [Nonempty κ] (v : κ → E) (x : E) : MeasurableSet (shifted v x) := by |
| change MeasurableSet {y : E | ∀ i, 0 ≤ ⟪v i,x+y⟫} |
| simpa only [ofPred_forall] using MeasurableSet.iInter (fun i => |
| measurableSet_le (measurable_const : Measurable (fun _ : E => (0:ℝ))) |
| (show Measurable (fun y : E => ⟪v i,x+y⟫) by fun_prop)) |
|
|
| lemma gaussian_open_pos {U : Set E} (hU : IsOpen U) (hne : U.Nonempty) : 0 < γ U := by |
| have hac : (volume : Measure E) ≪ γ := by |
| rw [Translation.stdGaussian_density] |
| exact withDensity_absolutelyContinuous' Translation.measurable_stdDensity.ennreal_ofReal.aemeasurable |
| (ae_of_all _ (fun x => ne_of_gt (ENNReal.ofReal_pos.mpr (Translation.stdDensity_pos x)))) |
| by_contra! hz |
| have heq : γ U = 0 := le_antisymm hz bot_le |
| exact ne_of_gt (hU.measure_pos volume hne) (hac heq) |
|
|
| lemma prob_mem (v : κ → E) (e : E) (he : ∀ i, 0 < ⟪v i,e⟫) (x : E) : |
| prob v x ∈ Ioo (0:ℝ) 1 := by |
| let U : Set E := {y | ∀ i, 0 < ⟪v i,x+y⟫} |
| have hU : IsOpen U := by |
| simpa only [U, ofPred_forall] using isOpen_iInter_of_finite (fun i => |
| isOpen_lt (continuous_const : Continuous (fun _ : E => (0:ℝ))) |
| (show Continuous (fun y : E => ⟪v i,x+y⟫) by fun_prop)) |
| have hn : U.Nonempty := ⟨e-x, by simpa only [U, mem_ofPred_eq, show x+(e-x)=e by abel] using he⟩ |
| have hs : U ⊆ shifted v x := fun y hy i => (hy i).le |
| have hp : 0 < (γ).real (shifted v x) := ENNReal.toReal_pos |
| (ne_of_gt ((gaussian_open_pos hU hn).trans_le (measure_mono hs))) (measure_ne_top _ _) |
| obtain ⟨i⟩ := ‹Nonempty κ› |
| have hv : v i ≠ 0 := by intro hh; simpa only [hh, inner_zero_left, lt_self_iff_false] using he i |
| let V : Set E := {y | ⟪v i,x+y⟫ < 0} |
| have hV : IsOpen V := isOpen_lt (by fun_prop) continuous_const |
| have hnV : V.Nonempty := by |
| refine ⟨-v i-x,?_⟩ |
| change ⟪v i,x+(-v i-x)⟫ < 0 |
| rw [show x+(-v i-x)= -v i by abel, inner_neg_right] |
| exact neg_neg_of_pos (real_inner_self_pos.mpr hv) |
| have hsV : V ⊆ (shifted v x)ᶜ := by |
| intro y hy hz |
| exact (not_lt_of_ge (hz i)) hy |
| have hc : 0 < (γ).real (shifted v x)ᶜ := ENNReal.toReal_pos |
| (ne_of_gt ((gaussian_open_pos hV hnV).trans_le (measure_mono hsV))) (measure_ne_top _ _) |
| rw [probReal_compl_eq_one_sub (measurableSet_shifted v x)] at hc |
| exact ⟨hp, by change (γ).real (shifted v x) < 1; linarith only [hc]⟩ |
|
|
| lemma gaussian_hyperplane_null (z : E) (hz : z ≠ 0) (c : ℝ) : |
| γ {x | ⟪z,x⟫ = c} = 0 := by |
| let L : StrongDual ℝ E := innerSL ℝ z |
| have hL : L ≠ 0 := by |
| intro heq |
| have hx : ⟪z,z⟫ = 0 := by |
| have h := DFunLike.congr_fun heq z |
| simpa [L] using h |
| exact hz (inner_self_eq_zero.mp hx) |
| have hv : (variance L γ).toNNReal ≠ 0 := by |
| rw [variance_dual_stdGaussian] |
| exact ne_of_gt (Real.toNNReal_pos.mpr (sq_pos_of_pos (norm_pos_iff.mpr hL))) |
| have instN : NullSingletonClass (gaussianReal (∫ x, L x ∂γ) (variance L γ).toNNReal) := |
| nullSingletonClass_gaussianReal hv |
| have h := congrArg (fun μ : Measure ℝ => μ {c}) (IsGaussian.map_eq_gaussianReal (μ := γ) L) |
| rw [Measure.map_apply L.continuous.measurable (measurableSet_singleton c), measure_singleton] at h |
| exact h |
|
|
| omit [Nonempty κ] in |
| lemma ae_no_boundary [Nonempty κ] (v : κ → E) (hv : ∀ i, v i ≠ 0) (x : E) : |
| ∀ᵐ y ∂γ, ∀ i, ⟪v i,x+y⟫ ≠ 0 := by |
| apply ae_all_iff.mpr |
| intro i |
| have hh := gaussian_hyperplane_null (v i) (hv i) (-⟪v i,x⟫) |
| have hh' : ∀ᵐ y ∂γ, ⟪v i,y⟫ ≠ -⟪v i,x⟫ := by |
| rw [ae_iff] |
| simpa only [not_not] using hh |
| filter_upwards [hh'] with y hy |
| rw [inner_add_right] |
| intro hzero |
| apply hy |
| linarith only [hzero] |
|
|
| lemma cdf_softmin_limit (v : κ → E) (y : E) (hy : ∀ i, ⟪v i,y⟫ ≠ 0) : |
| Tendsto (fun n : ℕ => Φ (SoftMin.value (fun i => (n:ℝ) • v i) y)) atTop |
| (𝓝 ((cone v).indicator (fun _ => (1:ℝ)) y)) := by |
| classical |
| by_cases hmem : y ∈ cone v |
| · rw [Set.indicator_of_mem hmem] |
| exact tendsto_Φ_atTop.comp (SoftMin.value_tendsto_atTop v y |
| (fun i => lt_of_le_of_ne (hmem i) (Ne.symm (hy i)))) |
| · rw [Set.indicator_of_notMem hmem] |
| simp only [cone, mem_ofPred_eq, not_forall, not_le] at hmem |
| obtain ⟨i,hi⟩ := hmem |
| exact tendsto_Φ_atBot.comp (SoftMin.value_tendsto_atBot v y i hi) |
|
|
| lemma prob_limit (v : κ → E) (hv : ∀ i, v i ≠ 0) (x : E) : |
| Tendsto (fun n : ℕ => HeatConvolution.heat |
| (fun y => Φ (SoftMin.value (fun i => (n:ℝ) • v i) y)) 1 x) atTop (𝓝 (prob v x)) := by |
| classical |
| have hlim := tendsto_integral_of_dominated_convergence (μ := γ) (bound := fun _ : E => (1:ℝ)) |
| (F := fun n : ℕ => fun y : E => Φ (SoftMin.value (fun i => (n:ℝ) • v i) (x+y))) |
| (f := fun y : E => (cone v).indicator (fun _ => (1:ℝ)) (x+y)) ?_ (integrable_const _) ?_ ?_ |
| · convert hlim using 1 |
| · funext n; simp only [HeatConvolution.heat, Real.sqrt_one, one_smul] |
| · rw [show (fun y : E => (cone v).indicator (fun _ => (1:ℝ)) (x+y)) = |
| (shifted v x).indicator (fun _ => (1:ℝ)) by |
| funext y; rfl] |
| rw [integral_indicator (measurableSet_shifted v x), setIntegral_const] |
| simp only [prob, smul_eq_mul, mul_one] |
| · intro n |
| have hc : Continuous (SoftMin.value (fun i => (n:ℝ) • v i)) := |
| continuous_iff_continuousAt.mpr (fun y => (SoftMin.value_fderiv _ y).continuousAt) |
| exact (continuous_Φ.comp (hc.comp (continuous_const.add continuous_id))).aestronglyMeasurable |
| · intro n |
| exact ae_of_all _ (fun y => SmoothCDF.cdf_bound _ _) |
| · filter_upwards [ae_no_boundary v hv x] with y hy |
| exact cdf_softmin_limit v (x+y) hy |
|
|
| lemma quantile_prob_concave (v : κ → E) (e : E) (he : ∀ i, 0 < ⟪v i,e⟫) : |
| ConcaveOn ℝ univ (fun x => q (prob v x)) := by |
| have hv : ∀ i, v i ≠ 0 := by |
| intro i hi; simpa only [hi, inner_zero_left, lt_self_iff_false] using he i |
| have hc (n : ℕ) : ConcaveOn ℝ univ (qheat (SoftMin.value (fun i => (n:ℝ) • v i)) 1) := by |
| let w : κ → E := fun i => (n:ℝ) • v i |
| let L : ℝ≥0 := ∑ i, ‖w i‖₊ |
| have hL (i : κ) : ‖w i‖ ≤ L := by |
| exact_mod_cast (Finset.single_le_sum (f := fun j => ‖w j‖₊) (fun j _ => bot_le) (Finset.mem_univ i)) |
| exact qheat_concave (SoftMin.value_fderiv w) (SoftMin.grad_fderiv w) |
| (SoftMin.continuous_hess w) (SoftMin.grad_bound w hL) |
| (SoftMin.hess_bound w L.coe_nonneg hL) (SoftMin.value_concave w) (by norm_num) |
| have ht (x : E) : Tendsto (fun n : ℕ => qheat (SoftMin.value (fun i => (n:ℝ) • v i)) 1 x) |
| atTop (𝓝 (q (prob v x))) := (continuousAt_q (prob_mem v e he x)).tendsto.comp (prob_limit v hv x) |
| refine ⟨convex_univ,?_⟩ |
| intro x _ y _ a b ha hb hab |
| exact le_of_tendsto_of_tendsto ((ht x).const_smul a |>.add ((ht y).const_smul b)) |
| (ht (a•x+b•y)) (Filter.Eventually.of_forall (fun n => (hc n).2 (mem_univ x) (mem_univ y) ha hb hab)) |
|
|
| end GaussianPropeller.EhrhardCone |
|
|
| namespace GaussianPropeller.Quantile |
|
|
| lemma Φ_eq_gaussianReal (a : ℝ) : Φ a = (gaussianReal 0 1).real (Iic a) := by |
| rw [measureReal_def, gaussianReal_apply_eq_integral 0 (by norm_num : (1:NNReal) ≠ 0), |
| ENNReal.toReal_ofReal (integral_nonneg (fun x => gaussianPDFReal_nonneg 0 1 x))] |
| unfold Φ |
| congr 1 |
| funext x |
| simp only [gaussianPDFReal, NNReal.coe_one, mul_one, sub_zero, φ] |
|
|
| lemma integrable_mul_φ : Integrable (fun x : ℝ => x*φ x) := by |
| have hi := (integrable_mul_exp_neg_mul_sq (by norm_num : (0:ℝ) < 1/2)).const_mul |
| ((Real.sqrt (2*Real.pi))⁻¹) |
| convert hi using 1 |
| funext x |
| simp only [φ] |
| rw [show -x^2/2 = -(1/2:ℝ)*x^2 by ring] |
| ring |
|
|
| lemma tendsto_φ_atBot : Tendsto φ atBot (𝓝 0) := by |
| have htop : Tendsto (fun x : ℝ => Real.exp (-(1/2:ℝ)*x^2)) atTop (𝓝 0) := |
| Real.tendsto_exp_atBot.comp ((tendsto_pow_atTop two_ne_zero).const_mul_atTop_of_neg (by norm_num : (-(1/2:ℝ)) < 0)) |
| have ht : Tendsto (fun x : ℝ => Real.exp (-(1/2:ℝ)*x^2)) atBot (𝓝 0) := by |
| simpa only [Function.comp_def, neg_sq] using htop.comp tendsto_neg_atBot_atTop |
| convert ht.const_mul ((Real.sqrt (2*Real.pi))⁻¹) using 1 |
| · funext x; unfold φ; congr 2; ring |
| · simp only [mul_zero] |
|
|
| lemma left_first_moment (a : ℝ) : ∫ x in Iic a, x*φ x = -φ a := by |
| have hd (x : ℝ) : HasDerivAt (fun x => -φ x) (x*φ x) x := by |
| convert (hasDerivAt_φ x).neg using 1 |
| ring |
| simpa only [neg_zero, sub_zero, Pi.neg_apply] using integral_Iic_of_hasDerivAt_of_tendsto |
| (continuous_φ.neg.continuousWithinAt) (fun x _ => hd x) |
| integrable_mul_φ.integrableOn (tendsto_φ_atBot.neg) |
|
|
| lemma integral_positive_cdf (a : ℝ) : |
| ∫ x : ℝ, max (a-x) 0 ∂gaussianReal 0 1 = a*Φ a+φ a := by |
| rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1:NNReal) ≠ 0)] |
| simp only [show ∀ x : ℝ, gaussianPDFReal 0 1 x = φ x by |
| intro x; simp only [gaussianPDFReal, φ, NNReal.coe_one, mul_one, sub_zero], smul_eq_mul] |
| have heq : ∫ x : ℝ, φ x * max (a-x) 0 = ∫ x in Iic a, (a-x)*φ x := by |
| rw [← setIntegral_eq_integral_of_forall_compl_eq_zero (s := Iic a)] |
| · exact setIntegral_congr_fun measurableSet_Iic (fun x hx => by |
| rw [max_eq_left (sub_nonneg.mpr hx)]; ring) |
| · intro x hx |
| simp only [mem_Iic, not_le] at hx |
| rw [max_eq_right (sub_nonpos.mpr hx.le), mul_zero] |
| rw [heq] |
| simp_rw [sub_mul] |
| rw [integral_sub (integrable_φ.const_mul a).integrableOn integrable_mul_φ.integrableOn, |
| integral_const_mul, left_first_moment] |
| simp only [Φ, sub_neg_eq_add] |
|
|
| lemma tail_integrable {X : Type uX} [MeasurableSpace X] {μ : Measure X} {f : X → ℝ} |
| (hi : Integrable f μ) (hn : 0 ≤ᵐ[μ] f) : |
| IntegrableOn (fun t : ℝ => μ.real {x | t ≤ f x}) (Ioi 0) := by |
| apply integrable_toReal_of_lintegral_ne_top |
| · exact (show Antitone (fun t : ℝ => μ {x | t ≤ f x}) from |
| fun _ _ hst => measure_mono (fun _ h => hst.trans h)).measurable.aemeasurable |
| · rw [← lintegral_eq_lintegral_meas_le μ hn hi.aemeasurable] |
| exact hi.lintegral_lt_top.ne |
|
|
| lemma integral_cdf_tail (a c : ℝ) (hc : 0 < c) : |
| IntegrableOn (fun u : ℝ => Φ (a-c*u)) (Ioi 0) ∧ |
| (∫ u in Ioi (0:ℝ), Φ (a-c*u)) = (a*Φ a+φ a)/c := by |
| let f : ℝ → ℝ := fun x => max (a-x) 0 / c |
| have hf : Integrable f (gaussianReal 0 1) := by |
| exact (((integrable_const a).sub (IsGaussian.integrable_id (μ := gaussianReal 0 1))).sup |
| (integrable_const 0)).div_const c |
| have hn : 0 ≤ᵐ[gaussianReal 0 1] f := ae_of_all _ (fun x => div_nonneg (le_max_right _ _) hc.le) |
| have heq (u : ℝ) (hu : 0 < u) : (gaussianReal 0 1).real {x | u ≤ f x} = Φ (a-c*u) := by |
| rw [Φ_eq_gaussianReal] |
| congr 1 |
| ext x |
| simp only [mem_ofPred_eq, mem_Iic, f, le_div_iff₀ hc, le_max_iff] |
| have hh : ¬ u*c ≤ 0 := not_le.mpr (mul_pos hu hc) |
| simp only [hh, or_false] |
| constructor <;> intro h <;> nlinarith only [h] |
| have hi : IntegrableOn (fun u : ℝ => Φ (a-c*u)) (Ioi 0) := |
| (tail_integrable hf hn).congr (ae_restrict_of_forall_mem measurableSet_Ioi (fun u hu => heq u hu)) |
| refine ⟨hi,?_⟩ |
| rw [← setIntegral_congr_fun measurableSet_Ioi heq, |
| ← hf.integral_eq_integral_meas_le hn] |
| rw [show f = fun x => max (a-x) 0 / c from rfl, integral_div, integral_positive_cdf] |
|
|
| end GaussianPropeller.Quantile |
|
|
| namespace GaussianPropeller.EhrhardCone |
| open GaussianPropeller.Quantile |
| variable {ι : Type uι} {κ : Type uκ} [Fintype ι] [Fintype κ] [Nonempty κ] |
| local notation "E" => EuclideanSpace ℝ ι |
| local notation "γ" => stdGaussian E |
|
|
| lemma measurableSet_cone (v : κ → E) : MeasurableSet (cone v) := by |
| simpa only [shifted, zero_add, ofPred_mem_eq] using measurableSet_shifted v 0 |
|
|
| omit [Fintype κ] [Nonempty κ] in |
| lemma prob_zero [Fintype κ] [Nonempty κ] (v : κ → E) : prob v 0 = (γ).real (cone v) := by |
| simp only [prob, shifted, zero_add, ofPred_mem_eq] |
|
|
| lemma prob_line_deriv (v : κ → E) (h : E) : |
| HasDerivAt (fun u : ℝ => prob v (-u • h)) |
| (-⟪h, ∫ x in cone v, x ∂γ⟫) 0 := by |
| have heq (u : ℝ) : prob v (-u • h) = |
| ((γ).map (fun x => x-u•h)).real (cone v) := by |
| unfold prob |
| rw [measureReal_def, measureReal_def, Measure.map_apply (by fun_prop) (measurableSet_cone v)] |
| apply congrArg ENNReal.toReal |
| apply congrArg γ |
| ext x |
| change (-u•h+x ∈ cone v) ↔ (x-u•h ∈ cone v) |
| rw [show -u•h+x = x-u•h by simp only [neg_smul, sub_eq_add_neg, add_comm]] |
| simp_rw [heq] |
| exact Translation.translation_derivative h (cone v) (measurableSet_cone v) |
|
|
| lemma prob_tangent (v : κ → E) (e : E) (he : ∀ i, 0 < ⟪v i,e⟫) (h : E) (u : ℝ) : |
| prob v (-u • h) ≤ Φ (q (prob v 0)- |
| (⟪h, ∫ x in cone v, x ∂γ⟫ / φ (q (prob v 0)))*u) := by |
| have hc : ConcaveOn ℝ univ (fun t : ℝ => q (prob v (-t•h))) := by |
| refine ⟨convex_univ,?_⟩ |
| intro s _ t _ a b ha hb hab |
| have hh := (quantile_prob_concave v e he).2 (mem_univ (-s•h)) (mem_univ (-t•h)) ha hb hab |
| change a * q (prob v (-s•h))+b * q (prob v (-t•h)) ≤ q (prob v (-(a*s+b*t)•h)) |
| rw [show -(a*s+b*t)•h = a•(-s•h)+b•(-t•h) by module] |
| exact hh |
| have hd := (hasDerivAt_q (prob_mem v e he (-(0:ℝ)•h))).comp (0:ℝ) (prob_line_deriv v h) |
| simp only [neg_zero, zero_smul] at hd |
| have ht := concave_le_tangent hc (mem_univ 0) (mem_univ u) hd |
| simp only [neg_zero, zero_smul, sub_zero] at ht |
| have ht' : q (prob v (-u•h)) ≤ q (prob v 0)- |
| (⟪h, ∫ x in cone v, x ∂γ⟫ / φ (q (prob v 0)))*u := by |
| calc |
| _ ≤ q (prob v 0) + (φ (q (prob v 0)))⁻¹ * (-⟪h, ∫ x in cone v, x ∂γ⟫) * u := ht |
| _ = _ := by ring |
| have hh := strictMono_Φ.monotone ht' |
| rwa [Φ_q (prob_mem v e he (-u•h))] at hh |
|
|
| lemma integral_prob_upper (v : κ → E) (e : E) (he : ∀ i, 0 < ⟪v i,e⟫) |
| (h : E) (hβ : 0 < ⟪h, ∫ x in cone v, x ∂γ⟫) : |
| (∫ u in Ioi (0:ℝ), prob v (-u • h)) ≤ |
| φ (q (prob v 0)) * (q (prob v 0)*prob v 0+φ (q (prob v 0))) / |
| ⟪h, ∫ x in cone v, x ∂γ⟫ := by |
| let a := q (prob v 0) |
| let c := ⟪h, ∫ x in cone v, x ∂γ⟫ / φ a |
| have hc : 0 < c := div_pos hβ (φ_pos a) |
| calc |
| _ ≤ ∫ u in Ioi (0:ℝ), Φ (a-c*u) := integral_mono_of_nonneg |
| (ae_of_all (volume.restrict (Ioi (0:ℝ))) (fun u => measureReal_nonneg)) |
| (integral_cdf_tail a c hc).1 |
| (ae_of_all _ (prob_tangent v e he h)) |
| _ = (a*Φ a+φ a)/c := (integral_cdf_tail a c hc).2 |
| _ = _ := by |
| dsimp [a,c] |
| rw [Φ_q (prob_mem v e he 0)] |
| simp only [div_eq_mul_inv, mul_inv_rev, inv_inv] |
| ring |
|
|
| end GaussianPropeller.EhrhardCone |
|
|
| end OAI |
|
|
