| import Mathlib |
| import OAI.Probability.SKRatio.Matrices.RandomMatrixBounded |
| import OAI.Probability.SKRatio.Gaussian.ExponentialBounds |
| import OAI.Probability.SKRatio.Matrices.GaussianSquareTails |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set Real |
| namespace SKRatio.MatrixNet |
| open SKRatioClock.Regression |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma independent_sum_upper_tail {Ω ι : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] (X : ι → Ω → ℝ) |
| (hXi : iIndepFun X μ) (hXm : ∀ i, Measurable (X i)) |
| (s : Finset ι) {t M : ℝ} (ht : 0<t) |
| (hi : ∀ i ∈ s, Integrable (fun ω => exp (t*X i ω)) μ) |
| (hmgf : ∀ i ∈ s, cgf (X i) μ t ≤ M) (hM : 0≤M) |
| {n : ℕ} (hn : 0<n) (hcard : s.card≤n) (a : ℝ) : |
| μ {ω | a ≤ (∑ i∈s, X i ω)/(n:ℝ)} ≤ ENNReal.ofReal (exp (-((t*a-M)*n))) := by |
| have hsum := hXi.integrable_exp_mul_sum hXm (s := s) hi |
| have hb := measure_ge_le_exp_cgf (μ := μ) (X := ∑ i∈s, X i) (a*n) ht.le hsum |
| rw [hXi.cgf_sum hXm hi] at hb |
| have hm : ∑ i∈s, cgf (X i) μ t ≤ (n:ℝ)*M := by |
| calc |
| _ ≤ ∑ _i∈s, M := Finset.sum_le_sum hmgf |
| _ = (s.card:ℝ)*M := by simp |
| _ ≤ (n:ℝ)*M := mul_le_mul_of_nonneg_right (by exact_mod_cast hcard) hM |
| have he : {ω | a*(n:ℝ) ≤ (∑ i∈s, X i) ω} = {ω | a ≤ (∑ i∈s, X i ω)/(n:ℝ)} := by |
| ext ω |
| simp only [Finset.sum_apply,mem_ofPred_eq,le_div_iff₀ (Nat.cast_pos.mpr hn : (0:ℝ)<n)] |
| rw [he] at hb |
| have h := ENNReal.ofReal_le_ofReal hb |
| simp only [Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] at h |
| refine h.trans (ENNReal.ofReal_le_ofReal (exp_le_exp.mpr ?_)) |
| linarith |
|
|
| lemma gaussian_squareTail_row_bound {a : ℝ} (ha : 0<a) : |
| ∃ R c : ℝ, 0<R ∧ 0<c ∧ |
| (∫ z, squareTail R z ∂gaussianReal 0 1) < a ∧ |
| ∀ {ι : Type} [Fintype ι] (s : Finset ι) {n : ℕ}, 0<n → s.card ≤ n → |
| standardArrayLaw ι {z | a ≤ (∑ i∈s, squareTail R (z i))/(n:ℝ)} ≤ |
| ENNReal.ofReal (exp (-c*n)) := by |
| have ht : Tendsto (fun R : ℕ => log (∫ z, exp (squareTail R z/4) ∂gaussianReal 0 1)) |
| atTop (𝓝 0) := by |
| simpa only [Function.comp_def,log_one] using (continuousAt_log (by norm_num : (1:ℝ)≠0)).tendsto.comp |
| (tendsto_squareTail_mgf gaussian_integrable_exp_quarter_sq) |
| have hsq : Integrable (fun z : ℝ => z^2) (gaussianReal 0 1) := by |
| simpa only [id_eq] using (IsGaussian.memLp_two_id (μ := gaussianReal 0 1)).integrable_sq |
| have hm : Tendsto (fun R : ℕ => ∫ z, squareTail R z ∂gaussianReal 0 1) atTop (𝓝 0) := by |
| have hb := tendsto_integral_of_dominated_convergence (fun z : ℝ => z^2) |
| (μ := gaussianReal 0 1) |
| (F := fun R : ℕ => squareTail R) (f := fun _ => (0:ℝ)) |
| (fun R => (measurable_squareTail R).aestronglyMeasurable) hsq |
| have h := hb (fun R => ae_of_all _ (fun z => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (squareTail_nonneg _ _)] |
| exact squareTail_le _ _)) (ae_of_all _ (fun z => by |
| apply tendsto_const_nhds.congr' |
| filter_upwards [eventually_ge_atTop ⌈|z|⌉₊] with R hR |
| have hz : |z| ≤ (R:ℝ) := (Nat.le_ceil _).trans (by exact_mod_cast hR) |
| simp [squareTail,not_lt.mpr hz])) |
| simpa using h |
| obtain ⟨R,hR,hmg,hmean⟩ := ((eventually_ge_atTop 1).and |
| ((ht.eventually (gt_mem_nhds (by positivity : (0:ℝ)<a/4))).and |
| (hm.eventually (gt_mem_nhds ha)))).exists |
| let M := log (∫ z, exp (squareTail R z/4) ∂gaussianReal 0 1) |
| have hM : 0≤M := by |
| apply log_nonneg |
| calc |
| 1 = ∫ _z : ℝ, (1:ℝ) ∂gaussianReal 0 1 := by simp |
| _ ≤ _ := integral_mono (integrable_const _) (integrable_exp_squareTail gaussian_integrable_exp_quarter_sq R) |
| (by intro z; exact one_le_exp_iff.mpr (div_nonneg (squareTail_nonneg _ _) (by norm_num))) |
| refine ⟨R,a/4-M,by exact_mod_cast hR,sub_pos.mpr hmg,hmean,?_⟩ |
| intro ι _ s n hn hs |
| have hc (i : ι) : cgf (fun z : ι → ℝ => squareTail R (z i)) (standardArrayLaw ι) (1/4) = M := by |
| unfold cgf mgf M |
| congr 1 |
| have hh := (coordinate_hasLaw i).integral_comp (f := fun z : ℝ => exp (squareTail R z/4)) (((measurable_squareTail R).div_const 4).exp.aestronglyMeasurable) |
| simpa only [one_div_mul_eq_div, Function.comp_def] using hh |
| have hi (i : ι) : Integrable (fun z : ι → ℝ => exp ((1/4)*squareTail R (z i))) (standardArrayLaw ι) := by |
| have hh := integrable_exp_squareTail gaussian_integrable_exp_quarter_sq (R:ℝ) |
| rw [←(coordinate_hasLaw i).map_eq] at hh |
| have h := (integrable_map_measure (((measurable_squareTail R).div_const 4).exp.aestronglyMeasurable) (coordinate_hasLaw i).aemeasurable).mp hh |
| simpa only [one_div_mul_eq_div, Function.comp_def] using h |
| have hb := independent_sum_upper_tail (fun i (z : ι → ℝ) => squareTail R (z i)) |
| (coordinates_independent.comp (fun _ => squareTail R) (fun _ => measurable_squareTail R)) |
| (fun i => (measurable_squareTail R).comp (measurable_pi_apply i)) s |
| (by norm_num : (0:ℝ)<1/4) (fun i _ => hi i) (fun i _ => le_of_eq (hc i)) hM hn hs a |
| simpa only [one_div_mul_eq_div,neg_mul] using hb |
|
|
| def incidentEdges {n : ℕ} (i : Fin n) : Finset (Edge n) := |
| Finset.univ.filter (fun e => e.1.1 = i ∨ e.1.2 = i) |
|
|
| lemma incidentEdges_card_le {n : ℕ} (i : Fin n) : (incidentEdges i).card ≤ n := by |
| let other (e : Edge n) : Fin n := if e.1.1 = i then e.1.2 else e.1.1 |
| have h := Finset.card_le_card_of_injOn other |
| (t := (Finset.univ : Finset (Fin n))) (s := incidentEdges i) |
| (fun _ _ => Finset.mem_univ _) (by |
| intro e he f hf hef |
| have he' : e.1.1 = i ∨ e.1.2 = i := (Finset.mem_filter.mp he).2 |
| have hf' : f.1.1 = i ∨ f.1.2 = i := (Finset.mem_filter.mp hf).2 |
| have ehe := e.2 |
| have ehf := f.2 |
| apply Subtype.ext |
| apply Prod.ext |
| all_goals dsimp [other] at hef; split_ifs at hef <;> rcases he' with he' | he' <;> |
| rcases hf' with hf' | hf' <;> simp_all <;> omega) |
| simpa using h |
|
|
| lemma sum_coupling_row {n : ℕ} (g : Disorder n) (i : Fin n) : |
| ∑ j, coupling g i j = ∑ e ∈ incidentEdges i, g e := by |
| have he := coupling_bilinear (fun j => if j=i then 1 else 0) (fun _ => 1) g |
| simp only [mul_one,ite_mul,one_mul,zero_mul,Finset.sum_ite_irrel,Finset.sum_const_zero,Finset.sum_ite_eq',Finset.mem_univ,ite_true] at he |
| rw [he,incidentEdges,Finset.sum_filter] |
| apply Finset.sum_congr rfl |
| intro e _ |
| have hh : e.1.1 ≠ e.1.2 := ne_of_lt e.2 |
| by_cases h1 : e.1.1=i <;> by_cases h2 : e.1.2=i <;> simp_all |
|
|
| lemma abs_coupling {n : ℕ} (g : Disorder n) (i j : Fin n) : |
| |coupling g i j| = coupling (fun e => |g e|) i j := by |
| unfold coupling |
| split_ifs <;> simp |
|
|
| lemma coupling_nonneg {n : ℕ} {g : Disorder n} (hg : ∀ e, 0 ≤ g e) (i j : Fin n) : |
| 0 ≤ coupling g i j := by |
| unfold coupling |
| split_ifs <;> first | exact hg _ | exact le_refl 0 |
|
|
| lemma coupling_add {n : ℕ} (g h : Disorder n) : |
| coupling (g+h) = coupling g + coupling h := by |
| funext i j |
| change coupling (g+h) i j = coupling g i j + coupling h i j |
| unfold coupling |
| split_ifs <;> simp_all |
|
|
| lemma coupling_sub {n : ℕ} (g h : Disorder n) : |
| coupling (g-h) = coupling g - coupling h := by |
| funext i j |
| change coupling (g-h) i j = coupling g i j - coupling h i j |
| unfold coupling |
| split_ifs <;> simp_all |
|
|
| lemma coupling_const_mul {n : ℕ} (c : ℝ) (g : Disorder n) : |
| coupling (fun e => c*g e) = c • coupling g := by |
| funext i j |
| change coupling (fun e => c*g e) i j = c*coupling g i j |
| unfold coupling |
| split_ifs <;> simp_all |
|
|
| lemma coupling_const_norm_le {n : ℕ} (c : ℝ) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (coupling (fun _ => c))‖ ≤ n*|c| := by |
| apply matrix_opNorm_le_schur _ (n*|c|) (by positivity) |
| all_goals |
| intro i |
| calc |
| _ ≤ ∑ _j : Fin n, |c| := Finset.sum_le_sum (fun j _ => by |
| unfold coupling; split_ifs <;> simp [abs_nonneg]) |
| _ = _ := by simp |
|
|
| lemma integrable_standard_square : Integrable (fun z : ℝ => z^2) (gaussianReal 0 1) := by |
| simpa only [id_eq] using (IsGaussian.memLp_two_id (μ := gaussianReal 0 1)).integrable_sq |
|
|
| lemma standard_square_integral : ∫ z : ℝ, z^2 ∂gaussianReal 0 1 = 1 := by |
| have h := variance_fun_id_gaussianReal (μ := 0) (v := 1) |
| rw [variance_eq_integral measurable_id'.aemeasurable] at h |
| simpa only [integral_id_gaussianReal,sub_zero,NNReal.coe_one] using h |
|
|
| def clippedSquare (R z : ℝ) : ℝ := min (z^2) (R^2) |
|
|
| lemma continuous_clippedSquare (R : ℝ) : Continuous (clippedSquare R) := by |
| unfold clippedSquare; fun_prop |
|
|
| lemma clippedSquare_nonneg (R z : ℝ) : 0 ≤ clippedSquare R z := le_min (sq_nonneg _) (sq_nonneg _) |
|
|
| lemma clippedSquare_le (R z : ℝ) : clippedSquare R z ≤ R^2 := min_le_right _ _ |
|
|
| lemma square_sub_clip_nonneg (R z : ℝ) : 0 ≤ z^2-clippedSquare R z := |
| sub_nonneg.mpr (min_le_left _ _) |
|
|
| lemma square_sub_clip_le {R : ℝ} (hR : 0 ≤ R) (z : ℝ) : |
| z^2-clippedSquare R z ≤ squareTail R z := by |
| by_cases hz : R < |z| |
| · simp only [squareTail,hz,ite_true] |
| linarith [clippedSquare_nonneg R z] |
| · have h : z^2 ≤ R^2 := by |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg z) hR).2 (le_of_not_gt hz) |
| simp only [clippedSquare,min_eq_left h,sub_self,squareTail,hz,ite_false,le_refl] |
|
|
| lemma clippedSquare_integrable (R : ℝ) : Integrable (clippedSquare R) (gaussianReal 0 1) := by |
| apply Integrable.of_bound (continuous_clippedSquare R).aestronglyMeasurable (R^2) |
| exact ae_of_all _ (fun z => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (clippedSquare_nonneg _ _)]; exact clippedSquare_le _ _) |
|
|
| lemma clippedSquare_mean_bounds (R : ℝ) : |
| 0 ≤ ∫ z, clippedSquare R z ∂gaussianReal 0 1 ∧ |
| (∫ z, clippedSquare R z ∂gaussianReal 0 1) ≤ R^2 := by |
| constructor |
| · exact integral_nonneg (clippedSquare_nonneg R) |
| · simpa using integral_mono (clippedSquare_integrable R) (integrable_const (R^2)) |
| (clippedSquare_le R) |
|
|
| lemma clippedSquare_mean_error {R : ℝ} (hR : 0 ≤ R) : |
| |(∫ z, clippedSquare R z ∂gaussianReal 0 1)-1| ≤ |
| ∫ z, squareTail R z ∂gaussianReal 0 1 := by |
| have hi : Integrable (squareTail R) (gaussianReal 0 1) := |
| integrable_standard_square.mono' (measurable_squareTail R).aestronglyMeasurable |
| (ae_of_all _ (fun z => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (squareTail_nonneg _ _)]; exact squareTail_le _ _)) |
| have hm : (∫ z, clippedSquare R z ∂gaussianReal 0 1) ≤ 1 := by |
| rw [← standard_square_integral] |
| exact integral_mono (clippedSquare_integrable R) integrable_standard_square (fun _ => min_le_left _ _) |
| have h := integral_mono (integrable_standard_square.sub (clippedSquare_integrable R)) hi |
| (square_sub_clip_le hR) |
| simp only [Pi.sub_apply] at h |
| rw [integral_sub integrable_standard_square (clippedSquare_integrable R),standard_square_integral] at h |
| rw [abs_of_nonpos (sub_nonpos.mpr hm)] |
| linarith only [h] |
|
|
| lemma centered_clip_subgaussian {ι : Type*} [Fintype ι] {R : ℝ} (hR : 0 < R) (i : ι) : |
| HasSubgaussianMGF |
| (fun z : ι → ℝ => clippedSquare R (z i)-(∫ x, clippedSquare R x ∂gaussianReal 0 1)) |
| ((Real.toNNReal (R^2))^2) (standardArrayLaw ι) := by |
| apply bounded_zero_subgaussian _ |
| (((continuous_clippedSquare R).measurable.comp (measurable_pi_apply i)).sub measurable_const).aemeasurable |
| · exact ae_of_all _ (fun z => by |
| simp only [Pi.sub_apply,Function.comp_def] |
| rw [Real.coe_toNNReal _ (sq_pos_of_pos hR).le,abs_le] |
| have h := clippedSquare_mean_bounds R |
| constructor <;> linarith [clippedSquare_le R (z i),clippedSquare_nonneg R (z i)]) |
| · have hc := (coordinate_hasLaw i).integral_comp |
| (f := fun x : ℝ => clippedSquare R x-(∫ y, clippedSquare R y ∂gaussianReal 0 1)) |
| (((continuous_clippedSquare R).sub continuous_const).aestronglyMeasurable) |
| simp only [Function.comp_def] at hc |
| simp only [Pi.sub_apply,Function.comp_def] |
| rw [hc,integral_sub (clippedSquare_integrable R) (integrable_const _)] |
| simp |
|
|
| lemma clipped_square_matrix_rare {R u : ℝ} (hR : 0 < R) (hu : 0 < u) : |
| ExponentiallyRare (fun n => standardArrayLaw (Edge n)) |
| (fun n => {z | u < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (coupling (fun e => (clippedSquare R (z e)-(∫ x, clippedSquare R x ∂gaussianReal 0 1))/n))‖}) := by |
| apply exponentiallyRare_of_geom_quadratic _ _ (C := 2) (b := 81) (a := (u/2)^2/(4*(R^2)^2)) |
| (by norm_num) (by positivity) (by norm_num) |
| filter_upwards [eventually_gt_atTop 0] with n hn |
| have hb := bounded_matrix_opNorm_tail hn |
| (fun e (z : Edge n → ℝ) => clippedSquare R (z e)-(∫ x, clippedSquare R x ∂gaussianReal 0 1)) |
| (coordinates_independent.comp |
| (fun _ x => clippedSquare R x-(∫ y, clippedSquare R y ∂gaussianReal 0 1)) |
| (fun _ => ((continuous_clippedSquare R).sub continuous_const).measurable)) |
| (show 0 < Real.toNNReal (R^2) from Real.toNNReal_pos.mpr (by positivity)) |
| (centered_clip_subgaussian hR) (u := u/2) (by positivity) |
| rw [show 2*(u/2) = u by ring,Real.coe_toNNReal _ (sq_nonneg _)] at hb |
| convert hb using 1 |
| congr 2 |
| ring_nf |
|
|
| lemma square_remainder_norm_le {n : ℕ} (hn : 0<n) (z : Disorder n) |
| {R a : ℝ} (hR : 0≤R) (ha : 0≤a) |
| (hrow : ∀ i, (∑ e ∈ incidentEdges i, squareTail R (z e))/(n:ℝ) ≤ a) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (coupling (fun e => (z e^2-clippedSquare R (z e))/n))‖ ≤ a := by |
| have hnR : (0:ℝ)<n := Nat.cast_pos.mpr hn |
| let g : Disorder n := fun e => (z e^2-clippedSquare R (z e))/n |
| have hg : ∀ e, 0≤g e := fun e => div_nonneg (square_sub_clip_nonneg _ _) hnR.le |
| have hr (i : Fin n) : ∑ j, |coupling g i j| ≤ a := by |
| simp_rw [abs_of_nonneg (coupling_nonneg hg _ _)] |
| rw [sum_coupling_row] |
| change (∑ e ∈ incidentEdges i, (z e^2-clippedSquare R (z e))/(n:ℝ)) ≤ a |
| rw [←Finset.sum_div] |
| exact (div_le_div_of_nonneg_right (Finset.sum_le_sum (fun e _ => square_sub_clip_le hR _)) hnR.le).trans (hrow i) |
| exact matrix_opNorm_le_schur (coupling g) a ha hr (fun j => by |
| simpa only [coupling_symm] using hr j) |
|
|
| lemma coupling_norm_add_three {n : ℕ} (A B D : Disorder n) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (coupling (A+B+D))‖ ≤ |
| (‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (coupling A)‖ + |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (coupling B)‖) + |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (coupling D)‖ := by |
| rw [coupling_add,coupling_add] |
| change ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| ((Matrix.of (coupling A)+Matrix.of (coupling B))+Matrix.of (coupling D))‖ ≤ _ |
| rw [map_add,map_add] |
| exact (norm_add_le _ _).trans (add_le_add_left (norm_add_le _ _) _) |
|
|
| lemma centered_square_mean_norm_le {n : ℕ} (hn : 0<n) {R : ℝ} (hR : 0≤R) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (coupling |
| (fun _ => ((∫ x, clippedSquare R x ∂gaussianReal 0 1)-1)/n))‖ ≤ |
| ∫ x, squareTail R x ∂gaussianReal 0 1 := by |
| let m := ∫ x, clippedSquare R x ∂gaussianReal 0 1 |
| have hh := coupling_const_norm_le (n := n) ((m-1)/n) |
| have hnR : (n:ℝ)≠0 := Nat.cast_ne_zero.mpr hn.ne' |
| have hm : (n:ℝ)*|(m-1)/n| = |m-1| := by |
| rw [abs_div,show |(n:ℝ)|=(n:ℝ) from abs_of_nonneg (Nat.cast_nonneg n)] |
| field_simp |
| rw [hm] at hh |
| exact hh.trans (clippedSquare_mean_error hR) |
|
|
| lemma centered_square_norm_le {n : ℕ} (hn : 0<n) (z : Disorder n) |
| {R a : ℝ} (hR : 0≤R) (ha : 0≤a) |
| (hrow : ∀ i, (∑ e ∈ incidentEdges i, squareTail R (z e))/(n:ℝ) ≤ a) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (coupling (fun e => (z e^2-1)/n))‖ ≤ |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (coupling (fun e => (clippedSquare R (z e)-(∫ x, clippedSquare R x ∂gaussianReal 0 1))/n))‖ + |
| (∫ x, squareTail R x ∂gaussianReal 0 1) + a := by |
| let m := ∫ x, clippedSquare R x ∂gaussianReal 0 1 |
| let A : Disorder n := fun e => (clippedSquare R (z e)-m)/n |
| let B : Disorder n := fun _ => (m-1)/n |
| let D : Disorder n := fun e => (z e^2-clippedSquare R (z e))/n |
| have he : (fun e => (z e^2-1)/(n:ℝ)) = A+B+D := by |
| funext e |
| dsimp only [A,B,D,Pi.add_apply] |
| ring |
| have hB := centered_square_mean_norm_le hn hR |
| have hD := square_remainder_norm_le hn z hR ha hrow |
| rw [he] |
| exact (coupling_norm_add_three A B D).trans (add_le_add (add_le_add_right hB _) hD) |
|
|
| theorem centered_gaussian_square_matrix_rare {u : ℝ} (hu : 0<u) : |
| ExponentiallyRare (fun n => standardArrayLaw (Edge n)) |
| (fun n => {z | u < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (coupling (fun e => (z e^2-1)/n))‖}) := by |
| obtain ⟨R,c,hR,hc,hmean,hb⟩ := gaussian_squareTail_row_bound (a := u/3) (by positivity) |
| have htail : ExponentiallyRare (fun n => standardArrayLaw (Edge n)) |
| (fun n => ⋃ i : Fin n, {z | u/3 ≤ (∑ e ∈ incidentEdges i, squareTail R (z e))/(n:ℝ)}) := by |
| apply exponentiallyRare_iUnion_polynomial _ _ hc (hC := zero_lt_one) (hD := zero_lt_one) 1 |
| · apply Eventually.of_forall |
| intro n |
| simp |
| · filter_upwards [eventually_gt_atTop 0] with n hn |
| intro i |
| simpa only [one_mul] using hb (incidentEdges i) hn (incidentEdges_card_le i) |
| apply ((clipped_square_matrix_rare hR (show 0<u/3 by positivity)).union htail).mono |
| filter_upwards [eventually_gt_atTop 0] with n hn |
| intro z hz |
| by_contra hnot |
| have hclip : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (coupling (fun e => (clippedSquare R (z e)-(∫ x, clippedSquare R x ∂gaussianReal 0 1))/n))‖ ≤ u/3 := |
| le_of_not_gt (fun h => hnot (Or.inl h)) |
| have hrows : ∀ i, (∑ e ∈ incidentEdges i, squareTail R (z e))/(n:ℝ) ≤ u/3 := by |
| intro i |
| exact (not_le.mp (fun hi => hnot (Or.inr (mem_iUnion.mpr ⟨i,hi⟩)))).le |
| have hh := centered_square_norm_le hn z hR.le (by positivity : 0≤u/3) hrows |
| change u < _ at hz |
| linarith |
|
|
| end SKRatio.MatrixNet |
|
|
| end |
| end |
|
|
| end OAI |
|
|
