| import Mathlib |
| import OAI.Probability.GaussianInformation.Inverse.Ellipsoids |
|
|
| namespace OAI |
| open MeasureTheory Metric |
| open scoped ENNReal |
| namespace AllScaleInverseVolume |
| open Set |
| open scoped Topology NNReal |
| noncomputable def synth {q d : ℕ} (e : Fin q → Ambient d) (x : Ambient q) : Ambient d := |
| ∑ j,x j • e j |
| noncomputable def coordinates {q d : ℕ} (e : Fin q → Ambient d) (x : Ambient d) : Ambient q := |
| WithLp.toLp 2 (fun j => inner ℝ (e j) x) |
| noncomputable def residual {q d : ℕ} (e : Fin q → Ambient d) (x : Ambient d) : Ambient d := |
| x-synth e (coordinates e x) |
|
|
| lemma synth_norm {q d : ℕ} (e : Fin q → Ambient d) (he : Orthonormal ℝ e) (x : Ambient q) : |
| ‖synth e x‖ = ‖x‖ := by |
| have hh := he.inner_sum (fun j => x j) (fun j => x j) Finset.univ |
| have hsq : ‖synth e x‖^2 = ‖x‖^2 := by |
| rw [EuclideanSpace.real_norm_sq_eq x] |
| simpa only [synth,real_inner_self_eq_norm_sq,starRingEnd_apply,star_trivial,pow_two] using hh |
| nlinarith [norm_nonneg (synth e x),norm_nonneg x] |
|
|
| lemma synth_sub {q d : ℕ} (e : Fin q → Ambient d) (x y : Ambient q) : |
| synth e (x-y) = synth e x-synth e y := by |
| simp [synth,sub_smul,Finset.sum_sub_distrib] |
| lemma synth_smul {q d : ℕ} (e : Fin q → Ambient d) (r : ℝ) (x : Ambient q) : |
| synth e (r • x) = r • synth e x := by |
| simp [synth,mul_smul,Finset.smul_sum] |
|
|
| def groupedTube {q d : ℕ} (e : Fin q → Ambient d) (g : Fin q → ℕ) (r : ℝ) : Set (Ambient d) := |
| {x | ‖residual e x‖ ≤ r ∧ ∀ h : ℕ, |
| ∑ j ∈ Finset.univ.filter (fun j => g j=h),(inner ℝ (e j) x)^2 ≤ r^2*(4:ℝ)^h} |
|
|
| lemma exists_groupedTube_cover {q d : ℕ} (e : Fin q → Ambient d) |
| (he : ∀ x : Ambient q,‖synth e x‖ ≤ ‖x‖) |
| (g : Fin q → ℕ) {r : ℝ} (hr : 0 < r) : |
| ∃ s : Finset (Ambient d), |
| groupedTube e g r ⊆ ⋃ y ∈ s,ball y (3*r) ∧ |
| (s.card:ℝ) ≤ 5^q*(4:ℝ)^(∑ j,g j) := by |
| classical |
| let c : Fin q → ℝ := fun j => (1/4:ℝ)^(g j) |
| have hc : ∀ j,0 < c j ∧ c j ≤ 1 := fun j => |
| ⟨pow_pos (by norm_num) _,pow_le_one₀ (by norm_num) (by norm_num)⟩ |
| obtain ⟨s,_,hcov,hcard⟩ := exists_ellipsoid_cover c hc |
| refine ⟨s.image (fun y => synth e (r • y)),?_,?_⟩ |
| · intro x hx |
| let a := r⁻¹ • coordinates e x |
| have ha : ∀ h : ℕ,∑ j ∈ Finset.univ.filter (fun j => g j=h),(a j)^2 ≤ (4:ℝ)^h := by |
| intro h |
| have hj : ∑ j ∈ Finset.univ.filter (fun j => g j=h),(a j)^2 = |
| (r⁻¹)^2 * ∑ j ∈ Finset.univ.filter (fun j => g j=h),(inner ℝ (e j) x)^2 := by |
| simp [a,coordinates,mul_pow,Finset.mul_sum] |
| rw [hj] |
| calc |
| _ ≤ (r⁻¹)^2*(r^2*4^h) := mul_le_mul_of_nonneg_left (hx.2 h) (sq_nonneg _) |
| _ = _ := by field_simp [hr.ne'] |
| obtain ⟨y,hy,hay⟩ := hcov a (weighted_group_bound g a ha) |
| refine mem_iUnion₂.mpr ⟨synth e (r • y),Finset.mem_image.mpr ⟨y,hy,rfl⟩,?_⟩ |
| have hra : r • a = coordinates e x := by simp [a,smul_smul,ne_of_gt hr] |
| have hdist : ‖synth e (coordinates e x)-synth e (r • y)‖ ≤ r := by |
| rw [← hra,← synth_sub,← smul_sub] |
| calc |
| _ ≤ ‖r • (a-y)‖ := he _ |
| _ = r*‖a-y‖ := by rw [norm_smul,Real.norm_eq_abs,abs_of_pos hr] |
| _ ≤ r := (mul_le_mul_of_nonneg_left (by simpa [dist_eq_norm] using hay) hr.le).trans_eq (mul_one r) |
| change ‖x-synth e (r • y)‖ < 3*r |
| have hadd := norm_add_le (residual e x) (synth e (coordinates e x)-synth e (r • y)) |
| have heq : residual e x+(synth e (coordinates e x)-synth e (r • y)) = x-synth e (r • y) := by |
| simp only [residual]; abel |
| rw [heq] at hadd |
| linarith [hx.1] |
| · calc |
| _ ≤ (s.card:ℝ) := by exact_mod_cast Finset.card_image_le |
| _ ≤ 5^q*(∏ j,c j)⁻¹ := hcard |
| _ = _ := by |
| simp only [c,← Finset.prod_pow_eq_pow_sum,Finset.prod_inv_distrib,one_div,inv_pow,inv_inv] |
| |
| lemma measure_groupedTube_le {q d : ℕ} (ν : Measure (Ambient d)) (hν : BallBound ν) |
| (u : Ambient d) (e : Fin q → Ambient d) |
| (he : ∀ x : Ambient q,‖synth e x‖ ≤ ‖x‖) |
| (g : Fin q → ℕ) {r : ℝ} (hr : 0 < r) : |
| ν {z | z-u ∈ groupedTube e g r} ≤ |
| ENNReal.ofReal (5^q*(4:ℝ)^(∑ j,g j)*(3*r)^(((d:ℝ)-1)/2)) := by |
| classical |
| obtain ⟨s,hcov,hcard⟩ := exists_groupedTube_cover e he g hr |
| have hsub : {z | z-u ∈ groupedTube e g r} ⊆ ⋃ y ∈ s,ball (u+y) (3*r) := by |
| intro z hz |
| obtain ⟨y,hy,hzy⟩ := mem_iUnion₂.mp (hcov hz) |
| refine mem_iUnion₂.mpr ⟨y,hy,?_⟩ |
| simpa only [mem_ball,dist_eq_norm,sub_add_eq_sub_sub] using hzy |
| calc |
| _ ≤ ν (⋃ y ∈ s,ball (u+y) (3*r)) := measure_mono hsub |
| _ ≤ ∑ y ∈ s,ν (ball (u+y) (3*r)) := measure_biUnion_finset_le _ _ |
| _ ≤ ∑ _y ∈ s,ENNReal.ofReal ((3*r)^(((d:ℝ)-1)/2)) := |
| Finset.sum_le_sum (fun y _ => hν (u+y) (3*r) (by positivity)) |
| _ = (s.card:ℝ≥0∞)*ENNReal.ofReal ((3*r)^(((d:ℝ)-1)/2)) := by simp |
| _ ≤ ENNReal.ofReal (5^q*(4:ℝ)^(∑ j,g j))*ENNReal.ofReal ((3*r)^(((d:ℝ)-1)/2)) := by |
| gcongr |
| simpa only [ENNReal.ofReal_natCast] using ENNReal.ofReal_le_ofReal hcard |
| _ = _ := (ENNReal.ofReal_mul (by positivity)).symm |
| |
| end AllScaleInverseVolume |
| |
| end OAI |
| |
