| import Mathlib |
| import OAI.Probability.GaussianInformation.Inverse.GramVolume |
|
|
| namespace OAI |
| open MeasureTheory Metric |
| open scoped ENNReal |
| namespace AllScaleInverseVolume |
| open Set |
| open scoped Topology NNReal |
| open InnerProductSpace |
| open scoped RealInnerProductSpace |
| open InnerProductSpace Matrix MeasureTheory Set |
| open scoped RealInnerProductSpace |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| def Greedy {α : Type*} {k : ℕ} (D : (Fin k → α) → Fin k → α → ℝ) (f : Fin k → α) : Prop := |
| ∀ i j : Fin k,i ≤ j → D f i (f j) ≤ D f i (f i) |
|
|
| lemma swap_preserves_lower_bound {k : ℕ} {b i j c : Fin k} |
| (hij : i ≤ j) (hbi : b ≤ i) (hbc : b ≤ c) : b ≤ Equiv.swap i j c := by |
| classical |
| by_cases hci : c=i |
| · subst c; simpa using hbi.trans hij |
| by_cases hcj : c=j |
| · subst c; simpa using hbi |
| simpa [Equiv.swap_apply_of_ne_of_ne hci hcj] using hbc |
|
|
| lemma exists_greedy_permutation {α : Type*} {k : ℕ} |
| (D : (Fin k → α) → Fin k → α → ℝ) |
| (hD : ∀ f g i,(∀ j,i > j → f j=g j) → ∀ x,D f i x=D g i x) |
| (f : Fin k → α) : ∃ π : Equiv.Perm (Fin k),Greedy D (f ∘ π) := by |
| classical |
| have build : ∀ m : ℕ,m ≤ k → ∃ π : Equiv.Perm (Fin k), |
| ∀ i j : Fin k,i.val < m → i ≤ j → D (f ∘ π) i (f (π j)) ≤ D (f ∘ π) i (f (π i)) := by |
| intro m |
| induction m with |
| | zero => |
| intro _ |
| exact ⟨Equiv.refl _,fun i j hi _ => (Nat.not_lt_zero _ hi).elim⟩ |
| | succ m ih => |
| intro hm |
| obtain ⟨π,hπ⟩ := ih (by omega) |
| let i : Fin k := ⟨m,by omega⟩ |
| obtain ⟨j,hj,hmax⟩ := Finset.exists_max_image (Finset.Ici i) |
| (fun j : Fin k => D (f ∘ π) i (f (π j))) (by simp) |
| have hij : i ≤ j := Finset.mem_Ici.mp hj |
| let τ := Equiv.swap i j |
| have hfix : ∀ a : Fin k,a < i → τ a=a := by |
| intro a ha |
| exact Equiv.swap_apply_of_ne_of_ne (ne_of_lt ha) (ne_of_lt (ha.trans_le hij)) |
| have hpref : ∀ b : Fin k,b ≤ i → ∀ a : Fin k,a < b → |
| (f ∘ (τ.trans π)) a = (f ∘ π) a := by |
| intro b hb a ha |
| simp only [Function.comp_apply,Equiv.trans_apply,hfix a (ha.trans_le hb)] |
| refine ⟨τ.trans π,?_⟩ |
| intro b c hb hbc |
| have hbi : b ≤ i := by change b.val ≤ m; omega |
| have hscore := hD (f ∘ (τ.trans π)) (f ∘ π) b (hpref b hbi) |
| by_cases hbeq : b=i |
| · subst b |
| simp only [Equiv.trans_apply] at hscore ⊢ |
| rw [hscore,hscore] |
| have hτi : τ i=j := Equiv.swap_apply_left i j |
| rw [hτi] |
| exact hmax (τ c) (Finset.mem_Ici.mpr (swap_preserves_lower_bound hij le_rfl hbc)) |
| · have hblt : b < i := lt_of_le_of_ne hbi hbeq |
| simp only [Equiv.trans_apply] at hscore ⊢ |
| rw [hscore,hscore,hfix b hblt] |
| exact hπ b (τ c) (by change b.val < m at hblt; exact hblt) |
| (swap_preserves_lower_bound hij hbi hbc) |
| obtain ⟨π,hπ⟩ := build k le_rfl |
| exact ⟨π,fun i j hij => hπ i j i.isLt hij⟩ |
| noncomputable def prefixResidual {d k : ℕ} (f : Fin k → Ambient d) (i : Fin k) |
| (x : Ambient d) : Ambient d := orthoResidual (gramSchmidtNormed ℝ f) (Finset.Iio i) x |
| noncomputable def prefixAxes {d k : ℕ} (f : Fin k → Ambient d) (i : Fin k) : Fin k → Ambient d := |
| fun j => if j < i then gramSchmidtNormed ℝ f j else 0 |
|
|
| lemma prefixAxes_suborthonormal {d k : ℕ} (f : Fin k → Ambient d) (i : Fin k) : |
| Suborthonormal (prefixAxes f i) := |
| (gramSchmidtNormed_suborthonormal f).cut (Set.Iio i) |
|
|
| lemma gramSchmidtNormed_term {d k : ℕ} (f : Fin k → Ambient d) (j : Fin k) (x : Ambient d) : |
| (inner ℝ (gramSchmidtNormed ℝ f j) x) • gramSchmidtNormed ℝ f j = |
| (inner ℝ (gramSchmidt ℝ f j) x / ‖gramSchmidt ℝ f j‖^2) • gramSchmidt ℝ f j := by |
| simp only [gramSchmidtNormed,real_inner_smul_left,smul_smul] |
| congr 1 |
| simp only [div_eq_mul_inv,RCLike.ofReal_real_eq_id,id_eq] |
| ring |
|
|
| lemma prefixResidual_self {d k : ℕ} (f : Fin k → Ambient d) (i : Fin k) : |
| prefixResidual f i (f i)=gramSchmidt ℝ f i := by |
| calc |
| _ = f i-∑ j ∈ Finset.Iio i,(inner ℝ (gramSchmidt ℝ f j) (f i) / ‖gramSchmidt ℝ f j‖^2) • gramSchmidt ℝ f j := by |
| simp only [prefixResidual,orthoResidual,gramSchmidtNormed_term] |
| _ = _ := (gramSchmidt_real_def f i).symm |
|
|
| lemma prefixResidual_eq {d k : ℕ} (f : Fin k → Ambient d) (i : Fin k) (x : Ambient d) : |
| residual (prefixAxes f i) x=prefixResidual f i x := by |
| classical |
| have hI : Finset.univ.filter (fun j : Fin k => j < i)=Finset.Iio i := by ext j; simp |
| simp only [residual,synth,coordinates,prefixAxes,prefixResidual,orthoResidual,WithLp.ofLp_toLp] |
| congr 1 |
| calc |
| _ = ∑ j : Fin k,if j < i then (inner ℝ (gramSchmidtNormed ℝ f j) x) • gramSchmidtNormed ℝ f j else 0 := by |
| apply Finset.sum_congr rfl |
| intro j _ |
| by_cases hj : j < i <;> simp [hj] |
| _ = _ := by rw [← Finset.sum_filter,hI] |
|
|
| lemma prefixResidual_prefix_congr {d k : ℕ} {f g : Fin k → Ambient d} (i : Fin k) |
| (hfg : ∀ j,j < i → f j=g j) (x : Ambient d) : prefixResidual f i x=prefixResidual g i x := by |
| unfold prefixResidual orthoResidual |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro j hj |
| have he : gramSchmidtNormed ℝ f j=gramSchmidtNormed ℝ g j := by |
| simp only [gramSchmidtNormed,gramSchmidt_prefix_congr j |
| (fun a ha => hfg a (ha.trans_lt (Finset.mem_Iio.mp hj)))] |
| rw [he] |
|
|
| lemma prefixAxes_prefix_congr {d k : ℕ} {f g : Fin k → Ambient d} (i : Fin k) |
| (hfg : ∀ j,j < i → f j=g j) : prefixAxes f i=prefixAxes g i := by |
| funext j |
| by_cases hj : j < i |
| · simp only [prefixAxes,ite_eq_left hj,gramSchmidtNormed,gramSchmidt_prefix_congr j |
| (fun a ha => hfg a (ha.trans_lt hj))] |
| · simp [prefixAxes,hj] |
|
|
| lemma measurable_gramSchmidtNormed {d k : ℕ} (j : Fin k) : |
| Measurable (fun f : Fin k → Ambient d => gramSchmidtNormed ℝ f j) := |
| (measurable_gramSchmidt j).norm.inv.smul (measurable_gramSchmidt j) |
|
|
| lemma measurable_prefixAxes {d k : ℕ} (i j : Fin k) : |
| Measurable (fun f : Fin k → Ambient d => prefixAxes f i j) := by |
| by_cases hj : j < i |
| · simpa [prefixAxes,hj] using measurable_gramSchmidtNormed (d := d) j |
| · simp [prefixAxes,hj] |
|
|
| lemma measurable_prefixResidual {d k : ℕ} (i : Fin k) : |
| Measurable (fun p : (Fin k → Ambient d) × Ambient d => prefixResidual p.1 i p.2) := by |
| unfold prefixResidual orthoResidual |
| apply measurable_snd.sub |
| apply Finset.measurable_sum |
| intro j _ |
| exact (((measurable_gramSchmidtNormed (d := d) j).comp measurable_fst).inner (𝕜 := ℝ) measurable_snd).smul |
| ((measurable_gramSchmidtNormed (d := d) j).comp measurable_fst) |
|
|
| noncomputable def residualScore {d k : ℕ} (f : Fin k → Ambient d) (i : Fin k) (x : Ambient d) : ℝ := |
| ‖prefixResidual f i x‖ |
|
|
| lemma exists_greedy_gram_order {d k : ℕ} (f : Fin k → Ambient d) : |
| ∃ π : Equiv.Perm (Fin k),Greedy residualScore (f ∘ π) := by |
| apply exists_greedy_permutation |
| intro f g i hfg x |
| exact congrArg norm (prefixResidual_prefix_congr i hfg x) |
|
|
| lemma greedy_lengths_antitone {d k : ℕ} {f : Fin k → Ambient d} |
| (hg : Greedy residualScore f) : Antitone (fun i => ‖gramSchmidt ℝ f i‖) := by |
| intro i j hij |
| calc |
| _ = ‖prefixResidual f j (f j)‖ := by rw [prefixResidual_self] |
| _ ≤ ‖prefixResidual f i (f j)‖ := |
| orthoResidual_norm_antitone (gramSchmidtNormed_suborthonormal f) (Finset.Iio_subset_Iio hij) _ |
| _ ≤ ‖prefixResidual f i (f i)‖ := hg i j hij |
| _ = _ := by rw [prefixResidual_self] |
|
|
| lemma measurableSet_greedy {d k : ℕ} : MeasurableSet {f : Fin k → Ambient d | Greedy residualScore f} := by |
| simp only [Greedy,residualScore,ofPred_forall] |
| apply MeasurableSet.iInter |
| intro i |
| apply MeasurableSet.iInter |
| intro j |
| apply MeasurableSet.iInter |
| intro _ |
| exact measurableSet_le |
| ((measurable_prefixResidual i).comp (measurable_id.prodMk (measurable_pi_apply j))).norm |
| ((measurable_prefixResidual i).comp (measurable_id.prodMk (measurable_pi_apply i))).norm |
|
|
| end AllScaleInverseVolume |
|
|
| end OAI |
|
|
