| import OAI.Probability.GaussianPropeller.Deletion |
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| namespace OAI |
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| universe uE uF uι |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
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| open scoped RealInnerProductSpace |
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| namespace GaussianPropeller.Gram |
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| variable {ι : Type uι} {E : Type uE} [Fintype ι] [DecidableEq ι] |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
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| omit [Fintype ι] [DecidableEq ι] in |
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| theorem independent_omit (z : ι → E) |
| (hneg : Pairwise (fun i j => ⟪z i, z j⟫ < 0)) (i : ι) : |
| LinearIndependent ℝ (fun j : {j : ι // j ≠ i} => z j) := by |
| apply LinearMap.BilinForm.linearIndependent_of_pairwise_le_zero (innerₗ E) |
| (fun x hx => real_inner_self_pos.mpr hx) (-(innerₗ E) (z i)) |
| · intro j |
| change 0 < -⟪z i, z j⟫ |
| exact neg_pos.mpr (hneg (Ne.symm j.property)) |
| · intro j l hjl |
| exact (hneg (fun h => hjl (Subtype.ext h))).le |
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| theorem sum_omit_eq {F : Type uF} [AddCommMonoid F] (f : ι → F) (i : ι) |
| (hi : f i = 0) : (∑ j : {j : ι |
| rw [← Finset.sum_subtype (Finset.univ.erase i) (by simp) f] |
| simpa [hi] using Finset.sum_erase_add Finset.univ f (Finset.mem_univ i) |
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| theorem coefficients_constant (z : ι → E) |
| (hneg : Pairwise (fun i j => ⟪z i, z j⟫ < 0)) (hsum : ∑ i, z i = 0) |
| (a : ι → ℝ) (ha : ∑ i, a i • z i = 0) (i j : ι) : a j = a i := by |
| have hrel : ∑ l, (a l - a i) • z l = 0 := by |
| simp only [sub_smul, Finset.sum_sub_distrib, ← Finset.smul_sum, hsum, ha, |
| smul_zero, sub_self] |
| have hrel' : ∑ l : {l : ι // l ≠ i}, (a l - a i) • z l = 0 := by |
| rw [sum_omit_eq (fun l => (a l - a i) • z l) i (by simp)] |
| exact hrel |
| by_cases hji : j = i |
| · rw [hji] |
| · have hzero := (Fintype.linearIndependent_iff.mp (independent_omit z hneg i)) |
| (fun l => a l - a i) hrel' ⟨j, hji⟩ |
| exact sub_eq_zero.mp hzero |
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| theorem affineIndependent (z : ι → E) |
| (hneg : Pairwise (fun i j => ⟪z i, z j⟫ < 0)) (hsum : ∑ i, z i = 0) : |
| AffineIndependent ℝ z := by |
| rw [affineIndependent_iff_of_fintype] |
| intro a ha hrel i |
| have hlin : ∑ j, a j • z j = 0 := by |
| simpa only [Finset.weightedVSub_eq_linear_combination Finset.univ ha] using hrel |
| have hconst : ∀ j, a j = a i := coefficients_constant z hneg hsum a hlin i |
| have hc : (Fintype.card ι : ℝ) * a i = 0 := by |
| simpa only [hconst, Finset.sum_const, Finset.card_univ, nsmul_eq_mul] using ha |
| have hcard : (Fintype.card ι : ℝ) ≠ 0 := by |
| have : Nonempty ι := ⟨i⟩ |
| exact_mod_cast Fintype.card_ne_zero |
| exact (mul_eq_zero.mp hc).resolve_left hcard |
| |
| theorem independent_differences (z : ι → E) |
| (hneg : Pairwise (fun i j => ⟪z i, z j⟫ < 0)) (hsum : ∑ i, z i = 0) |
| (i : ι) : LinearIndependent ℝ (fun j : {j : ι // j ≠ i} => z j - z i) := by |
| exact (affineIndependent_iff_linearIndependent_vsub ℝ z i).mp |
| (affineIndependent z hneg hsum) |
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| theorem span_omit_eq (z : ι → E) (hsum : ∑ i, z i = 0) (i : ι) : |
| Submodule.span ℝ (Set.range (fun j : {j : ι // j ≠ i} => z j)) = |
| Submodule.span ℝ (Set.range z) := by |
| classical |
| apply le_antisymm |
| · exact Submodule.span_mono (by rintro _ ⟨j, rfl⟩; exact ⟨j, rfl⟩) |
| · apply Submodule.span_le.mpr |
| rintro _ ⟨j, rfl⟩ |
| by_cases hji : j ≠ i |
| · exact Submodule.subset_span ⟨⟨j, hji⟩, rfl⟩ |
| · have hji' : j = i := not_ne_iff.mp hji |
| subst j |
| let V := Submodule.span ℝ (Set.range (fun j : {j : ι // j ≠ i} => z j)) |
| have hi : z i = -(∑ j ∈ Finset.univ.erase i, z j) := by |
| have h := Finset.sum_erase_add Finset.univ z (Finset.mem_univ i) |
| rw [hsum] at h |
| exact eq_neg_of_add_eq_zero_right h |
| rw [hi] |
| apply V.neg_mem |
| apply V.sum_mem |
| intro j hj |
| exact Submodule.subset_span ⟨⟨j, (Finset.mem_erase.mp hj).1⟩, rfl⟩ |
| |
| theorem finrank_span (z : ι → E) |
| (hneg : Pairwise (fun i j => ⟪z i, z j⟫ < 0)) (hsum : ∑ i, z i = 0) |
| (i : ι) : Module.finrank ℝ (Submodule.span ℝ (Set.range z)) = |
| Fintype.card ι - 1 := by |
| rw [← span_omit_eq z hsum i, finrank_span_eq_card (independent_omit z hneg i)] |
| simp [Fintype.card_subtype_compl (fun j : ι => j = i)] |
| |
| theorem span_differences_eq (z : ι → E) (hsum : ∑ i, z i = 0) (i : ι) : |
| Submodule.span ℝ (Set.range (fun j : {j : ι // j ≠ i} => z j - z i)) = |
| Submodule.span ℝ (Set.range z) := by |
| classical |
| let W := Submodule.span ℝ (Set.range (fun j : {j : ι // j ≠ i} => z j - z i)) |
| have hdiff (j : ι) : z j - z i ∈ W := by |
| by_cases hji : j = i |
| · simp [hji] |
| · exact Submodule.subset_span ⟨⟨j, hji⟩, rfl⟩ |
| have hi : z i ∈ W := by |
| have hh : (∑ j, (z j - z i)) = -(Fintype.card ι : ℝ) • z i := by |
| simp [Finset.sum_sub_distrib, hsum, ← Nat.cast_smul_eq_nsmul ℝ, neg_smul] |
| have hm : -(Fintype.card ι : ℝ) • z i ∈ W := by |
| rw [← hh] |
| exact W.sum_mem (fun j _ => hdiff j) |
| have hcard : -(Fintype.card ι : ℝ) ≠ 0 := by |
| have : Nonempty ι := ⟨i⟩ |
| exact neg_ne_zero.mpr (by exact_mod_cast Fintype.card_ne_zero) |
| exact (W.smul_mem_iff hcard).mp hm |
| apply le_antisymm |
| · apply Submodule.span_le.mpr |
| rintro _ ⟨j, rfl⟩ |
| exact Submodule.sub_mem _ (Submodule.subset_span ⟨j, rfl⟩) |
| (Submodule.subset_span ⟨i, rfl⟩) |
| · apply Submodule.span_le.mpr |
| rintro _ ⟨j, rfl⟩ |
| change z j ∈ W |
| have h := W.add_mem (hdiff j) hi |
| simpa only [sub_add_cancel] using h |
| |
| omit [Fintype ι] [DecidableEq ι] in |
| |
| theorem exists_inner_values [FiniteDimensional ℝ E] (v : ι → E) |
| (hv : LinearIndependent ℝ v) (a : ι → ℝ) : |
| ∃ h ∈ Submodule.span ℝ (Set.range v), ∀ i, ⟪h, v i⟫ = a i := by |
| let W := Submodule.span ℝ (Set.range v) |
| let b := Module.Basis.span hv |
| let L : StrongDual ℝ W := LinearMap.toContinuousLinearMap (b.constr ℝ a) |
| let h : W := (InnerProductSpace.toDual ℝ W).symm L |
| refine ⟨h, h.property, fun i => ?_⟩ |
| have hi : ⟪h, b i⟫ = a i := by |
| rw [InnerProductSpace.toDual_symm_apply] |
| exact b.constr_basis ℝ a i |
| change ⟪(h : E), ((b i) : E)⟫ = a i at hi |
| simpa only [b, Module.Basis.coe_span_apply] using hi |
| |
| theorem exists_translation [FiniteDimensional ℝ E] (z : ι → E) |
| (hneg : Pairwise (fun i j => ⟪z i, z j⟫ < 0)) (hsum : ∑ i, z i = 0) |
| (i : ι) : ∃ h ∈ Submodule.span ℝ (Set.range z), |
| (∀ j, j ≠ i → ⟪h, z i - z j⟫ = 1) ∧ |
| ⟪h, z i⟫ = ((Fintype.card ι : ℝ) - 1) / (Fintype.card ι : ℝ) := by |
| obtain ⟨h, hh, hpair⟩ := exists_inner_values |
| (fun j : {j : ι // j ≠ i} => z j - z i) |
| (independent_differences z hneg hsum i) (fun _ => -1) |
| have hpair' (j : ι) (hji : j ≠ i) : ⟪h, z i - z j⟫ = 1 := by |
| have hp := hpair ⟨j, hji⟩ |
| simp only [inner_sub_right] at hp ⊢ |
| linarith |
| refine ⟨h, ?_, hpair', ?_⟩ |
| · rwa [span_differences_eq z hsum i] at hh |
| · have hsum' : ∑ j : {j : ι |
| (Fintype.card ι : ℝ) - 1 := by |
| simp only [show ∀ j : {j : ι |
| fun j => hpair' j j.property, Finset.sum_const, Finset.card_univ, |
| nsmul_eq_mul, mul_one] |
| have hcard : Fintype.card {j : ι |
| simp [Fintype.card_subtype_compl (fun j : ι => j = i)] |
| rw [hcard, Nat.cast_sub (Fintype.card_pos_iff.mpr ⟨i⟩)] |
| simp |
| have hsum'' : (Fintype.card ι : ℝ) * ⟪h, z i⟫ = |
| (Fintype.card ι : ℝ) - 1 := by |
| rw [sum_omit_eq (fun j => ⟪h, z i - z j⟫) i (by simp)] at hsum' |
| simpa only [inner_sub_right, Finset.sum_sub_distrib, Finset.sum_const, |
| Finset.card_univ, nsmul_eq_mul, ← inner_sum, hsum, inner_zero_right, |
| sub_zero] using hsum' |
| have hn : (Fintype.card ι : ℝ) ≠ 0 := by |
| have : Nonempty ι := ⟨i⟩ |
| exact_mod_cast Fintype.card_ne_zero |
| exact (eq_div_iff hn).mpr (by simpa only [mul_comm] using hsum'') |
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| end GaussianPropeller.Gram |
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| end OAI |
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