| import OAI.Probability.PosteriorReplicas.Geometry.LabelEllipsoids |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Filter Metric |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace PosteriorReplicas |
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| lemma affine_infDist_orthogonal {d : ℕ} (p x : Vec d) (U : Submodule ℝ (Vec d)) : |
| Metric.infDist x (AffineSubspace.mk' p U : Set (Vec d)) = ‖Uᗮ.starProjection (x-p)‖ := by |
| rw [PosteriorRegression.infDist_affine_mk', Submodule.starProjection_orthogonal_val, |
| Submodule.starProjection_minimal, Metric.infDist_eq_iInf] |
| simp only [dist_eq_norm] |
| rfl |
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| def tupleDistance {d m : ℕ} (u : Tuple d m) (x : Vec d) : ℝ := |
| ‖(differenceSpan u)ᗮ.starProjection (x-u 0)‖ |
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| lemma tupleDistance_eq_infDist {d m : ℕ} (u : Tuple d m) (x : Vec d) : |
| tupleDistance u x = Metric.infDist x (AffineSubspace.mk' (u 0) (differenceSpan u) : Set (Vec d)) := |
| (affine_infDist_orthogonal _ _ _).symm |
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| @[fun_prop] lemma measurable_tupleDistance_fixed {d m : ℕ} (u : Tuple d m) : |
| Measurable (tupleDistance u) := by unfold tupleDistance; fun_prop |
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| lemma differenceSpan_finrank_le {d m : ℕ} (u : Tuple d m) : |
| Module.finrank ℝ (differenceSpan u) ≤ m := by |
| change Module.finrank ℝ (Submodule.span ℝ (range (differences u))) ≤ m |
| simpa only [Set.finrank, Fintype.card_fin] using |
| finrank_range_le_card (R := ℝ) (differences u) |
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| lemma tupleDistance_tube {d k m : ℕ} (hd : k+m+2 ≤ d) (u : Tuple d m) |
| {f : Vec d → ℝ≥0∞} {L : ℝ≥0∞} (hfL : ∀ x, f x ≤ L) |
| {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : |
| (sphereLaw d).withDensity f {x | tupleDistance u x ≤ r} ≤ |
| (L*ENNReal.ofReal ((72:ℝ)^d))*ENNReal.ofReal (r^(k+1)) := by |
| let : NeZero d := ⟨by omega⟩ |
| simp_rw [tupleDistance_eq_infDist] |
| apply spherical_tilt_affine_tube hfL _ _ hr hr1 |
| rw [AffineSubspace.direction_mk'] |
| have h := differenceSpan_finrank_le u |
| omega |
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| lemma tupleDistance_pos_ae {d m : ℕ} (hd : 3 ≤ d) (hm : m < d) (u : Tuple d m) |
| (f : Vec d → ℝ≥0∞) : |
| ∀ᵐ x ∂(sphereLaw d).withDensity f, 0 < tupleDistance u x := by |
| let S := AffineSubspace.mk' (u 0) (differenceSpan u) |
| have hz : sphereLaw d S = 0 := by |
| apply base_affine_null hd .spherical S |
| rw [AffineSubspace.direction_mk'] |
| exact (differenceSpan_finrank_le u).trans_lt hm |
| have hn : ∀ᵐ x ∂sphereLaw d, x ∉ S := by |
| rw [ae_iff] |
| simp only [not_not] |
| convert hz using 2 |
| ext x |
| rfl |
| filter_upwards [(withDensity_absolutelyContinuous (sphereLaw d) f).ae_le hn] with x hx |
| rw [tupleDistance_eq_infDist] |
| exact (S.closed_of_finiteDimensional.notMem_iff_infDist_pos |
| ⟨u 0, AffineSubspace.self_mem_mk' _ _⟩).mp hx |
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|
| lemma labels_sub {d k : ℕ} (A : Rows d k) (x y : Vec d) : |
| labels A (x-y) = labels A x - labels A y := by |
| ext i |
| simp [labels, inner_sub_right] |
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| lemma constrained_label_lift_le {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (U : Submodule ℝ (Vec d)) (hU : ∀ i, A i ∈ Uᗮ) (p x : Vec d) : |
| ‖liftLabel A (labels A x - labels A p)‖ ≤ ‖Uᗮ.starProjection (x-p)‖ := by |
| have he : labels A (Uᗮ.starProjection (x-p)) = labels A (x-p) := by |
| ext i |
| change ⟪A i,Uᗮ.starProjection (x-p)⟫ = ⟪A i,x-p⟫ |
| rw [← Uᗮ.inner_starProjection_left_eq_right, |
| Uᗮ.starProjection_eq_self_iff.mpr (hU i)] |
| rw [← labels_sub, ← he, liftLabel_eq_pairingLift] |
| change ‖pairingLift A (fun i => ⟪A i,Uᗮ.starProjection (x-p)⟫)‖ ≤ _ |
| rw [← starProjection_eq_pairingLift hA] |
| exact Submodule.norm_starProjection_apply_le _ _ |
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|
| lemma constrained_tube_le_ellipsoid {d k m : ℕ} (ν : Measure (Vec d)) |
| (u : Tuple d m) {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (hU : ∀ i, A i ∈ (differenceSpan u)ᗮ) (r : ℝ) : |
| ν {x | tupleDistance u x ≤ r} ≤ |
| ν {x | ‖liftLabel A (labels A x - labels A (u 0))‖ ≤ r} := by |
| apply measure_mono |
| intro x hx |
| exact (constrained_label_lift_le hA _ hU _ _).trans hx |
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| lemma density_inverse_moment_bins {d k m : ℕ} (hd : k+m+2 ≤ d) |
| (u : Tuple d m) {f : Vec d → ℝ≥0∞} {L : ℝ≥0∞} (hfL : ∀ x, f x ≤ L) |
| [IsProbabilityMeasure ((sphereLaw d).withDensity f)] (J : ℕ) |
| (hcut : (L*ENNReal.ofReal ((72:ℝ)^d))*ENNReal.ofReal ((1/2:ℝ)^J)* |
| (1+(2:ℝ≥0∞)^(k+1)) ≤ 1) |
| {A : Rows d k} (hA : LinearIndependent ℝ A) (hU : ∀ i, A i ∈ (differenceSpan u)ᗮ) : |
| (∫⁻ x, ENNReal.ofReal ((tupleDistance u x)⁻¹^k) ∂(sphereLaw d).withDensity f) ≤ |
| 2 + ∑ j ∈ Finset.range J, ENNReal.ofReal ((2:ℝ)^(k*(j+1))) * |
| (sphereLaw d).withDensity f {x | ‖liftLabel A (labels A x-labels A (u 0))‖ ≤ (1/2:ℝ)^j} := by |
| apply (inverse_moment_finite_bins _ (measurable_tupleDistance_fixed u) |
| (fun x => norm_nonneg _) k J _ (fun r hr hr1 => tupleDistance_tube hd u hfL hr hr1) hcut).trans |
| apply add_le_add le_rfl |
| apply Finset.sum_le_sum |
| intro j hj |
| exact mul_le_mul' le_rfl (constrained_tube_le_ellipsoid _ u hA hU _) |
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|
| end PosteriorReplicas |
| end |
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| end OAI |
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