| import OAI.Probability.PosteriorReplicas.Measures.BaseDisintegration |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Set Matrix InnerProductSpace Metric EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace PosteriorReplicas |
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| def baseProjectionConstant (base : BaseLaw) (d k : ℕ) : ℝ := |
| match base with |
| | .gaussian => (2 * Real.pi) ^ (-(k : ℝ) / 2) |
| | .spherical => Real.Gamma ((d : ℝ) / 2) / |
| (Real.pi ^ ((k : ℝ) / 2) * Real.Gamma (((d : ℝ) - k) / 2)) |
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| lemma baseProjectionConstant_pos {d k : ℕ} (hk : k < d) (base : BaseLaw) : |
| 0 < baseProjectionConstant base d k := by |
| have hd : 0 < (d : ℝ) := by exact_mod_cast (by omega : 0 < d) |
| have hdk : 0 < (d : ℝ) - k := sub_pos.mpr (by exact_mod_cast hk) |
| cases base <;> unfold baseProjectionConstant <;> positivity |
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| lemma baseDensity_le_sup {d k : ℕ} (hk : k + 2 ≤ d) (base : BaseLaw) |
| (A : Rows d k) (y : Labels k) : baseDensity base A y ≤ |
| ENNReal.ofReal (baseProjectionConstant base d k / Real.sqrt (rowGram A).det) := by |
| have hC := (baseProjectionConstant_pos (by omega : k < d) base).le |
| by_cases hA : LinearIndependent ℝ A |
| · rw [baseDensity.eq_def, ite_eq_left hA] |
| cases base |
| · dsimp only [baseProjectionConstant] at hC ⊢ |
| split_ifs with hy |
| · apply ENNReal.ofReal_le_ofReal |
| apply div_le_div_of_nonneg_right _ (Real.sqrt_nonneg _) |
| have hb : 0 ≤ 1 - ‖liftLabel A y‖^2 := by nlinarith [norm_nonneg (liftLabel A y)] |
| have he : 0 ≤ ((d : ℝ) - k - 2) / 2 := by |
| have : (k : ℝ) + 2 ≤ d := by exact_mod_cast hk |
| linarith |
| exact mul_le_of_le_one_right hC (Real.rpow_le_one hb (by nlinarith [sq_nonneg ‖liftLabel A y‖]) he) |
| · exact bot_le |
| · dsimp only [baseProjectionConstant] at hC ⊢ |
| apply ENNReal.ofReal_le_ofReal |
| apply div_le_div_of_nonneg_right _ (Real.sqrt_nonneg _) |
| exact mul_le_of_le_one_right hC (Real.exp_le_one_iff.mpr (div_nonpos_of_nonpos_of_nonneg |
| (neg_nonpos.mpr (sq_nonneg _)) (by norm_num))) |
| · rw [baseDensity.eq_def, ite_eq_right hA] |
| exact bot_le |
|
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| lemma baseDensity_le_orthonormal {d k : ℕ} (hk : k + 2 ≤ d) (base : BaseLaw) |
| {A : Rows d k} (hA : Orthonormal ℝ A) (y : Labels k) : |
| baseDensity base A y ≤ ENNReal.ofReal (baseProjectionConstant base d k) := by |
| simpa only [rowGram_eq_gram, Matrix.gram_eq_one_iff_orthonormal.mpr hA, |
| Matrix.det_one, Real.sqrt_one, div_one] using baseDensity_le_sup hk base A y |
|
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| lemma projected_base_measure {d k : ℕ} {A : Rows d k} (hA : LinearIndependent ℝ A) |
| (hk : k < d) (base : BaseLaw) : (base.measure d).map (labels A) = |
| volume.withDensity (baseDensity base A) := by |
| have h := projected_measure hA hk base (f := fun _ => 1) measurable_const |
| have hμ : (base.measure d).withDensity (fun _ => 1) = base.measure d := by |
| exact withDensity_one |
| have hp : projectedDensity base (fun _ => 1) A = baseDensity base A := by |
| funext y |
| simp only [projectedDensity, tiltMass, lintegral_one, measure_univ, mul_one] |
| rw [hμ, hp] at h |
| exact h |
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| lemma orthonormal_in_orthogonal {d q : ℕ} (U : Submodule ℝ (Vec d)) |
| (hq : q ≤ Module.finrank ℝ Uᗮ) : |
| ∃ v : Fin q → Vec d, Orthonormal ℝ v ∧ ∀ i, v i ∈ Uᗮ := by |
| let b := stdOrthonormalBasis ℝ Uᗮ |
| let v : Fin q → Uᗮ := fun i => b (Fin.castLE hq i) |
| refine ⟨fun i => (v i : Vec d), ?_, fun i => (v i).2⟩ |
| exact (b.orthonormal.comp (Fin.castLE hq) (Fin.castLE_injective hq)).comp_linearIsometry Uᗮ.subtypeₗᵢ |
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| lemma pairing_le_infDist {d : ℕ} (L : AffineSubspace ℝ (Vec d)) [Nonempty L] |
| {p v : Vec d} (hp : p ∈ L) (hv : v ∈ L.directionᗮ) (hvnorm : ‖v‖ = 1) (x : Vec d) : |
| |⟪v, x - p⟫| ≤ Metric.infDist x L := by |
| have ho : ⟪v, (orthogonalProjection L x : Vec d) - p⟫ = 0 := |
| Submodule.inner_left_of_mem_orthogonal (L.vsub_mem_direction (orthogonalProjection_mem x) hp) hv |
| have he : ⟪v, x - p⟫ = ⟪v, x - (orthogonalProjection L x : Vec d)⟫ := by |
| rw [inner_sub_right] at ho |
| simp only [inner_sub_right] |
| linarith |
| rw [he, ← dist_orthogonalProjection_eq_infDist, dist_eq_norm] |
| simpa only [hvnorm, one_mul] using abs_real_inner_le_norm v (x - (orthogonalProjection L x : Vec d)) |
|
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| lemma base_affine_tube {d q : ℕ} (hq : q + 2 ≤ d) (base : BaseLaw) |
| (L : AffineSubspace ℝ (Vec d)) [Nonempty L] |
| (hL : Module.finrank ℝ L.direction + q ≤ d) {r : ℝ} (hr : 0 < r) : |
| (base.measure d) {x | Metric.infDist x L ≤ r} ≤ |
| (ENNReal.ofReal (baseProjectionConstant base d q) * 2^q) * ENNReal.ofReal (r^q) := by |
| have hdim : q ≤ Module.finrank ℝ L.directionᗮ := by |
| have he := L.direction.finrank_add_finrank_orthogonal |
| simp only [finrank_euclideanSpace_fin] at he |
| omega |
| obtain ⟨v, hv, hvin⟩ := orthonormal_in_orthogonal L.direction hdim |
| obtain ⟨p, hp⟩ := Set.nonempty_coe_sort.mp (inferInstance : Nonempty L) |
| let c := labels v p |
| let s : Set (Labels q) := Set.pi univ (fun i => Icc (c i - r) (c i + r)) |
| have hs : MeasurableSet s := MeasurableSet.univ_pi (fun i => measurableSet_Icc) |
| have hsub : {x | Metric.infDist x L ≤ r} ⊆ (labels v) ⁻¹' s := by |
| intro x hx |
| simp only [Set.mem_preimage, s, Set.mem_pi, Set.mem_univ, true_implies, Set.mem_Icc] |
| intro i |
| have hi := (pairing_le_infDist L hp (hvin i) (hv.norm_eq_one i) x).trans hx |
| rw [inner_sub_right, abs_le] at hi |
| exact ⟨by dsimp [c, labels]; linarith [hi.1], by dsimp [c, labels]; linarith [hi.2]⟩ |
| calc |
| _ ≤ (base.measure d) ((labels v) ⁻¹' s) := measure_mono hsub |
| _ = ((base.measure d).map (labels v)) s := |
| (Measure.map_apply (by unfold labels; fun_prop) hs).symm |
| _ = ∫⁻ y in s, baseDensity base v y := by |
| rw [projected_base_measure hv.linearIndependent (by omega) base, withDensity_apply _ hs] |
| _ ≤ ∫⁻ _y in s, ENNReal.ofReal (baseProjectionConstant base d q) := |
| setLIntegral_mono' hs (fun y _ => baseDensity_le_orthonormal hq base hv y) |
| _ = ENNReal.ofReal (baseProjectionConstant base d q) * ENNReal.ofReal (2*r)^q := by |
| rw [lintegral_const, Measure.restrict_apply_univ] |
| congr 1 |
| change (Measure.pi (fun _ : Fin q => volume)) s = _ |
| rw [Measure.pi_pi] |
| simp only [Real.volume_Icc, show ∀ i, c i + r - (c i - r) = 2*r from fun _ => by ring] |
| simp |
| _ = _ := by |
| rw [ENNReal.ofReal_mul (by norm_num), mul_pow, ENNReal.ofReal_ofNat, |
| ENNReal.ofReal_pow hr.le, mul_assoc] |
|
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| end PosteriorReplicas |
| end |
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| end OAI |
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