| import Mathlib |
| import OAI.Probability.LogConcave.LowerBounds.FrameTransport |
| import OAI.Probability.LogConcave.Sampling.StateLaw |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section LowerProof |
| open Matrix Topology TopologicalSpace ProbabilityTheory Classical WithLp |
| open scoped Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory |
|
|
| namespace LogConcaveSampling.LowerBound |
|
|
| def roundFrames {d k : ℕ} (hk : k+1 < d) (Λ : Point d →L[ℝ] Point d) |
| (F : Frames d) (x : Point d) (r : Point d × Point d × Point d) : Frames d := |
| let F₁ := inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x r.1 |
| let u := (frameTransport F₁).symm x |
| let y := Λ u |
| forwardFrameUpdate hk F₁ (filledObservation (k := k+1) F₁.2 y r.2.1) r.2.2 |
|
|
| @[fun_prop] lemma measurable_inverse_frameTransport {d : ℕ} : |
| Measurable (fun p : Frames d × Point d => (frameTransport p.1).symm p.2) := by |
| simp only [← frameTransport_swap] |
| have h := (measurable_frameTransport (d := d)).comp |
| (show Measurable (fun p : Frames d × Point d => (p.1.swap,p.2)) by fun_prop) |
| exact h |
|
|
| @[fun_prop] lemma measurable_roundFrames {d k : ℕ} (hk : k+1 < d) (Λ : Point d →L[ℝ] Point d) : |
| Measurable (fun p : (Frames d × Point d) × (Point d × Point d × Point d) => |
| roundFrames hk Λ p.1.1 p.1.2 p.2) := by |
| let Z := (Frames d × Point d) × (Point d × Point d × Point d) |
| let f : Z → Frames d := fun p => inverseFrameUpdate (Nat.lt_of_succ_lt hk) p.1.1 p.1.2 p.2.1 |
| have hf : Measurable f := (measurable_inverseFrameUpdate (Nat.lt_of_succ_lt hk)).comp |
| (f := fun p : Z => (p.1.1,p.1.2,p.2.1)) ((measurable_fst.fst).prodMk (measurable_fst.snd.prodMk measurable_snd.fst)) |
| let u : Z → Point d := fun p => (frameTransport (f p)).symm p.1.2 |
| have hu : Measurable u := (measurable_inverse_frameTransport (d := d)).comp |
| (f := fun p : Z => (f p,p.1.2)) (hf.prodMk measurable_fst.snd) |
| let y : Z → Point d := fun p => Λ (u p) |
| have hy : Measurable y := Λ.continuous.measurable.comp (f := u) hu |
| let z : Z → Point d := fun p => filledObservation (k := k+1) (f p).2 (y p) p.2.2.1 |
| have hz : Measurable z := (measurable_filledObservation (d := d) (k := k+1)).comp |
| (f := fun p : Z => ((f p).2,y p,p.2.2.1)) (hf.snd.prodMk (hy.prodMk measurable_snd.snd.fst)) |
| change Measurable (fun p : Z => forwardFrameUpdate hk (f p) (z p) p.2.2.2) |
| exact (measurable_forwardFrameUpdate hk).comp |
| (f := fun p : Z => (f p,z p,p.2.2.2)) (hf.prodMk (hz.prodMk measurable_snd.snd.snd)) |
|
|
| lemma filledForward_law {d k : ℕ} (hk : k < d) (F : Frames d) (y : Point d) : |
| (((stdGaussian (Point d)).prod (stdGaussian (Point d))).prod |
| (remainingHaar (prefixSpace d (k+1)))).map |
| (fun p => frameCompletion (forwardFrameUpdate hk F |
| (filledObservation (k := k) F.2 y p.1.1) p.1.2) p.2) = |
| (remainingHaar (prefixSpace d k)).map (frameCompletion F) := by |
| have ha := Measure.prodAssoc_prod (μ := stdGaussian (Point d)) (ν := stdGaussian (Point d)) |
| (τ := remainingHaar (prefixSpace d (k+1))) |
| calc |
| _ = ((stdGaussian (Point d)).prod ((stdGaussian (Point d)).prod |
| (remainingHaar (prefixSpace d (k+1))))).map |
| (fun p => frameCompletion (forwardFrameUpdate hk F |
| (filledObservation (k := k) F.2 y p.1) p.2.1) p.2.2) := by |
| rw [← ha, Measure.map_map (by fun_prop) MeasurableEquiv.prodAssoc.measurable] |
| rfl |
| _ = _ := by |
| apply map_prod_eq_of_constant_fiber_law (by fun_prop) |
| intro g |
| exact forwardFrameUpdate_law hk F (filledObservation (k := k) F.2 y g) |
|
|
| theorem roundFrames_law {d k : ℕ} (hk : k+1 < d) (Λ : Point d →L[ℝ] Point d) |
| (F : Frames d) (x : Point d) : |
| (((stdGaussian (Point d)).prod ((stdGaussian (Point d)).prod (stdGaussian (Point d)))).prod |
| (remainingHaar (prefixSpace d (k+2)))).map |
| (fun p => frameCompletion (roundFrames hk Λ F x p.1) p.2) = |
| (remainingHaar (prefixSpace d k)).map (frameCompletion F) := by |
| let f : (Point d × Point d × Point d) × Rotations d → Rotations d := |
| fun p => frameCompletion (roundFrames hk Λ F x p.1) p.2 |
| let g : Point d × Rotations d → Rotations d := |
| fun p => frameCompletion (inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x p.1) p.2 |
| have hf : Measurable f := (measurable_frameCompletion (d := d)).comp |
| (f := fun p : (Point d × Point d × Point d) × Rotations d => |
| (roundFrames hk Λ F x p.1,p.2)) |
| (((measurable_roundFrames hk Λ).comp |
| (f := fun p : (Point d × Point d × Point d) × Rotations d => ((F,x),p.1)) |
| (measurable_const.prodMk measurable_fst)).prodMk measurable_snd) |
| have hg : Measurable g := by unfold g; fun_prop |
| have hh (z : Point d) : |
| (((stdGaussian (Point d)).prod (stdGaussian (Point d))).prod |
| (remainingHaar (prefixSpace d (k+2)))).map (fun p => f ((z,p.1),p.2)) = |
| (remainingHaar (prefixSpace d (k+1))).map (fun h => g (z,h)) := by |
| exact filledForward_law hk (inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x z) |
| (Λ ((frameTransport (inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x z)).symm x)) |
| have he := eliminate_fresh_residual (μ := stdGaussian (Point d)) |
| (γ := (stdGaussian (Point d)).prod (stdGaussian (Point d))) |
| (η := remainingHaar (prefixSpace d (k+1))) |
| (η' := remainingHaar (prefixSpace d (k+2))) |
| (step := id) measurable_id hf hg hh |
| rw [Measure.map_id] at he |
| exact he.trans (inverseFrameUpdate_law (Nat.lt_of_succ_lt hk) F x) |
| |
| lemma roundFrames_compatible {d k : ℕ} (hk : k+1 < d) (Λ : Point d →L[ℝ] Point d) |
| (F : Frames d) (x : Point d) (r : Point d × Point d × Point d) : |
| CompatibleFrames (prefixSpace d k) F (roundFrames hk Λ F x r) := by |
| unfold roundFrames |
| exact compatibleFrames_trans (compatibleFrames_inverseUpdate _ _ _ _) |
| (compatibleFrames_mono (prefixSpace_mono d (Nat.le_succ k)) |
| (compatibleFrames_forwardUpdate _ _ _ _)) |
| |
| end LogConcaveSampling.LowerBound |
| |
| namespace LogConcaveSampling.LowerBound |
| open scoped RealInnerProductSpace |
| |
| def linearReply {d : ℕ} (Λ : Point d →L[ℝ] Point d) (O : Rotations d) (x : Point d) : |
| ℝ × Point d := |
| let u := (rotationIsometry O).symm x |
| (⟪u,Λ u⟫ / 2, rotationIsometry O (Λ u)) |
| |
| @[fun_prop] lemma measurable_linearReply {d : ℕ} (Λ : Point d →L[ℝ] Point d) : |
| Measurable (fun p : Rotations d × Point d => linearReply Λ p.1 p.2) := by |
| unfold linearReply |
| simp only [← rotationIsometry_inv_apply] |
| fun_prop |
| |
| def roundCoordinate {d k : ℕ} (hk : k+1 < d) (F : Frames d) (x g : Point d) : Point d := |
| (frameTransport (inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x g)).symm x |
| |
| @[fun_prop] lemma measurable_roundCoordinate {d k : ℕ} (hk : k+1 < d) : |
| Measurable (fun p : (Frames d × Point d) × Point d => roundCoordinate hk p.1.1 p.1.2 p.2) := by |
| let Z := (Frames d × Point d) × Point d |
| let f : Z → Frames d := fun p => inverseFrameUpdate (Nat.lt_of_succ_lt hk) p.1.1 p.1.2 p.2 |
| have hf : Measurable f := (measurable_inverseFrameUpdate (Nat.lt_of_succ_lt hk)).comp |
| (measurable_fst.fst.prodMk (measurable_fst.snd.prodMk measurable_snd)) |
| change Measurable (fun p : Z => (frameTransport (f p)).symm p.1.2) |
| exact (measurable_inverse_frameTransport (d := d)).comp |
| (f := fun p : Z => (f p,p.1.2)) (hf.prodMk measurable_fst.snd) |
| |
| lemma roundCoordinate_mem {d k : ℕ} (hk : k+1 < d) (F : Frames d) (x g : Point d) : |
| roundCoordinate hk F x g ∈ |
| frameSpace (inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x g).2 (prefixSpace d (k+1)) := by |
| apply frameTransport_preimage_mem |
| exact mem_frameSpace_extendFrame _ _ _ |
| |
| lemma replyCoordinate_mem {d k : ℕ} (hk : k+1 < d) (Λ : Point d →L[ℝ] Point d) |
| (F : Frames d) (x : Point d) (r : Point d × Point d × Point d) : |
| Λ (roundCoordinate hk F x r.1) ∈ frameSpace (roundFrames hk Λ F x r).2 (prefixSpace d (k+2)) := |
| mem_frameSpace_filledExtend hk _ _ _ |
| |
| theorem roundReply_correct {d k : ℕ} (hk : k+1 < d) (Λ : Point d →L[ℝ] Point d) |
| (F : Frames d) (x : Point d) (r : Point d × Point d × Point d) |
| {H : Rotations d} (hH : H ∈ subspaceStabilizer (prefixSpace d (k+2))) : |
| linearReply Λ (frameCompletion (roundFrames hk Λ F x r) H) x = |
| (⟪roundCoordinate hk F x r.1, Λ (roundCoordinate hk F x r.1)⟫ / 2, |
| frameTransport (roundFrames hk Λ F x r) (Λ (roundCoordinate hk F x r.1))) := by |
| let F₁ := inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x r.1 |
| have hc : CompatibleFrames (prefixSpace d (k+1)) F₁ (roundFrames hk Λ F x r) := |
| compatibleFrames_forwardUpdate hk _ _ _ |
| obtain ⟨H',hH',he⟩ := compatibleFrames_reconstruct _ hc |
| (subspaceStabilizer_anti (prefixSpace_mono d (Nat.le_succ (k+1))) hH) |
| have hq : (rotationIsometry (frameCompletion (roundFrames hk Λ F x r) H)).symm x = |
| roundCoordinate hk F x r.1 := by |
| rw [he] |
| exact frameCompletion_symm_eq_transport _ F₁ hH' (mem_frameSpace_extendFrame _ _ _) |
| unfold linearReply |
| simp only [hq] |
| rw [frameCompletion_eq_transport _ _ hH (replyCoordinate_mem hk Λ F x r)] |
|
|
| theorem roundFrames_span_growth {d k : ℕ} (hk : k+1 < d) (Λ : Point d →L[ℝ] Point d) |
| (F : Frames d) (x : Point d) (r : Point d × Point d × Point d) |
| (hq : (prefixSpace d k)ᗮ.orthogonalProjectionOnto (rotationIsometry F.2⁻¹ r.1) ≠ 0) |
| (hr : (prefixSpace d (k+1))ᗮ.orthogonalProjectionOnto |
| (rotationIsometry (inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x r.1).2⁻¹ r.2.1) ≠ 0) : |
| frameSpace (roundFrames hk Λ F x r).2 (prefixSpace d (k+2)) ≤ |
| ((frameSpace F.2 (prefixSpace d k) ⊔ ℝ ∙ r.1) ⊔ |
| (frameSpace F.2 (prefixSpace d k) ⊔ ℝ ∙ r.1).map Λ.toLinearMap) ⊔ ℝ ∙ r.2.1 := by |
| let F₁ := inverseFrameUpdate (Nat.lt_of_succ_lt hk) F x r.1 |
| let K := frameSpace F.2 (prefixSpace d k) ⊔ ℝ ∙ r.1 |
| have hK : frameSpace F₁.2 (prefixSpace d (k+1)) ≤ K := |
| frameSpace_extendFrame_le _ _ _ hq |
| have hu : roundCoordinate hk F x r.1 ∈ K := hK (roundCoordinate_mem hk F x r.1) |
| have hΛ : Λ (roundCoordinate hk F x r.1) ∈ K.map Λ.toLinearMap := ⟨_,hu,rfl⟩ |
| have hy : ℝ ∙ Λ (roundCoordinate hk F x r.1) ≤ K.map Λ.toLinearMap := by |
| apply Submodule.span_le.mpr |
| intro y hy |
| rcases Set.mem_singleton_iff.mp hy with rfl |
| exact hΛ |
| exact (frameSpace_filledExtend_le hk F₁.2 _ _ hr).trans |
| (sup_le_sup (sup_le_sup hK hy) le_rfl) |
|
|
| end LogConcaveSampling.LowerBound |
|
|
| end LowerProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
