| import Mathlib |
| import OAI.Probability.LogConcave.LowerBounds.PrefixSpace |
|
|
| 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 |
|
|
| namespace LogConcaveSampling.LowerBound |
|
|
| def frameTransport {d : ℕ} (F : Frames d) : Point d ≃ₗᵢ[ℝ] Point d := |
| rotationIsometry (frameCompletion F 1) |
|
|
| @[fun_prop] lemma measurable_frameTransport {d : ℕ} : |
| Measurable (fun p : Frames d × Point d => frameTransport p.1 p.2) := by |
| have continuous_completion : Continuous |
| (fun pair : Frames d × Point d => frameCompletion pair.1 1) := by |
| unfold frameCompletion |
| exact (continuous_fst.fst.mul continuous_const).mul continuous_fst.snd.inv |
| exact ((continuous_rotation_apply (d := d)).comp |
| (continuous_completion.prodMk continuous_snd)).measurable |
|
|
| lemma frameCompletion_apply {d : ℕ} (F : Frames d) (H : Rotations d) (x : Point d) : |
| rotationIsometry (frameCompletion F H) x = |
| rotationIsometry F.1 (rotationIsometry H (rotationIsometry F.2⁻¹ x)) := by |
| simp only [frameCompletion, rotationIsometry_mul] |
|
|
| lemma frameCompletion_eq_transport {d : ℕ} (E : Submodule ℝ (Point d)) |
| (F : Frames d) {H : Rotations d} (hH : H ∈ subspaceStabilizer E) |
| {x : Point d} (hx : x ∈ frameSpace F.2 E) : |
| rotationIsometry (frameCompletion F H) x = frameTransport F x := by |
| rw [frameCompletion_apply, hH _ ((mem_frameSpace F.2 E x).mp hx)] |
| simp only [frameTransport, frameCompletion_apply, rotationIsometry_one] |
|
|
| lemma frameTransport_swap {d : ℕ} (F : Frames d) (x : Point d) : |
| frameTransport F.swap x = (frameTransport F).symm x := by |
| unfold frameTransport |
| rw [← rotationIsometry_inv_apply] |
| have h : frameCompletion F.swap 1 = (frameCompletion F 1)⁻¹ := by |
| simpa only [inv_one] using frameCompletion_swap F 1 |
| rw [h] |
|
|
| lemma frameCompletion_symm_eq_transport {d : ℕ} (E : Submodule ℝ (Point d)) |
| (F : Frames d) {H : Rotations d} (hH : H ∈ subspaceStabilizer E) |
| {x : Point d} (hx : x ∈ frameSpace F.1 E) : |
| (rotationIsometry (frameCompletion F H)).symm x = (frameTransport F).symm x := by |
| rw [← rotationIsometry_inv_apply, ← frameCompletion_swap] |
| rw [frameCompletion_eq_transport E F.swap ((subspaceStabilizer E).inv_mem hH) hx, |
| frameTransport_swap] |
|
|
| lemma frameTransport_preimage_mem {d : ℕ} (E : Submodule ℝ (Point d)) |
| (F : Frames d) {x : Point d} (hx : x ∈ frameSpace F.1 E) : |
| (frameTransport F).symm x ∈ frameSpace F.2 E := by |
| rw [mem_frameSpace] at hx ⊢ |
| rw [← frameTransport_swap, frameTransport, frameCompletion_apply, rotationIsometry_one] |
| simp only [rotationIsometry_inv_apply] |
| change (rotationIsometry F.2).symm (rotationIsometry F.2 ((rotationIsometry F.1).symm x)) ∈ E |
| rw [LinearIsometryEquiv.symm_apply_apply] |
| simpa only [rotationIsometry_inv_apply] using hx |
|
|
| lemma extendFrame_relative_mem {d k : ℕ} (hk : k < d) (O : Rotations d) (x : Point d) : |
| O⁻¹ * extendFrame hk O x ∈ subspaceStabilizer (prefixSpace d k) := by |
| simpa only [extendFrame, inv_mul_cancel_left] using |
| (complementSection (prefixSpace d k) (prefixUnit hk) |
| (flagDirection hk (rotationIsometry O⁻¹ x))).property |
|
|
| def CompatibleFrames {d : ℕ} (E : Submodule ℝ (Point d)) (F F' : Frames d) : Prop := |
| F.1⁻¹ * F'.1 ∈ subspaceStabilizer E ∧ F.2⁻¹ * F'.2 ∈ subspaceStabilizer E |
|
|
| lemma compatibleFrames_reconstruct {d : ℕ} (E : Submodule ℝ (Point d)) |
| {F F' : Frames d} (h : CompatibleFrames E F F') {H : Rotations d} |
| (hH : H ∈ subspaceStabilizer E) : |
| ∃ H' ∈ subspaceStabilizer E, frameCompletion F' H = frameCompletion F H' := by |
| refine ⟨(F.1⁻¹*F'.1)*H*(F.2⁻¹*F'.2)⁻¹, |
| (subspaceStabilizer E).mul_mem ((subspaceStabilizer E).mul_mem h.1 hH) |
| ((subspaceStabilizer E).inv_mem h.2),?_⟩ |
| unfold frameCompletion |
| group |
|
|
| lemma compatibleFrames_inverseUpdate {d k : ℕ} (hk : k < d) (F : Frames d) (x g : Point d) : |
| CompatibleFrames (prefixSpace d k) F (inverseFrameUpdate hk F x g) := by |
| rw [inverseFrameUpdate_eq] |
| exact ⟨extendFrame_relative_mem hk _ _, extendFrame_relative_mem hk _ _⟩ |
|
|
| lemma compatibleFrames_forwardUpdate {d k : ℕ} (hk : k < d) (F : Frames d) (y g : Point d) : |
| CompatibleFrames (prefixSpace d k) F (forwardFrameUpdate hk F y g) := by |
| rw [forwardFrameUpdate_eq] |
| exact ⟨extendFrame_relative_mem hk _ _, extendFrame_relative_mem hk _ _⟩ |
|
|
| lemma subspaceStabilizer_anti {d : ℕ} {E F : Submodule ℝ (Point d)} (h : E ≤ F) : |
| subspaceStabilizer F ≤ subspaceStabilizer E := by |
| intro O hO x hx |
| exact hO x (h hx) |
|
|
| lemma compatibleFrames_mono {d : ℕ} {E G : Submodule ℝ (Point d)} (h : E ≤ G) |
| {F F' : Frames d} (hc : CompatibleFrames G F F') : CompatibleFrames E F F' := |
| ⟨subspaceStabilizer_anti h hc.1, subspaceStabilizer_anti h hc.2⟩ |
|
|
| lemma compatibleFrames_trans {d : ℕ} {E : Submodule ℝ (Point d)} {F F' F'' : Frames d} |
| (h : CompatibleFrames E F F') (h' : CompatibleFrames E F' F'') : |
| CompatibleFrames E F F'' := by |
| constructor |
| · have hm := (subspaceStabilizer E).mul_mem h.1 h'.1 |
| simpa only [mul_assoc, mul_inv_cancel_left] using hm |
| · have hm := (subspaceStabilizer E).mul_mem h.2 h'.2 |
| simpa only [mul_assoc, mul_inv_cancel_left] using hm |
|
|
| end LogConcaveSampling.LowerBound |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| end LowerProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
