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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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
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