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2f887db | 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 | import OAI.Computability.PerfectCompleteness.Foundations.ManyGoodRowsLemmas
import OAI.Computability.PerfectCompleteness.Foundations.RecursiveSpaceEquivLemmas
import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayTransportLemmas
namespace OAI
section
namespace PerfectCompleteness.GoodRowSampling
noncomputable section
open scoped BigOperators Classical
open UniqueGamesTheorem.Fourier.MatrixCharacters (F2)
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
open PerfectCompleteness.ManyGoodRows
attribute [local instance] linearMapFintype
variable {E F Y : Type*}
[AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F]
[FiniteDimensional F2 E] [FiniteDimensional F2 F]
[Fintype E] [Fintype F] [Fintype Y]
abbrev RowFiber {r : Nat} (A : RowMap F r) (U : E →ₗ[F2] (Fin r → F2)) :=
{X : E →ₗ[F2] F // A.comp X = U}
def rowFiberEquiv {r : Nat} (A : RowMap F r) (U : E →ₗ[F2] (Fin r → F2))
(T : E →ₗ[F2] F) (hT : A.comp T = U) :
MatrixSlice.Slice (⊥ : Submodule F2 E) (LinearMap.ker A) T ≃ RowFiber A U where
toFun X := ⟨X.val, (RowAdvice.slice_row_eq ⊥ A T X).trans hT⟩
invFun X := ⟨X.val, by
apply LinearMap.ext
intro x
have hx : (x : E) = 0 := (Submodule.mem_bot F2).mp x.property
simp [hx], (RowAdvice.comp_eq_iff_ker_quotient A X.val T).mp (X.property.trans hT.symm)⟩
left_inv X := by apply Subtype.ext; rfl
right_inv X := by apply Subtype.ext; rfl
omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype E] [Fintype F] in
@[simp] theorem rowFiberEquiv_val {r : Nat} (A : RowMap F r)
(U : E →ₗ[F2] (Fin r → F2)) (T : E →ₗ[F2] F) (hT : A.comp T = U)
(X : MatrixSlice.Slice (⊥ : Submodule F2 E) (LinearMap.ker A) T) :
(rowFiberEquiv A U T hT X).val = X.val := rfl
omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype Y] in
theorem witness_rowFiber_nonempty {f : (E →ₗ[F2] F) → Option Y}
{r : Nat} {ρ : ℝ} {A : RowMap F r} {U : E →ₗ[F2] (Fin r → F2)}
(w : Witness f r ρ A U) : Nonempty (RowFiber A U) :=
⟨⟨w.base, w.row_value⟩⟩
omit [Fintype Y] in
theorem witness_column_probability {f : (E →ₗ[F2] F) → Option Y}
{r : Nat} {ρ : ℝ} {A : RowMap F r} {U : E →ₗ[F2] (Fin r → F2)}
(w : Witness f r ρ A U) :
letI : Nonempty (RowFiber A U) := witness_rowFiber_nonempty w
1 / (2 : ℝ) ^ (r * Module.finrank F2 F) ≤
(FiniteDistribution.uniform (RowFiber A U)).probability
(fun X => decide (X.val.domRestrict w.columnSpace = w.base.domRestrict w.columnSpace)) := by
let : Nonempty (RowFiber A U) := witness_rowFiber_nonempty w
have h := SliceMass.probability_columnConstraint_ge_of_finrank_le
w.columnSpace (LinearMap.ker A) w.base r w.column_count
have he := UniformConditioning.uniform_probability_equiv
(rowFiberEquiv A U w.base w.row_value)
(fun X : RowFiber A U =>
decide (X.val.domRestrict w.columnSpace = w.base.domRestrict w.columnSpace))
apply h.trans_eq
refine Eq.trans ?_ he
congr 1
funext X
exact decide_eq_decide.mpr Iff.rfl
omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype Y] in
theorem witness_row_probability_pos {f : (E →ₗ[F2] F) → Option Y}
{r : Nat} {ρ : ℝ} {A : RowMap F r} {U : E →ₗ[F2] (Fin r → F2)}
(w : Witness f r ρ A U) :
0 < (FiniteDistribution.uniform (E →ₗ[F2] F)).probability
(fun X => decide (A.comp X = U)) := by
let : Nonempty (RowFiber A U) := witness_rowFiber_nonempty w
apply lt_of_lt_of_eq
(UniformConditioning.uniform_probability_pos (fun X : E →ₗ[F2] F => A.comp X = U))
congr 1
funext X
exact decide_eq_decide.mpr Iff.rfl
omit [Fintype Y] in
theorem witness_conditional_probability {f : (E →ₗ[F2] F) → Option Y}
{r : Nat} {ρ : ℝ} {A : RowMap F r} {U : E →ₗ[F2] (Fin r → F2)}
(w : Witness f r ρ A U) :
1 / (2 : ℝ) ^ (r * Module.finrank F2 F) ≤
((FiniteDistribution.uniform (E →ₗ[F2] F)).condition
(fun X => decide (A.comp X = U)) (witness_row_probability_pos w)).probability
(fun X => decide (X.domRestrict w.columnSpace = w.base.domRestrict w.columnSpace)) := by
let : Nonempty (RowFiber A U) := witness_rowFiber_nonempty w
have hevent : (fun X : E →ₗ[F2] F => decide (A.comp X = U)) =
(fun X : E →ₗ[F2] F => @decide (A.comp X = U) (Classical.propDecidable _)) := by
funext X
exact decide_eq_decide.mpr Iff.rfl
simp only [hevent, UniformConditioning.probability_condition_uniform]
convert witness_column_probability w using 1
congr!
end
end PerfectCompleteness.GoodRowSampling
end
end OAI
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