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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import OAI.Computability.PerfectCompleteness.Sampling.BucketResamplingLaw
import OAI.Computability.PerfectCompleteness.Sampling.BucketUniformLemmas
namespace OAI
namespace PerfectCompleteness.BucketMatrixResampling
noncomputable section
open scoped Classical
open UniqueGamesTheorem.Foundations.Games
open BucketSampler (F2 Direction Tape)
variable {Ω : Type*} (V : Submodule F2 (Ω → F2))
abbrev Matrix (ℓ : Nat) := Module.Dual F2 V →ₗ[F2] (Fin ℓ → F2)
def assembledMatrix (ℓ : Nat) (t : Tape ℓ V) : Matrix V ℓ :=
EvaluationMatrix.ofRows V (BucketSampler.evaluate ℓ (id : V → V) t)
theorem assembledMatrix_update (ℓ : Nat) (t : Tape ℓ V)
(a : Direction ℓ) (new : V) :
assembledMatrix V ℓ (Function.update t a new) =
EvaluationMatrix.shift V (assembledMatrix V ℓ t) a.val (new - t a) := by
unfold assembledMatrix
rw [BucketSampler.evaluate_update]
apply LinearMap.ext
intro q
funext i
change q (BucketSampler.evaluate ℓ (id : V → V) t i + a.val i • (new - t a)) =
q (BucketSampler.evaluate ℓ (id : V → V) t i) + q (new - t a) * a.val i
simp only [map_add, map_smul, smul_eq_mul, mul_comm]
private theorem pushforward_id {A : Type*} [Fintype A] (μ : FiniteDistribution A) :
μ.pushforward (fun a => a) = μ := by
apply FiniteDistribution.eq_of_weight_eq
intro a
simp [FiniteDistribution.pushforward]
variable (ℓ : Nat) [Fintype V] [Fintype (Matrix V ℓ)]
theorem matrix_difference_law (μ : FiniteDistribution (Tape ℓ V)) (a : Direction ℓ) :
(μ.product (FiniteDistribution.uniform V)).pushforward
(fun p => (assembledMatrix V ℓ p.1, p.2 - p.1 a)) =
(μ.pushforward (assembledMatrix V ℓ)).product (FiniteDistribution.uniform V) := by
rw [UniformDifference.observed_difference_law μ (fun t => t a)
(assembledMatrix V ℓ)]
have hp := FiniteDistribution.product_pushforward μ
(FiniteDistribution.uniform V) (assembledMatrix V ℓ) (id : V → V)
have hi : (FiniteDistribution.uniform V).pushforward (id : V → V) =
FiniteDistribution.uniform V :=
pushforward_id (FiniteDistribution.uniform V)
rw [hi] at hp
simpa only [id_eq] using hp
theorem resampled_matrix_pair_law (μ : FiniteDistribution (Tape ℓ V)) (a : Direction ℓ) :
(μ.product (FiniteDistribution.uniform V)).pushforward
(fun p => (assembledMatrix V ℓ p.1,
assembledMatrix V ℓ (Function.update p.1 a p.2))) =
((μ.pushforward (assembledMatrix V ℓ)).product (FiniteDistribution.uniform V)).pushforward
(fun p => (p.1, EvaluationMatrix.shift V p.1 a.val p.2)) := by
have h := congrArg
(fun ν : FiniteDistribution (Matrix V ℓ × V) => ν.pushforward
(fun p => (p.1, EvaluationMatrix.shift V p.1 a.val p.2)))
(matrix_difference_law V ℓ μ a)
rw [FiniteDistribution.pushforward_comp] at h
simpa only [assembledMatrix_update] using h
variable [FiniteDimensional F2 V]
theorem uniform_assembledMatrix_law :
(BucketSampler.tapeLaw ℓ (FiniteDistribution.uniform V)).pushforward
(assembledMatrix V ℓ) = FiniteDistribution.uniform (Matrix V ℓ) :=
BucketUniform.uniform_matrix_law V ℓ
theorem uniform_matrix_difference_law (a : Direction ℓ) :
((BucketSampler.tapeLaw ℓ (FiniteDistribution.uniform V)).product
(FiniteDistribution.uniform V)).pushforward
(fun p => (assembledMatrix V ℓ p.1, p.2 - p.1 a)) =
(FiniteDistribution.uniform (Matrix V ℓ)).product (FiniteDistribution.uniform V) := by
rw [matrix_difference_law, uniform_assembledMatrix_law]
theorem uniform_resampled_matrix_pair_law (a : Direction ℓ) :
((BucketSampler.tapeLaw ℓ (FiniteDistribution.uniform V)).product
(FiniteDistribution.uniform V)).pushforward
(fun p => (assembledMatrix V ℓ p.1,
assembledMatrix V ℓ (Function.update p.1 a p.2))) =
((FiniteDistribution.uniform (Matrix V ℓ)).product
(FiniteDistribution.uniform V)).pushforward
(fun p => (p.1, EvaluationMatrix.shift V p.1 a.val p.2)) := by
rw [resampled_matrix_pair_law, uniform_assembledMatrix_law]
theorem uniform_resampled_event_probability (a : Direction ℓ)
(event : Matrix V ℓ × Matrix V ℓ → Bool) :
((BucketSampler.tapeLaw ℓ (FiniteDistribution.uniform V)).product
(FiniteDistribution.uniform V)).probability
(fun p => event (assembledMatrix V ℓ p.1,
assembledMatrix V ℓ (Function.update p.1 a p.2))) =
((FiniteDistribution.uniform (Matrix V ℓ)).product
(FiniteDistribution.uniform V)).probability
(fun p => event (p.1, EvaluationMatrix.shift V p.1 a.val p.2)) := by
have h := congrArg (fun ν => FiniteDistribution.probability ν event)
(uniform_resampled_matrix_pair_law V ℓ a)
simpa only [FiniteDistribution.probability_pushforward] using h
end
end PerfectCompleteness.BucketMatrixResampling
end OAI
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