File size: 1,672 Bytes
f205175 | 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 | import OAI.Probability.CubeShuffle.FourierBound
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace CubeShuffle.WeightedSweep
open Specht UnitaryFinite
def dimension (μ : YoungDiagram) : ℝ := Module.finrank ℂ (space μ)
def sweepLaw (d : ℕ) : Equiv.Perm (Card d) → ℝ := sampleLaw (run d d)
lemma sweepLaw_nonneg (d : ℕ) (g : Equiv.Perm (Card d)) : 0 ≤ sweepLaw d g :=
sampleLaw_nonneg _ _
lemma sweepLaw_sum (d : ℕ) : ∑ g, sweepLaw d g = 1 := sampleLaw_sum _
def fourierSweep (d : ℕ) (μ : Shapes (2^d)) :
hilbertSpace μ.1 →L[ℂ] hilbertSpace μ.1 :=
sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d d)
lemma fourierSweep_eq_group_sum (d : ℕ) (μ : Shapes (2 ^ d)) :
fourierSweep d μ = ∑ g : Equiv.Perm (Card d), (sweepLaw d g : ℂ) •
continuousRepresentation (V := hilbertSpace μ.1) (cardRepresentation d μ) g := by
let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance
let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance
let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance
exact sampleOperator_weighted (V := hilbertSpace μ.1) (cardRepresentation d μ) (run d d)
def schattenMoment {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] (A : V →L[ℂ] V) (p : ℝ) : ℝ :=
∑ i : Fin (Module.finrank ℂ V), (A.toLinearMap.singularValues i) ^ p
def schattenNorm {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] (A : V →L[ℂ] V) (p : ℝ) : ℝ :=
schattenMoment A p ^ (1 / p)
end CubeShuffle.WeightedSweep
end
end OAI
|