File size: 4,036 Bytes
a89d282 | 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 | import OAI.Probability.InvariantIsing.Spectral.SelectedTypePressure
import OAI.Probability.InvariantIsing.Spectral.CompactPartitionPressure
namespace OAI
/-! Spectral rounding for a sequence of selected dimensions. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology Classical Function
namespace InvariantIsing
theorem selected_partition_pressure_tendsto
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(hpub : PanchenkoTalagrandFieldPairInput) {ι : Type*} [Fintype ι]
(μ : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (μ N)]
[∀ N, (μ N).IsMulRightInvariant]
(D : ℕ → ℕ) (hD : ∀ k, 0 < D k) (hDlim : Tendsto D atTop atTop)
(eig : (k : ℕ) → Fin (D k) → ℝ) (ν : ProbabilityMeasure ℝ)
(hweak : Tendsto (fun k => empiricalSpectralLaw (hD k) (eig k)) atTop (𝓝 ν))
(S : ι → Set ℝ) (hcover : ∀ x, ∃ i, x∈S i) (hdis : Pairwise (Disjoint on S))
(hS : ∀ i, MeasurableSet (S i)) (hbd : ∀ i, (ν : Measure ℝ) (frontier (S i))=0)
(lam : ι → ℝ) (fallback aMax : {i // 0 < (ν : Measure ℝ).real (S i)})
(hMax : ∀ a : {i // 0 < (ν : Measure ℝ).real (S i)}, lam a ≤ lam aMax) :
Tendsto (fun k => ∫ V, rotatedPressure
(fun i => lam (positiveSpectralLabel (fun a => (ν : Measure ℝ).real (S a)) fallback
(spectralPartitionIndex S hcover (eig k i))))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ (D k)) atTop
(𝓝 (variationalFunctional (measureR
(finiteSpectralMeasure (fun i : {i // 0 < (ν : Measure ℝ).real (S i)} =>
(ν : Measure ℝ).real (S i)) (fun i => lam i)) (lam aMax))).toReal) := by
let w := fun a => (ν : Measure ℝ).real (S a)
let g := fun k (i : Fin (D k)) => spectralPartitionIndex S hcover (eig k i)
have hc := spectral_label_counts_partition D hD eig ν hweak S hS hbd g
(fun k a i => spectralPartitionIndex_eq_iff S hcover hdis (eig k i) a)
have hp := positive_spectral_counts_tendsto D g w (fun _ => measureReal_nonneg) fallback hc
exact selected_finite_type_pressure_tendsto hhaar hgauss hpub μ
(fun a : {i // 0 < w i} => lam a) D hD hDlim
(fun k => positiveSpectralLabel w fallback ∘ g k) (fun a : {i // 0 < w i} => w a)
(fun a => a.property) (positiveSpectralWeights_sum w (fun _ => measureReal_nonneg)
(spectralPartitionWeights_sum (ν : Measure ℝ) S hcover hdis hS)) hp aMax hMax
namespace CompactSpectralPartition
lemma selected_pressure_tendsto {ν : ProbabilityMeasure ℝ} {a b δ : ℝ}
(P : CompactSpectralPartition ν a b δ)
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(hpub : PanchenkoTalagrandFieldPairInput)
(μ : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (μ N)]
[∀ N, (μ N).IsMulRightInvariant]
(D : ℕ → ℕ) (hD : ∀ k, 0 < D k) (hDlim : Tendsto D atTop atTop)
(eig : (k : ℕ) → Fin (D k) → ℝ)
(hweak : Tendsto (fun k => empiricalSpectralLaw (hD k) (eig k)) atTop (𝓝 ν)) :
Tendsto (fun k => ∫ V, rotatedPressure (fun i => P.lowerValue (eig k i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ (D k)) atTop
(𝓝 (variationalFunctional (measureR (P.law : Measure ℝ) P.edge)).toReal) ∧
Tendsto (fun k => ∫ V, rotatedPressure (fun i => P.upperValue (eig k i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ (D k)) atTop
(𝓝 (variationalFunctional (measureR (P.law : Measure ℝ) P.edge)).toReal) := by
have hh (fallback : P.Positive) := selected_partition_pressure_tendsto hhaar hgauss hpub μ
D hD hDlim eig ν hweak P.cells (spectralBallCell_cover _) (spectralBallCell_disjoint _)
(spectralBallCell_measurable _ (fun _ => measurableSet_ball))
(spectralBallCell_null_boundary (ν : Measure ℝ) _ P.null_boundary)
P.value fallback P.upper P.le_upper
exact ⟨hh P.lower,hh P.upper⟩
end CompactSpectralPartition
end InvariantIsing
end
end OAI
|