| import Mathlib |
| import OAI.Computability.VertexCover.Analysis.ZeroMemIncidenceHull |
| import OAI.Computability.VertexCover.Analysis.DistanceMinusSelect |
| import OAI.Computability.VertexCover.Analysis.AeRegular |
|
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| namespace OAI |
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| namespace VertexCover.LabelCover |
| open MeasureTheory |
|
|
| theorem separator_continuous (Φ : LabelCover) {d : ℕ} |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : |
| Continuous (Φ.separator A hA) := |
| ((Φ.distanceTo_continuous A hA).comp continuous_neg |>.sub |
| (Φ.distanceTo_continuous A hA)).div_const 2 |
|
|
| noncomputable def batchFunction (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) : ℝ := |
| Φ.separator A hA (c0 + Φ.batchSum seed J s) |
|
|
| theorem batchFunction_continuous (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : |
| Continuous (Φ.batchFunction seed J c0 A hA) := |
| (Φ.separator_continuous A hA).comp |
| (continuous_const.add (Φ.batchSumCLM seed J).continuous) |
|
|
| theorem batchFunction_hasFDerivAt (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) |
| (hs : VertexCover.AffineSelector.Regular (Φ.affineSlope seed J) |
| (Φ.affineOffset c0 (A := A)) s) : |
| ∃ im ip : Φ.AffineIndex A, HasFDerivAt (Φ.batchFunction seed J c0 A hA) |
| ((1/2 : ℝ) • (Φ.affineSlope seed J im - Φ.affineSlope seed J ip)) s := by |
| have hc : Continuous (fun s : Φ.BatchWeights J => c0 + Φ.batchSum seed J s) := |
| continuous_const.add (Φ.batchSumCLM seed J).continuous |
| obtain ⟨ip, hp⟩ := VertexCover.AffineSelector.hasFDerivAt _ _ |
| ((Φ.distanceTo_continuous A hA).comp hc) |
| (Φ.distance_plus_select seed J c0 A hA) hs |
| obtain ⟨im, hm⟩ := VertexCover.AffineSelector.hasFDerivAt _ _ |
| ((Φ.distanceTo_continuous A hA).comp hc.neg) |
| (Φ.distance_minus_select seed J c0 A hA) hs |
| refine ⟨im, ip, ?_⟩ |
| have heq : Φ.batchFunction seed J c0 A hA = (fun y => (1/2 : ℝ) * |
| (Φ.distanceTo A hA (-(c0 + Φ.batchSum seed J y)) - |
| Φ.distanceTo A hA (c0 + Φ.batchSum seed J y))) := by |
| funext y |
| unfold batchFunction separator |
| ring |
| rw [heq] |
| exact (hm.sub hp).const_mul (1/2 : ℝ) |
|
|
| def BatchGradientValid (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) (g : Φ.BatchWeights J) : Prop := |
| ∃ G ∈ Φ.incidenceHull seed, ∀ (j : J) k, g j k = G j k |
|
|
| theorem signed_incidence_mem (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| {I : Finset (Φ.Coordinate d)} (hI : Φ.Compatible I) (e : Bool) : |
| affineSign e • Φ.incidence seed I ∈ Φ.incidenceHull seed := by |
| apply subset_convexHull ℝ |
| cases e |
| · exact ⟨I, hI, Or.inr (by simp)⟩ |
| · exact ⟨I, hI, Or.inl (by simp)⟩ |
|
|
| theorem neg_signed_incidence_mem (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| {I : Finset (Φ.Coordinate d)} (hI : Φ.Compatible I) (e : Bool) : |
| -(affineSign e • Φ.incidence seed I) ∈ Φ.incidenceHull seed := by |
| apply subset_convexHull ℝ |
| cases e |
| · exact ⟨I, hI, Or.inl (by simp)⟩ |
| · exact ⟨I, hI, Or.inr (by simp)⟩ |
|
|
| theorem batchGradient_affineSlope (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) {A : Finset (Φ.Coordinate d → ℝ)} |
| (i : Φ.AffineIndex A) (j : J) (k : Fin (Φ.WeightDimension d)) : |
| Φ.batchGradient J (Φ.affineSlope seed J i) j k = |
| affineSign i.2.2.2 * Φ.incidence seed i.2.1.val j k := by |
| change affineSign i.2.2.2 * |
| Φ.batchGradient J ((Φ.selectionForm i.2.1.val).comp (Φ.batchSumCLM seed J)) j k = _ |
| rw [Φ.batchGradient_form seed J ((Φ.mem_compatibleSelections _).mp i.2.1.property)] |
|
|
| theorem batchGradient_half_difference_valid (Φ : LabelCover) {d : ℕ} |
| (seed : Φ.Seeds d) (J : Finset (Fin d)) {A : Finset (Φ.Coordinate d → ℝ)} |
| (im ip : Φ.AffineIndex A) : |
| Φ.BatchGradientValid seed J (Φ.batchGradient J |
| ((1/2 : ℝ) • (Φ.affineSlope seed J im - Φ.affineSlope seed J ip))) := by |
| let gm := affineSign im.2.2.2 • Φ.incidence seed im.2.1.val |
| let gp := -(affineSign ip.2.2.2 • Φ.incidence seed ip.2.1.val) |
| have hm := Φ.signed_incidence_mem seed |
| ((Φ.mem_compatibleSelections _).mp im.2.1.property) im.2.2.2 |
| have hp := Φ.neg_signed_incidence_mem seed |
| ((Φ.mem_compatibleSelections _).mp ip.2.1.property) ip.2.2.2 |
| refine ⟨(1/2 : ℝ) • gm + (1/2 : ℝ) • gp, |
| (convex_convexHull ℝ _) hm hp (by norm_num) (by norm_num) (by norm_num), ?_⟩ |
| intro j k |
| change (1/2 : ℝ) * (Φ.batchGradient J (Φ.affineSlope seed J im) j k - |
| Φ.batchGradient J (Φ.affineSlope seed J ip) j k) = _ |
| rw [Φ.batchGradient_affineSlope, Φ.batchGradient_affineSlope] |
| simp only [gm, gp, Pi.add_apply, Pi.smul_apply, Pi.neg_apply, smul_eq_mul] |
| ring |
|
|
| noncomputable def canonicalGradient (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) : |
| Φ.BatchWeights J := by |
| classical |
| exact if VertexCover.AffineSelector.Regular (Φ.affineSlope seed J) |
| (Φ.affineOffset c0 (A := A)) s then |
| Φ.batchGradient J (fderiv ℝ (Φ.batchFunction seed J c0 A hA) s) else 0 |
|
|
| theorem canonicalGradient_valid (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) : |
| Φ.BatchGradientValid seed J (Φ.canonicalGradient seed J c0 A hA s) := by |
| classical |
| unfold canonicalGradient |
| split_ifs with hs |
| · obtain ⟨im, ip, hder⟩ := Φ.batchFunction_hasFDerivAt seed J c0 A hA s hs |
| rw [hder.fderiv] |
| exact Φ.batchGradient_half_difference_valid seed J im ip |
| · exact ⟨0, Φ.zero_mem_incidenceHull seed, fun _ _ => rfl⟩ |
|
|
| end VertexCover.LabelCover |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
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| end |
| end |
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| end |
| end |
| end |
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
