| import Mathlib | |
| import OAI.Probability.SKGap.Localization.QuadraticCoeff | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set | |
| open scoped NNReal ENNReal | |
| open Filter | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Real Set Filter | |
| open MeasureTheory.Measure | |
| open scoped ENNReal NNReal MeasureTheory Topology | |
| open MeasureTheory | |
| noncomputable section | |
| noncomputable section | |
| open MeasureTheory Set NormedSpace | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Set Real | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Set Filter | |
| open scoped Topology Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Matrix NormedSpace ContinuousLinearMap | |
| open scoped Matrix.Norms.Frobenius | |
| noncomputable section | |
| open Matrix | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Real Set | |
| open scoped ENNReal NNReal | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set | |
| open scoped NNReal ENNReal | |
| noncomputable section | |
| open scoped BigOperators | |
| open MeasureTheory ProbabilityTheory | |
| namespace SKGap | |
| section DiagonalGeometry | |
| variable {ι : Type*} [Fintype ι] | |
| def diagonalSqrt (a : ι → ℝ) (p : EuclideanSpace ℝ ι) : EuclideanSpace ℝ ι := | |
| WithLp.toLp 2 (fun i => Real.sqrt (a i) * p i) | |
| lemma diagonalSqrt_norm_sq (a : ι → ℝ) (ha : ∀ i, 0 ≤ a i) (p : EuclideanSpace ℝ ι) : | |
| ‖diagonalSqrt a p‖^2 = ∑ i, a i * (p i)^2 := by | |
| rw [EuclideanSpace.norm_sq_eq] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| simp only [diagonalSqrt, WithLp.ofLp_toLp, Real.norm_eq_abs, sq_abs, mul_pow, | |
| Real.sq_sqrt (ha i)] | |
| lemma diagonalSqrt_norm_sq_le {a : ι → ℝ} {A : ℝ} | |
| (ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A) (p : EuclideanSpace ℝ ι) : | |
| ‖diagonalSqrt a p‖^2 ≤ A * ‖p‖^2 := by | |
| rw [diagonalSqrt_norm_sq a ha, ← sum_coord_sq, Finset.mul_sum] | |
| exact Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_right (hA i) (sq_nonneg _)) | |
| lemma diagonalSqrt_norm_le_sqrt {a : ι → ℝ} {A : ℝ} | |
| (ha : ∀ i, 0 ≤ a i) (hA0 : 0 ≤ A) (hA : ∀ i, a i ≤ A) | |
| {p : EuclideanSpace ℝ ι} (hp : ‖p‖ ≤ 1) : | |
| ‖diagonalSqrt a p‖ ≤ Real.sqrt A := by | |
| have h := diagonalSqrt_norm_sq_le ha hA p | |
| have hp2 : ‖p‖^2 ≤ 1 := by nlinarith [norm_nonneg p] | |
| have hh := mul_le_mul_of_nonneg_left hp2 hA0 | |
| have hs := Real.sq_sqrt hA0 | |
| have hs0 := Real.sqrt_nonneg A | |
| nlinarith [norm_nonneg (diagonalSqrt a p)] | |
| omit [Fintype ι] in | |
| lemma continuous_diagonalSqrt (a : ι → ℝ) : Continuous (diagonalSqrt a) := by | |
| apply (PiLp.continuous_toLp 2 (fun _ : ι => ℝ)).comp | |
| exact continuous_pi (fun i => continuous_const.mul (PiLp.continuous_apply 2 (fun _ => ℝ) i)) | |
| lemma diagonal_sum_mul_le_norm (a : ι → ℝ) (p g : EuclideanSpace ℝ ι) : | |
| (∑ i, diagonalSqrt a p i * g i) ≤ ‖p‖ * ‖diagonalSqrt a g‖ := by | |
| have he : (∑ i, diagonalSqrt a p i * g i) = inner ℝ p (diagonalSqrt a g) := by | |
| rw [PiLp.inner_apply] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| simp only [diagonalSqrt, WithLp.ofLp_toLp, RCLike.inner_apply, conj_trivial] | |
| ring | |
| rw [he] | |
| exact real_inner_le_norm _ _ | |
| lemma quadraticCoeff_norm {r : ℝ} (hr : 0 ≤ r) (w : EuclideanSpace ℝ ι) : | |
| ‖WithLp.toLp 2 (quadraticCoeff r w)‖ = Real.sqrt (2*r) * ‖w‖^2 := by | |
| have hsum : (∑ k, (quadraticCoeff r w k)^2) = 2*r*‖w‖^4 := by | |
| have h := quadraticCoeff_increment r hr w 0 | |
| have h0 : quadraticCoeff r (0 : EuclideanSpace ℝ ι) = 0 := by | |
| funext k | |
| cases k <;> simp [quadraticCoeff] | |
| simpa only [h0, Pi.zero_apply, sub_zero, norm_zero, zero_pow (by norm_num : (4 : ℕ) ≠ 0), | |
| WithLp.ofLp_zero, zero_mul, mul_zero, Finset.sum_const_zero, zero_pow (by norm_num : (2 : ℕ) ≠ 0), | |
| add_zero] using h | |
| have he : ‖WithLp.toLp 2 (quadraticCoeff r w)‖^2 = | |
| (Real.sqrt (2*r) * ‖w‖^2)^2 := by | |
| rw [← sum_coord_sq, hsum, mul_pow, Real.sq_sqrt (by positivity)] | |
| ring | |
| have hnon : 0 ≤ Real.sqrt (2*r) * ‖w‖^2 := by positivity | |
| nlinarith [norm_nonneg (WithLp.toLp 2 (quadraticCoeff r w))] | |
| end DiagonalGeometry | |
| lemma quadratic_margin {d b : ℝ} (_hd : 0 ≤ d) (hdb : d ≤ b) (hb : b ≤ 1) : | |
| 2*d-d^2 ≤ 1-(1-b)^2 := by | |
| nlinarith [mul_nonneg (sub_nonneg.mpr hdb) (show 0 ≤ 2-b-d by linarith)] | |
| end SKGap | |
| open Real | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |