| import OAI.Probability.InvariantIsing.Gaussian.GaussianCompactVariance |
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| namespace OAI |
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| /-! Compact scalar cutoffs preserving the exact Lipschitz constant. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
| namespace InvariantIsing |
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| def gaussianCone {d : ℕ} (L : ℝ≥0) (R : ℝ) (x : EuclideanSpace ℝ (Fin d)) : ℝ := |
| max ((L : ℝ)*(R-‖x‖)) 0 |
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| def gaussianConeCutoff {d : ℕ} (L : ℝ≥0) (R : ℝ) |
| (f : EuclideanSpace ℝ (Fin d) → ℝ) (x : EuclideanSpace ℝ (Fin d)) : ℝ := |
| max (-gaussianCone L R x) (min (f x) (gaussianCone L R x)) |
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| lemma gaussianCone_nonneg {d : ℕ} (L : ℝ≥0) (R : ℝ) |
| (x : EuclideanSpace ℝ (Fin d)) : 0 ≤ gaussianCone L R x := le_max_right _ _ |
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| lemma gaussianCone_lipschitz {d : ℕ} (L : ℝ≥0) (R : ℝ) : |
| LipschitzWith L (gaussianCone (d := d) L R) := by |
| have h : LipschitzWith L (fun x : EuclideanSpace ℝ (Fin d) => (L : ℝ)*(R-‖x‖)) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| rw [Real.dist_eq] |
| have he : (L : ℝ)*(R-‖x‖)-(L : ℝ)*(R-‖y‖) = (L : ℝ)*(‖y‖-‖x‖) := by ring |
| rw [he, abs_mul, abs_of_nonneg L.coe_nonneg] |
| apply mul_le_mul_of_nonneg_left _ L.coe_nonneg |
| simpa only [Real.dist_eq, NNReal.coe_one, one_mul, dist_comm y x] using |
| (lipschitzWith_one_norm (E := EuclideanSpace ℝ (Fin d))).dist_le_mul y x |
| exact h.max_const 0 |
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| lemma gaussianConeCutoff_lipschitz {d : ℕ} {L : ℝ≥0} {R : ℝ} |
| {f : EuclideanSpace ℝ (Fin d) → ℝ} (hf : LipschitzWith L f) : |
| LipschitzWith L (gaussianConeCutoff L R f) := by |
| unfold gaussianConeCutoff |
| simpa only [Pi.neg_apply, max_self] using |
| (gaussianCone_lipschitz (d := d) L R).neg.max (hf.min (gaussianCone_lipschitz L R)) |
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| lemma gaussianConeCutoff_zero {d : ℕ} (L : ℝ≥0) (R : ℝ) |
| (f : EuclideanSpace ℝ (Fin d) → ℝ) (x : EuclideanSpace ℝ (Fin d)) |
| (hx : R ≤ ‖x‖) : gaussianConeCutoff L R f x = 0 := by |
| have hc : gaussianCone L R x = 0 := |
| max_eq_right (mul_nonpos_of_nonneg_of_nonpos L.coe_nonneg (sub_nonpos.mpr hx)) |
| simp only [gaussianConeCutoff, hc, neg_zero, max_eq_left (min_le_right _ _)] |
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| lemma gaussianConeCutoff_compact {d : ℕ} (L : ℝ≥0) (R : ℝ) |
| (f : EuclideanSpace ℝ (Fin d) → ℝ) : HasCompactSupport (gaussianConeCutoff L R f) := by |
| apply HasCompactSupport.of_support_subset_isCompact (isCompact_closedBall (0 : EuclideanSpace ℝ (Fin d)) R) |
| intro x hx |
| by_contra hn |
| have he : R ≤ ‖x‖ := le_of_lt (by simpa only [Metric.mem_closedBall, dist_zero_right, not_le] using hn) |
| exact hx (gaussianConeCutoff_zero L R f x he) |
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| lemma gaussianConeCutoff_norm_le {d : ℕ} (L : ℝ≥0) (R : ℝ) |
| (f : EuclideanSpace ℝ (Fin d) → ℝ) (x : EuclideanSpace ℝ (Fin d)) : |
| ‖gaussianConeCutoff L R f x‖ ≤ ‖f x‖ := by |
| have ha := gaussianCone_nonneg L R x |
| rw [Real.norm_eq_abs, Real.norm_eq_abs] |
| unfold gaussianConeCutoff |
| by_cases hf : 0 ≤ f x |
| · have hlo : 0 ≤ max (-gaussianCone L R x) (min (f x) (gaussianCone L R x)) := |
| (le_min hf ha).trans (le_max_right _ _) |
| rw [abs_of_nonneg hlo, abs_of_nonneg hf] |
| exact max_le (by linarith) (min_le_left _ _) |
| · have hf' : f x ≤ 0 := (lt_of_not_ge hf).le |
| rw [min_eq_left (hf'.trans ha), abs_of_nonpos (max_le (by linarith) hf'), abs_of_nonpos hf'] |
| exact neg_le_neg (le_max_right _ _) |
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| lemma gaussianConeCutoff_eq {d : ℕ} {L : ℝ≥0} {R : ℝ} |
| {f : EuclideanSpace ℝ (Fin d) → ℝ} (hf : LipschitzWith L f) (h0 : f 0 = 0) |
| (x : EuclideanSpace ℝ (Fin d)) (hx : 2*‖x‖ ≤ R) : |
| gaussianConeCutoff L R f x = f x := by |
| have hb : |f x| ≤ (L : ℝ)*‖x‖ := by |
| simpa only [h0, dist_zero_right, Real.norm_eq_abs] using hf.dist_le_mul x 0 |
| have hh : (L : ℝ)*‖x‖ ≤ (L : ℝ)*(R-‖x‖) := |
| mul_le_mul_of_nonneg_left (by linarith) L.coe_nonneg |
| have ha : |f x| ≤ gaussianCone L R x := hb.trans (hh.trans (le_max_left _ _)) |
| unfold gaussianConeCutoff |
| rw [min_eq_left ((le_abs_self _).trans ha), max_eq_right] |
| linarith [neg_le_abs (f x)] |
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| end InvariantIsing |
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| end |
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| end OAI |
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