Line A: y = 26x/3 - 118
L_A(lambda) = 2.75 + lambda
passes through k = 2 points
P-adic Regularisation Counterexample
In this dataset, increasing regularisation can increase the number of exactly-fitted points. That violates Euclidean intuition and highlights a geometry-specific effect.
Line B: y = 12
L_B(lambda) = 3.00
passes through k = 3 points
Interpretation
The crossover is at lambda* = 0.25. For lower values, the lower-data-loss line wins. For higher values, the regularised line wins and fits more points exactly.
| Lambda region | Winner | Exactly fitted points |
|---|---|---|
| lambda < 0.25 | Line A | k = 2 |
| lambda = 0.25 | Tie | 2 and 3 |
| lambda > 0.25 | Line B | k = 3 |