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    $\begingroup$ Considering the units of measurement of $\lambda,$ there cannot possibly be such a rule of thumb. $\endgroup$ Commented Feb 22, 2024 at 20:10
  • $\begingroup$ @whuber what do you mean by the units of measurement of $\lambda$? And why does that mean there can't be a rule of thumb? $\endgroup$ Commented Feb 23, 2024 at 10:57
  • $\begingroup$ Suppose, for example, $X$ is a weight measured in Kg and $y$ is a length in m. For $X\beta-y$ to make any sense, $\beta$ must have units of m/Kg, whence the objective function you are minimizing is in units of m^2. Equating that with the right hand side, we deduce $\lambda$ is in units of m*Kg. Now, when you change the units of measurement, the numerical value of $\lambda$ will change. Your question is tantamount, say, to asking for a rule of thumb to determine when a person is overweight. If the answer is they weigh over 100, the issue is 100 what? Pounds, kilos, grams, stone? $\endgroup$ Commented Feb 23, 2024 at 13:16
  • $\begingroup$ @whuber When I said a rule of thumb I didn't mean a fixed and universal value. I meant a value that could be derived from $X$ and $y$ that is more or less true. For example, if $\lambda$ is bigger than the first eigenvalue of $X$ then you would expect around 50% of zeroes (this is a completely made up example.). Does it make sense? $\endgroup$ Commented Feb 25, 2024 at 22:00