A negative source correlation shrinks by magnitude, then copies the negative sign back into the exact numerator.
Make the source negative
Use the same x values 1, 2, and 3, but flip the y values to -2, -4, and -6. The source sum is sum xy=-28 and sum x squared is still 14.
∑ x y = − 28 , ∑ x 2 = 14 \sum xy=-28,\quad \sum x^2=14 ∑ x y = − 28 , ∑ x 2 = 14 Negative Source Keeps Its Sign Exact sign-copy shrinkage for a negative source correlation. negative source keeps its sign quantity exact value why it is shown Σxy -28 negative source Σx^2 14 denominator |Σxy| 28 shrink magnitude source sign(Σxy) -1 copy sign after shrink max(|Σxy|-λ,0) 21 positive magnitude numerator -21 sign times magnitude w -3/2 divide by Σx^2 sign(Σxy)*max(|Σxy|-λ,0)/Σx^2 sign copied after shrinking magnitude one exact sign/shrinkage example λ is chosen before the solve; L1 can hit exact zero NOT convergence; NOT training; NOT learning NOT general feature selection; NOT generalization; NOT future-data proof NOT accuracy; NOT calibration; NOT probability truth; NOT model quality
Shrink the magnitude
With λ=7, shrink the magnitude: max(|-28| minus 7, 0)=21. The sign has not been copied back yet.
max ( ∣ − 28 ∣ − 7 , 0 ) = 21 \max(|-28| - 7,0)=21 max ( ∣ − 28∣ − 7 , 0 ) = 21 Negative Source Keeps Its Sign Exact sign-copy shrinkage for a negative source correlation. negative source keeps its sign quantity exact value why it is shown Σxy -28 negative source Σx^2 14 denominator |Σxy| 28 shrink magnitude source sign(Σxy) -1 copy sign after shrink max(|Σxy|-λ,0) 21 positive magnitude numerator -21 sign times magnitude w -3/2 divide by Σx^2 sign(Σxy)*max(|Σxy|-λ,0)/Σx^2 sign copied after shrinking magnitude one exact sign/shrinkage example λ is chosen before the solve; L1 can hit exact zero NOT convergence; NOT training; NOT learning NOT general feature selection; NOT generalization; NOT future-data proof NOT accuracy; NOT calibration; NOT probability truth; NOT model quality
Copy the sign back
Because the source sum is negative, the numerator is -1 times 21 = -21. Divide by 14 to get w=-3/2.
w = − 21 14 = − 3 / 2 w={-21\over 14}=-3/2 w = 14 − 21 = − 3/2 Negative Source Keeps Its Sign Exact sign-copy shrinkage for a negative source correlation. negative source keeps its sign quantity exact value why it is shown Σxy -28 negative source Σx^2 14 denominator |Σxy| 28 shrink magnitude source sign(Σxy) -1 copy sign after shrink max(|Σxy|-λ,0) 21 positive magnitude numerator -21 sign times magnitude w -3/2 divide by Σx^2 sign(Σxy)*max(|Σxy|-λ,0)/Σx^2 sign copied after shrinking magnitude one exact sign/shrinkage example λ is chosen before the solve; L1 can hit exact zero NOT convergence; NOT training; NOT learning NOT general feature selection; NOT generalization; NOT future-data proof NOT accuracy; NOT calibration; NOT probability truth; NOT model quality
What this example says
This is one exact sign/shrinkage example. Lambda is chosen before the solve. It is NOT convergence, NOT training, NOT learning, NOT general feature selection, NOT generalization, NOT future-data proof, NOT accuracy, NOT calibration, NOT probability truth, and NOT model quality.
sign ( ∑ x y ) max ( ∣ ∑ x y ∣ − λ , 0 ) / ∑ x 2 \operatorname{sign}(\sum xy)\max(|\sum xy|-\lambda,0)/\sum x^2 sign ( ∑ x y ) max ( ∣ ∑ x y ∣ − λ , 0 ) / ∑ x 2 Negative Source Keeps Its Sign Exact sign-copy shrinkage for a negative source correlation. negative source keeps its sign quantity exact value why it is shown Σxy -28 negative source Σx^2 14 denominator |Σxy| 28 shrink magnitude source sign(Σxy) -1 copy sign after shrink max(|Σxy|-λ,0) 21 positive magnitude numerator -21 sign times magnitude w -3/2 divide by Σx^2 sign(Σxy)*max(|Σxy|-λ,0)/Σx^2 sign copied after shrinking magnitude one exact sign/shrinkage example λ is chosen before the solve; L1 can hit exact zero NOT convergence; NOT training; NOT learning NOT general feature selection; NOT generalization; NOT future-data proof NOT accuracy; NOT calibration; NOT probability truth; NOT model quality