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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Nonunique Lasso coefficients with a unique fitted value

Example

Take one observation, two identical columns $X = [1 1]$ , response $y = 1$ , and Lasso penalty $τ = 1 / 4$ in Lasso estimator for fixed design. The minimizing coefficient set is exactly ${(θ_{1}, θ_{2}) : θ_{1}, θ_{2} \geq 0, θ_{1} + θ_{2} = 3 / 4},$ so the coefficients are not unique while the fitted value is always $X θ = 3 / 4$ .

Facts & Assumptions

Given: $n = 1$ , $d = 2$ , $X = [1 1]$ , $y = 1$ , and $τ = 1 / 4$ .

[F1]

Lasso estimator for fixed design: $Q_{y} (θ) = (1 - θ_{1} - θ_{2})^{2} + 2 τ (∣ θ_{1} ∣ + ∣ θ_{2} ∣)$ .

[F2]

The Lasso objective has a nonempty compact convex minimizer set: for this positive-penalty fixed-design Lasso, the global minimizer set is nonempty.

[F3]

The one-coordinate absolute-value quadratic minimizer: $\frac{1}{2} s^{2} - b s + λ ∣ s ∣$ has unique minimizer $S_{λ} (b)$ when $λ > 0$ .

[F4]

Soft-thresholding map: $S_{λ} (b) = b - λ$ for $b > λ$ .

[F5]

The triangle inequality: $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ for real $u, v$ .

Proof

1.1F1F3F4F5givenalgebra

Put $s = θ_{1} + θ_{2}$ . [F5] gives $∣ θ_{1} ∣ + ∣ θ_{2} ∣ \geq ∣ s ∣$ , hence by [F1], $Q_{y} (θ) \geq (1 - s)^{2} + \frac{1}{2} ∣ s ∣ = 2 (\frac{1}{2} s^{2} - s + \frac{1}{4} ∣ s ∣) + 1$ . By [F3]--[F4], the scalar bracket is uniquely minimized at $s = S_{1 / 4} (1) = 3 / 4$ .

2.1F1F2step 1.1algebra∎

Equality in $∣ θ_{1} ∣ + ∣ θ_{2} ∣ \geq ∣ θ_{1} + θ_{2} ∣$ holds when the two coefficients have the same sign or one is zero. Since the minimizing sum is positive, every nonnegative pair summing to $3 / 4$ attains the lower bound from step 1.1; every minimizer must have that sum and equality, so the displayed set is exact. It contains distinct points $(3 / 4, 0)$ and $(0, 3 / 4)$ , and both fit to $3 / 4$ .

Remarks

Finite check assertion: for the one-row design with columns 1 and 1 and coefficient direction [1,-1], the fitted direction is zero, so moving along this null direction preserves the fitted value while the Gram matrix is rank deficient.

Depends on

Used by

Nothing in the library uses this result yet.

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Sources

Stephen J. Wright, Coordinate Descent Algorithms, §2 (standard reference, not scraped)
Clément Royer, Regularized, Large-Scale and Distributed Optimization, Lecture 3, §3.3 (standard reference, not scraped)