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Lasso on an orthonormal design is coordinatewise soft-thresholding

Statement

Let $n, p \geq 1$ , $X \in R^{n \times p}$ satisfy $X^{T} X = I_{p}$ , and let $y \in R^{n}$ and $λ > 0$ . The objective $\frac{1}{2 n} ∥ y - X β ∥_{2}^{2} + λ ∥ β ∥_{1}$ has the unique minimizer ${\hat{β}}_{j} = S_{n λ} ((X^{T} y)_{j})$ , where $S_{a} (t) = sgn (t) (∣ t ∣ - a)_{+}$ , $(u)_{+} = \max {u, 0}$ , and $sgn (0) = 0$ . Thus the threshold is $n λ$ under this loss normalization.

Statement references

The Elements of Statistical Learning, §3.4.1 and §18.4

Rigollet and Hütter, §§2.3–2.4, soft thresholding and the Lasso objective

Facts & Assumptions

Given: $n, p \geq 1$ , $X^{T} X = I_{p}$ , $y \in R^{n}$ , and $λ > 0$ .

[F1]

The fixed-design empirical squared loss is $∥ y - X β ∥_{2}^{2} / (2 n)$ and the Lasso penalty is $λ ∥ β ∥_{1}$ (Fixed-design least-squares, ridge, lasso, group, and fused objectives).

Proof

1.1F1givenalgebra

Put $z = X^{T} y$ . Expanding the squared norm and using $X^{T} X = I_{p}$ gives $\frac{1}{2 n} ∥ y - X β ∥_{2}^{2} + λ ∥ β ∥_{1} = \frac{∥ y ∥_{2}^{2} - ∥ z ∥_{2}^{2}}{2 n} + \sum_{j = 1}^{p} (\frac{(β_{j} - z_{j})^{2}}{2 n} + λ ∣ β_{j} ∣)$ . The first term is constant in $β$ , so the minimization separates into scalar strictly convex objectives.

2.1step 1.1algebra

For a scalar $z$ , the derivative of $(t - z)^{2} / (2 n) + λ ∣ t ∣$ on $t > 0$ is $(t - z) / n + λ$ , and on $t < 0$ it is $(t - z) / n - λ$ . The positive stationary point is $t = z - n λ$ exactly when $z > n λ$ ; the negative stationary point is $t = z + n λ$ exactly when $z < - n λ$ . When $∣ z ∣ \leq n λ$ , the one-sided derivatives at zero satisfy $(- z / n - λ) \leq 0 \leq (- z / n + λ)$ , so zero minimizes. Strict convexity makes each minimizer unique, and these cases are precisely $S_{n λ} (z)$ .

3.1step 1.1step 2.1∎

Applying step 2.1 to every coordinate yields the stated vector formula and uniqueness.

Depends on

Fixed-design least-squares, ridge, lasso, group, and fused objectives

Used by

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Sources

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, 2nd ed., corrected 12th printing, §3.4.1 and §18.4 (standard reference, not scraped)
Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, §§2.3–2.4 (standard reference, not scraped)