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Ridgeless minimum-norm least-squares regression

Definition

For positive integers $n, p$ , a real design matrix $X \in R^{n \times p}$ , and a response vector $y \in R^{n}$ , the ridgeless minimum-norm least-squares estimator is

$\hat{β} : = X^{+} y .$

Equivalently, it is the unique vector of smallest Euclidean norm among all minimisers of $u \mapsto ∥ X u - y ∥_{2}^{2}$ . The word ridgeless means that this least-squares objective has no added positive quadratic penalty $λ ∥ u ∥_{2}^{2}$ .

Definition references

The Moore--Penrose pseudoinverse $A^{+}$ as the solution of the four Penrose equations

Every finite real or complex matrix has a unique Moore--Penrose pseudoinverse

For every right-hand side $b$ , $A^{+} b$ is the unique least-squares solution of minimum Euclidean norm

Hastie, Montanari, Rosset, and Tibshirani, §2.1

Facts & Assumptions

Given: Positive integers $n, p$ , $X \in R^{n \times p}$ , and $y \in R^{n}$ .

[F1]

Every finite real matrix $X \in R^{n \times p}$ has a unique Moore--Penrose pseudoinverse $X^{+} \in R^{p \times n}$ satisfying the four Penrose equations (The Moore--Penrose pseudoinverse $A^{+}$ as the solution of the four Penrose equations, Every finite real or complex matrix has a unique Moore--Penrose pseudoinverse).

[F2]

For every finite real matrix $X$ and vector $y$ , $X^{+} y$ is the unique least-squares minimiser of minimum Euclidean norm (For every right-hand side $b$ , $A^{+} b$ is the unique least-squares solution of minimum Euclidean norm).

Verification

1.1givenF1

By [F1], $X^{+}$ exists uniquely and has size $p \times n$ , so $X^{+} y$ is a well-defined vector in $R^{p}$ .

2.1givenF2∎

By [F2], this vector is exactly the unique minimum-norm least-squares solution, which verifies the equivalent characterization.

Depends on

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hastie, Montanari, Rosset, and Tibshirani, Surprises in High-Dimensional Ridgeless Least Squares Interpolation (standard reference, not scraped)