Problems

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Linear Regression Closed Form

Linear AlgebraClassic ML

Medium

Linear regression finds the weight vector that minimizes the sum of squared errors between predictions and targets. The closed-form solution (the normal equation) computes the optimal weights directly using matrix algebra, without iterative optimization.

Given a feature matrix $X$ and a target vector $y$ , compute the weight vector $w$ using the normal equation.

Formula

w = (X^T X)^{-1} X^T y

Where $X$ is the $n \times d$ feature matrix ( $n$ samples, $d$ features), $y$ is the $n$ -dimensional target vector, and $w$ is the $d$ -dimensional weight vector.

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Examples

Input: $X = [[1], [2], [3]]$ , $y = [2, 4, 6]$

Output: $[2.0]$

The line $y = 2x$ perfectly fits the data.

Input: $X = [[1, 1], [1, 2], [1, 3]]$ , $y = [3, 5, 7]$

Output: $[1.0, 2.0]$

The first column acts as a bias. The model $y = 1 + 2x$ fits perfectly.

Hint 1

Look into $\texttt{np.linalg.inv}$ for computing $(X^T X)^{-1}$ .

Hint 2

NumPy's $\texttt{@}$ operator chains matrix multiplications, so the entire formula can fit in one expression.

Requirements

Use numpy to convert $X$ and $y$ to arrays
Compute the transpose $X^T$
Compute the products $X^T X$ and $X^T y$
Compute the inverse $(X^T X)^{-1}$
Multiply the inverse by $X^T y$ to get $w$
Return a list of floats or a numpy array

Constraints

$X$ has at least as many rows as columns $(n \geq d)$
$X^T X$ is always invertible
Return length must equal $d$ , the number of columns in $X$
numpy is available as $\texttt{np}$
Time limit: 300 ms

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