Low rank matrix approximation

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102. Low rank matrix approximation

easy

General

senior

Low-rank matrix approximation is a simple way to compress a matrix while keeping most of its information, often done using Singular Value Decomposition (SVD). Your task is to compute a rank-(k) approximation (A_k) of a given matrix (A) that minimizes reconstruction error. The Eckart-Young-Mirsky theorem states that the optimal rank- $k$ approximation in the Frobenius norm is given by the truncated SVD:

A_k = \sum_{i=1}^k \sigma_i u_i v_i^T = U_k \Sigma_k V_k^T

where $U_k$ and $V_k$ contain the first $k$ columns of $U$ and rows of $V^T$ (columns of $V$ ), and $\Sigma_k$ contains the top $k$ singular values.

Requirements

Implement the function

python

Rules:

Use SVD to approximate (A) as (A \approx U \Sigma V^\top), then keep only the top (k) singular values/vectors.
Return (A_k) as a NumPy array.
Do not use any helper libraries beyond NumPy and Python built-ins.

Example

python

Output:

python

Input Signature

Argument	Type
A	np.ndarray
k	int

Output Signature

Return Name	Type
value	np.ndarray

Constraints

Use NumPy only; no other libraries.
Return ndarray.
Use SVD with top-k components only.

Hint 1

Use np.linalg.svd(A, full_matrices=False) to get U, s, Vt where s is a 1D array of singular values.

Hint 2

Form the rank-k reconstruction via slicing: U[:, :k] @ np.diag(s[:k]) @ Vt[:k, :].

Roles

ML Engineer

AI Engineer

Companies

General

Levels

senior

entry

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