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Optimal Low-Rank Approximation

Hard · Linear Algebra & Machine Learning · Quant Trader interview question · linear-algebra, singular-value-decomposition, low-rank-approximation, frobenius-norm, factor-model

You are building a factor model to predict stock returns. You have a large matrix $A$ representing the historical returns of $n$ stocks over $m$ days. You want to reduce the dimensionality of this data by finding a rank-$k$ approximation $A_k$ of $A$ that minimizes the Frobenius norm of the difference between $A$ and $A_k$. Which of the following methods guarantees finding the optimal $A_k$?