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Factor models have been widely used in economics and finance. However, theheavy-tailed nature of macroeconomic and financial data is often neglected inthe existing literature. To address this issue and achieve robustness, wepropose an approach to estimate factor loadings and scores by minimizing theHuber loss function, which is motivated by the equivalence of conventionalPrincipal Component Analysis (PCA) and the constrained least squares method inthe factor model. We provide two algorithms that use different penalty forms.The first algorithm, which we refer to as Huber PCA, minimizes the$\ell_2$-norm-type Huber loss and performs PCA on the weighted samplecovariance matrix. The second algorithm involves an element-wise type Huberloss minimization, which can be solved by an iterative Huber regressionalgorithm. Our study examines the theoretical minimizer of the element-wiseHuber loss function and demonstrates that it has the same convergence rate asconventional PCA when the idiosyncratic errors have bounded second moments. Wealso derive their asymptotic distributions under mild conditions. Moreover, wesuggest a consistent model selection criterion that relies on rank minimizationto estimate the number of factors robustly. We showcase the benefits of HuberPCA through extensive numerical experiments and a real financial portfolioselection example. An R package named ``HDRFA" has been developed to implementthe proposed robust factor analysis.

Huber Principal Component Analysis for Large-dimensional Factor Models