Automatic few-shot font generation aims to solve a well-defined, real-world problem because manual font designs are expensive and sensitive to the expertise of designers. Existing methods learn to disentangle style and content elements by developing a universal style representation for each font style. However, this approach limits the model in representing diverse local styles, because it is unsuitable for complicated letter systems, for example, Chinese, whose characters consist of a varying number of components (often called "radical") -with a highly complex structure. In this paper, we propose a novel font generation method that learns localized styles, namely component-wise style representations, instead of universal styles. The proposed style representations enable the synthesis of complex local details in text designs. However, learning component-wise styles solely from a few reference glyphs is infeasible when a target script has a large number of components, for example, over 200 for Chinese. To reduce the number of required reference glyphs, we represent component-wise styles by a product of component and style factors, inspired by low-rank matrix factorization. Owing to the combination of strong representation and a compact factorization strategy, our method shows remarkably better few-shot font generation results (with only eight reference glyphs) than other state-of-the-art methods. Moreover, strong locality supervision, for example, location of each component, skeleton, or strokes, was not utilized. The source code is available at https://github.com/clovaai/lffont and https://github.com/clovaai/fewshot-font-generation.