Interpreting data is central to modern research. Large language models (LLMs) show promise in providing such natural language interpretations of data, yet simple feature extraction methods such as prompting often fail to produce accurate and versatile descriptions for diverse datasets and lack control over granularity and scale. To address these limitations, we propose a domain-agnostic method for dataset featurization that provides precise control over the number of features extracted while maintaining compact and descriptive representations comparable to human labeling. Our method optimizes the selection of informative binary features by evaluating the ability of an LLM to reconstruct the original data using those features. We demonstrate its effectiveness in dataset modeling tasks and through two case studies: (1) Constructing a feature representation of jailbreak tactics that compactly captures both the effectiveness and diversity of a larger set of human-crafted attacks; and (2) automating the discovery of features that align with human preferences, achieving accuracy and robustness comparable to human-crafted features. Moreover, we show that the pipeline scales effectively, improving as additional features are sampled, making it suitable for large and diverse datasets.