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Nanophotonics finds ever broadening applications requiring complex components with many parameters to be simultaneously designed. Recent methodologies employing optimization algorithms commonly focus on a single performance objective, provide isolated designs, and do not describe how the design parameters influence the device behaviour. Here we propose and demonstrate a machine-learning-based approach to map and characterize the multi-parameter design space of nanophotonic components. Pattern recognition is used to reveal the relationship between an initial sparse set of optimized designs through a significant reduction in the number of characterizing parameters. This defines a design sub-space of lower dimensionality that can be mapped faster by orders of magnitude than the original design space. The behavior for multiple performance criteria is visualized, revealing the interplay of the design parameters, highlighting performance and structural limitations, and inspiring new design ideas. This global perspective on high-dimensional design problems represents a major shift in modern nanophotonic design and provides a powerful tool to explore complexity in next-generation devices.