Enabled by technological improvements, photonic devices and circuits are becoming increasingly more complex. Non-trivial geometries are designed to reduce device footprint, improve performance, and introduce novel functionalities. However, the number of design variables required to properly represent these geometries quickly grows, limiting the effectiveness of classical design approaches. Moreover, parameters are often strongly interdependent, restricting the use of sequential optimizations or independent parameter sweeps. Although several optimization techniques can be effective for multi-parameter design, they commonly allow to optimize for a single or a handful designs and the optimization process needs to be repeated if new performance criteria are introduced. In contrast to classical design approaches, the in uences of the design parameters remain hidden as well as the general behavior of the design space. In this paper we present an extension of our recent work on the application of machine learning pattern recognition to the design of multi-parameter photonic devices. In particular, we propose using a combination of local optimization based on the adjoint method and the use of dimensionality reduction. Adjoint optimization is used multiple times to generate a small set of different designs with high performance. Dimensionality reduction is applied to analyze the relationship between these degenerate designs and identify a lower-dimensional design sub-space that includes all alternative good designs. This sub-space can be mapped for any performance criteria thus enabling informed decisions based on the relative priorities of all relevant performance specifications. As a proof of concept, we demonstrate a ten-parameter design of an integrated photonic power splitter using silicon-on-insulator technology. We identify a region of possible high performance design solutions and select two design candidates either maximizing the splitter efficiency or minimizing back-reflection.