Measuring Subtle HD Data Representation and Multimodal Imaging Phenotype Embedding for Precision Medicine

High-dimensional (HD) imaging biomarkers offer enhanced imaging characterization, yet their abundance impedes system performance. The methods that embed HD onto lower dimensional (LD) space address this challenge effectively, particularly for multimodal scenarios. In this study, we proposed different embedding approaches to tackle such a problem. The shortest path algorithm for isometric mapping (Isomap) is modified by an additional constraint using the Parzen-Rosenblatt (PR) density function. This aids in maintaining the uniformity of the graphs through HD to LD projections via Isomap. Then, we proposed Gaussian and Kaniadakis entropy-driven, $\kappa$ , Gaussian embedding techniques to interact with multimodal imaging phenotypic biomarkers. We comprehensively tested our methods on a combination of multiple imaging datasets related to various diseases across multiple modalities, i.e., computed tomography (CT), positron emission tomography (PET), X-ray, magnetic resonance imaging (MRI), ultrasound, and thermography, for an overall of 5158 cases. Our findings demonstrate that our embedding methods effectively transform HD to LD attributes comparatively while retaining vital information and performing phenotypic interactions. Our embedding techniques achieved the maximum accuracies of 78.5% (±4.4), 88.4% (±1.4), and 61.4% (±11.4), 80.9% (±5.8), 80.3% (±5.5), 82.9% (±2.3), and 63.2% (±7.7) for three lung cancer data, pneumonia, breast cancer ultrasound, thermography, and glioblastoma (GBM) diseases, respectively. This accuracy improved with $\kappa$ -Gaussian and Gaussian embedding yielded 79.7%(±2.7) and 80.4%(±3.7) for lung cancer and 65.01% (±3.7) and 62.6% (±7.8) for GBM, respectively. The results of survival models and Kaplan-Meier survival curve also indicate the notable ability of our embedding approaches to distinguish between patients with different median hazards indicating the preservation of HD multimodal imaging characteristics for precision medicine.