Surpassing Cosine Similarity for Multidimensional Comparisons: Dimension Insensitive Euclidean Metric (DIEM)

The advancement in computational power and hardware efficiency enabled the tackling of increasingly complex and high-dimensional problems. While artificial intelligence (AI) achieved remarkable results, the interpretability of high-dimensional solutions remains challenging. A critical issue is the comparison of multidimensional quantities, which is essential in techniques like Principal Component Analysis (PCA), or k-means clustering. Common metrics such as cosine similarity, Euclidean distance, and Manhattan distance are often used for such comparisons - for example in muscular synergies of the human motor control system. However, their applicability and interpretability diminish as dimensionality increases. This paper provides a comprehensive analysis of the effects of dimensionality on these metrics. Our results reveal significant limitations of cosine similarity, particularly its dependency on the dimensionality of the vectors, leading to biased and less interpretable outcomes. To address this, we introduce the Dimension Insensitive Euclidean Metric (DIEM) which demonstrates superior robustness and generalizability across dimensions. DIEM maintains consistent variability and eliminates the biases observed in traditional metrics, making it a reliable tool for high-dimensional comparisons. This novel metric has the potential to replace cosine similarity, providing a more accurate and insightful method to analyze multidimensional data in fields ranging from neuromotor control to machine and deep learning.