Improving Matrix Exponential for Generative AI Flows: A Taylor-Based Approach Beyond Paterson--Stockmeyer

The matrix exponential is a fundamental operator in scientific computing and system simulation, with applications ranging from control theory and quantum mechanics to modern generative machine learning. While Padé approximants combined with scaling and squaring have long served as the standard, recent Taylor-based methods, which utilize polynomial evaluation schemes that surpass the classical Paterson--Stockmeyer technique, offer superior accuracy and reduced computational complexity. This paper presents an optimized Taylor-based algorithm for the matrix exponential, specifically designed for the high-throughput requirements of generative AI flows. We provide a rigorous error analysis and develop a dynamic selection strategy for the Taylor order and scaling factor to minimize computational effort under a prescribed error tolerance. Extensive numerical experiments demonstrate that our approach provides significant acceleration and maintains high numerical stability compared to existing state-of-the-art implementations. These results establish the proposed method as a highly efficient tool for large-scale generative modeling.

F0 analysis of Ghanaian pop singing reveals progressive alignment with equal temperament over the past three decades: a case study

Contemporary Ghanaian popular singing combines European and traditional Ghanaian influences. We hypothesize that access to technology embedded with equal temperament catalyzed a progressive alignment of Ghanaian singing with equal-tempered scales over time. To test this, we study the Ghanaian singer Daddy Lumba, whose work spans from the earliest Ghanaian electronic style in the late 1980s to the present. Studying a singular musician as a case study allows us to refine our analysis without over-interpreting the findings. We curated a collection of his songs, distributed between 1989 and 2016, to extract F0 values from isolated vocals. We used Gaussian mixture modeling (GMM) to approximate each song's scale and found that the pitch variance has been decreasing over time. We also determined whether the GMM components follow the arithmetic relationships observed in equal-tempered scales, and observed that Daddy Lumba's singing better aligns with equal temperament in recent years. Together, results reveal the impact of exposure to equal-tempered scales, resulting in lessened microtonal content in Daddy Lumba's singing. Our study highlights a potential vulnerability of Ghanaian musical scales and implies a need for research that maps and archives singing styles.