I Was Wrong. Sedenions are not where the road ends and can be used for Practical Purposes in Neural Nets.

Despite their algebraic imitations Sedenions can function efficiently in neural nets

Understanding Sedenions and Their Role in Neural Networks

Introduction to Quaternions

As a mathematician, one must appreciate the elegance of quaternions, numbers that extend our understanding of algebra into four dimensions. Quaternions, are composed of real components and , alongside imaginary units and that obey specific multiplication rules. These properties make quaternions invaluable for representing rotations in three-dimensional space, finding utility in areas such as robotics and computer graphics.

From Quaternions to Octonions

Progressing beyond quaternions, we encounter octonions, which expand the concept into eight dimensions. Octonions are real numbers, and are imaginary units with unique multiplication rules. Unlike quaternions, octonion multiplication is non-associative, a property that challenges conventional algebraic intuition. Despite this, their structure proves useful in theoretical physics, notably in string theory and models of higher-dimensional space.

What Are Sedenions?

Sedenions extend this sequence into 16 dimensions. Sedenions lack certain algebraic properties such as division, making them appear, at first, to be purely theoretical constructs.

Historical Challenges with Sedenions

The historical skepticism surrounding sedenions stems from their departure from familiar algebraic properties. Unlike quaternions and octonions, sedenions do not have the division property, meaning you cannot reliably invert a sedenion to perform division. Additionally, their multiplication introduces a lack of alternativity — a property maintained by octonions but lost in sedenions — further complicating their algebraic structure. These characteristics relegated sedenions to the realm of mathematical curiosity for many years.

Another challenge was their limited applicability in conventional mathematical and physical models. Without clear use cases, sedenions were often seen as an over-complication, with their higher-dimensional structure dismissed as unnecessary for practical applications. This perception persisted until advancements in computational techniques and data representation highlighted their potential.

Why Use Sedenions?

Initially dismissed as impractical, sedenions have found unexpected relevance in fields requiring the encoding of highly complex, multidimensional data. One notable application is in time-series forecasting, where the goal is to predict future data points based on historical patterns. The 16-dimensional structure of sedenions allows for the efficient encoding and analysis of relationships between multiple variables, offering advantages over traditional approaches.

Sedenions in Neural Networks

The potential of sedenions is exemplified in the Metacognitive Sedenion-Valued Neural Network (Mc-SVNN). This innovative framework integrates two key components:

1. Sedenion-Valued Representations: By leveraging sedenions, the network processes intricate, multidimensional datasets, uncovering patterns that simpler representations might miss.
2. Metacognition: The network’s metacognitive capabilities enable it to regulate its learning process, determining which data to prioritize and which to disregard. This self-awareness optimizes the network’s focus and efficiency.

Applications of the Mc-SVNN include currency exchange rate forecasting, power demand prediction, and temperature analysis. In these tasks, the network’s use of sedenions has consistently yielded accurate predictions while reducing computational complexity.

Conclusion

The rediscovery of sedenions as practical tools illustrates the profound interplay between abstract mathematics and real-world applications. Through innovations like the Mc-SVNN, sedenions enable the modeling of complex systems, bridging the gap between high-dimensional mathematics and applied science. What was once a mathematical curiosity has become an interesting modern computational techniques, affirming the enduring value of exploring even the most abstract ideas.

Reference:

Saoud LS, Al-Marzouqi H. Metacognitive sedenion-valued neural network and its learning algorithm. IEEE access. 2020 Aug 6;8:144823–38.