  We introduce randomized algorithms to Clifford's Geometric Algebra,
generalizing randomized linear algebra to hypercomplex vector spaces. This
novel approach has many implications in machine learning, including training
neural networks to global optimality via convex optimization. Additionally, we
consider fine-tuning large language model (LLM) embeddings as a key application
area, exploring the intersection of geometric algebra and modern AI techniques.
In particular, we conduct a comparative analysis of the robustness of transfer
learning via embeddings, such as OpenAI GPT models and BERT, using traditional
methods versus our novel approach based on convex optimization. We test our
convex optimization transfer learning method across a variety of case studies,
employing different embeddings (GPT-4 and BERT embeddings) and different text
classification datasets (IMDb, Amazon Polarity Dataset, and GLUE) with a range
of hyperparameter settings. Our results demonstrate that convex optimization
and geometric algebra not only enhances the performance of LLMs but also offers
a more stable and reliable method of transfer learning via embeddings.
