Making AI Materially Better: Mathematics for Contemporary AI Needs

Dear Colleagues,

Artificial Intelligence (AI) has been recently undergoing a kind of renaissance. We can find AI across sectors in virtually every domain, e.g., chips, semi-, and autonomous vehicles, seamlessly in written and visual information as generative AI and dramatically improved deep learning as Kolmogorov-Arnold Networks (KAN). Public and private outlets have begun to construct energy resources devoted to delivering power to these energy-hungry algorithms that are comparatively more expensive to run than legacy algorithms. And AI, it is said, will likely be managing its energy consumption. The decrease in the cost of memory and CPUs, with the miniaturization of sensors and the ubiquitous internet, has also changed where AI has traditionally lived. Initially, AI addressed NP problems using relatively small data. As we have moved into the zettabyte age, even AI algorithms in the so-called tractable P, like OLS, face such voluminous data that only fractions can be used.

As both its presence and use become ubiquitous, a sobering observation becomes apparent: we still have no formal—mathematical—characterizations of AI that are as important as AI itself. We describe elements here—some are obvious, and some are not. Aside from a few examples, like support vector machines, AI algorithms have occurred, like most engineering, before the mathematics that describes them has made outcomes surprising and catastrophic at best.

We can all agree that fairness and ethics should be present in a way that makes AI as human-like as possible. While we know what an AI algorithm does behavior-wise, there does not exist a foundation for robustness: AI typically exists in at least moderately inclement computational conditions. What limited functional viability should be expected when using AI in a critical problem? We should be able to characterize adversity, including security threats as well. Most of the public’s enthusiasm for AI is likely due to the current success of generative AI. Yet even the creators readily claim emergent behavior (EB) is not described by the models, and likely EB is what will make generative AI so powerful. Interestingly, a crude concept called “hallucination” describes when the LLM (large language models) produce spurious answers. Taken from brain sciences, the term describes perceptions driven by distorted or false sensory inputs—a malfunctioning system. On the other hand, LLMs are simply rife with multivalent nodes triggered by prompts where the paths then describe different truths. A debate is growing around what a mathematical model is considering the size of LLMs. Partnered with this is a mathematical model for neural network generalization and overparameterization. Considering architecture, we must understand mathematical optimization for scalable and distributed machine learning. Given Feynman’s observation that the compute time ratio to data traffic time goes toward zero over time, distributed computing will eventually become the principal driver. Another area rife with opportunities is the geometric principles in deep learning: graphs, manifolds, and relational Data. The significance of the KAN relying on real analysis shows how elegantly this partnership can unfold. Representation learning, where we rely on the agent to construct features, needs attention. Without much speculation, we wonder why an information-theoretic approach can boost this increasingly important area. Sparse and low-rank methods for efficient machine learning as a parsimony concept need mathematics to bring together elements of convexity, reduction, representation, and approximation. AI cannot and does not improve simply with more data.

This Special Issue aims to seek compelling and timely mathematics partnered with contemporary AI algorithms and applications from all communities, which can fill in the missing elements of AI that will allow its use and impact to reflect our expectations. The submitted work must be original and unpublished, addressing contemporary AI needs. Any code and data must be made available to readers. We use AI in the broadest sense—to include machine learning.

Prof. Dr. Mehmet M. Dalkilic
Guest Editor

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• supervised learning
• unsupervised
• reinforcement
• deep
• representation
• generative
• reactive
• hybrid
• scalable and distributed
• big data
• emergent behavior