Solving a 250 Year Old Math Problem

Hardware architecture

by Vassil Dimitrov

7 Sep

2023

When I was in my teens, I developed an emotional appreciation of mathematics. When I see terms like 'prime numbers', 'Diophantine equations', 'polynomial complexity', 'discrete geometry'' I see beauty, purity, and clarity. There is no other field of science where the truth is so crystal clear. However, when I was young, I was surrounded by old school engineers who looked down at math as something too abstract and far removed from reality to be useful. I desperately wanted to show that this is horribly wrong.

Ironically, when I went to work, I was the only mathematician placed as a computer engineer in the design department. It turned out I was a good programmer and was able to combine these skills to solve problems in unique ways involving novel mathematics. In the world of engineering this essentially involved creating new number formats and algorithms to enable new mathematics that unlocked performance and efficiency benefits.

One of the most rewarding instances of this was in the creation of the first completely-in-canal hearing aid. A digital hearing aid is very computation intensive, requiring high precision DSP to deliver high quality amplification, in a tiny form factor with ultra low power consumption. I had previously developed something insanely beautiful and potentially very valuable, a multi dimensional logarithmic number system. Immediately, I applied the golden principle of mathematicians- “shut up and calculate.” I did a huge number of numerical experiments aimed at unveiling the goodness of this number system to deliver the same quality amplification at the necessary size and power footprints.

The same winning combination of novel mathematics, algorithm development and programming skills were put to use to deliver innovations in the field of cryptography where I had countless contributions to further the field into what it is today. Some of the numerical innovations, which initially may have seemed exotic or unconventional, have stood the test of time and are still the basis of modern day cryptography.

It’s in this same vein that Lemurian Labs was founded. I found a kindred spirit in Jay, a fellow mathematician and programmer that believes in the purity and power of mathematics to reimagine computing, this time in the field of AI. While the novel mathematics that is the foundation of Lemurian’s value can trace lineage back to the completely-in-canal logarithmic number system, Lemurian Labs parallel adaptive logarithm (PAL) is a culmination of decades of experience in mathematics, algorithms and programming. Very briefly, PAL applies three major principles to deliver increased performance and power efficiency:

Parallel: number mapping using different bases to express values as logarithms

Adaptive: choice of bases and number of bits to optimize memory, processing bandwidth and precision

Logarithm: mathematics to execute matrix multiplication in a significantly more efficient way

The beauty of PAL is that parallelizing the arithmetic using multi dimensional encoding delivers significant performance while maintaining accuracy.

Now as we are on the cusp of delivering this to the industry, I know this time is the right time. Conventional methods of improving performance and efficiency have rapidly diminishing returns and now the exotic and unconventional are ripe areas of exploration. With PAL, we have solved a 250 year old math problem and it feels great! Finally to create beauty with mathematics and, most important to me, to finally feel useful.

The Lemurian Labs Origin Story

Since the age of 15, I have held the belief that AI and robotics together would have a transformational impact on society as we know it. They would function as catalysts, enabling us to do more than we could ever imagine. For almost 65 years we have been trying to build AI systems and intelligent robots that can operate at near or surpass human levels, but we have consistently fallen short. Advancements in deep learning and reinforcement learning made it feel like we would soon be able to realize autonomous robotics, but after years of working in AI, it was starting to feel like we were at a standstill, and the current approaches would be insufficient. We still did not have AI models that could learn through interactions in the real world, maintain temporal context to enable better predictions, and effectively deal with a changing world. But we did know that larger models trained on more data would outperform smaller models. At around this time, I had begun to explore transformer based architectures and it felt like they may be the missing piece. So, in 2018, Vassil, my co-founder, Vassil, and I got together and started Lemurian to build a hybrid transformer-convolution based foundation model for autonomous robotics, and a platform to manage the end-to-end lifecycle of these models so that all robotics companies could more easily leverage AI. But in our pursuit, we very quickly realized that in order to build the kind of model we wanted, it would need to be a much larger model than we originally thought. Training it would have taken over a month on more than ten thousand GPUs. We went out to speak with other AI developers and started noticing that companies were currently training models that required exaflops of compute, and were already planning future models runs which would require zettaflops of compute. It became quite clear that these models were going to grow a lot larger, and within the decade would require a yottaflop to train. This was alarming because the first exascale computer wouldn’t be available for another 3 years.

Solving a 250 Year Old Math Problem

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The Lemurian Labs Origin Story

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