Dear QF Community,
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The Mathematical Stack of Modern Machine Learning
For most mainstream machine learning work, Linear Algebra, Calculus, and basic Probability are more than sufficient. They can take you remarkably far in practice, and they form the backbone of nearly all standard ML pipelines.
However, once you step into advanced and emerging areas such as Geometric Deep Learning and Topological Data Analysis (TDA), that toolkit is no longer enough. These fields demand a far richer mathematical machinery. You are no longer working only with vectors, matrices, and loss functions. You are working with geometry, structure, and shape.
At this point, the nature of the problem changes. The question is no longer just how to optimise a model, but how to understand and design the space in which the model lives. This creates a very interesting divide in terms of mathematical capabilities:
If you are curious, check out:
𝗚𝗲𝗼𝗺𝗲𝘁𝗿𝗶𝗰 𝗗𝗲𝗲𝗽 𝗟𝗲𝗮𝗿𝗻𝗶𝗻𝗴: 𝗚𝗿𝗶𝗱𝘀, 𝗚𝗿𝗼𝘂𝗽𝘀, 𝗚𝗿𝗮𝗽𝗵𝘀, 𝗚𝗲𝗼𝗱𝗲𝘀𝗶𝗰𝘀, 𝗮𝗻𝗱 𝗚𝗮𝘂𝗴𝗲𝘀 by Michael Bronstein et al (https://geometricdeeplearning.com/).
𝗧𝗼𝗽𝗼𝗹𝗼𝗴𝗶𝗰𝗮𝗹 𝗗𝗲𝗲𝗽 𝗟𝗲𝗮𝗿𝗻𝗶𝗻𝗴: 𝗚𝗼𝗶𝗻𝗴 𝗕𝗲𝘆𝗼𝗻𝗱 𝗚𝗿𝗮𝗽𝗵 𝗗𝗮𝘁𝗮 by Mustafa Hajij et al (https://arxiv.org/abs/2206.00606).
We have also put together an accessible 14-page guide that explores this divide in more depth. You can download a sample excerpt here:
The full version is available below for our paid Substack members and Academy subscribers. Your support is a powerful signal for us to keep investing in rigorous, long-form thinking and in producing work that goes beyond surface-level commentary.
A free podcast narration version of this post will also be uploaded to Spotify (here) and YouTube (here).
Narration note: The narration may occasionally drift due to technical details, but it captures the core ideas accurately. Any small technical inaccuracies are minimal and do not impact the overall message for a general audience.
Wishing you a wonderful rest of the week ahead.
QF Academy team
Full Download Resources
You can download the complete guide below. We encourage you to read through it carefully and share your thoughts with us. Your feedback is invaluable in helping us refine the material and continue developing content that is both rigorous and genuinely useful for deepening mathematical understanding.
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