Answers.Body: <p>Apart from common math, any advanced math would be potentially helpful for an AI practitioner, especially for research positions related to deep learning which still has quite a few unproven theoretical conjectures albeit many seemingly convincing empirical results have been accumulated over the years. For example there's a famous <a href="https://en.wikipedia.org/wiki/Manifold_hypothesis" rel="nofollow noreferrer">manifold hypothesis and information geometry of statistical manifolds</a> which is related to differential geometry, and today's learned representations by deep learning are often assumed to be on a manifold, but the precise geometry and topology of these learned manifolds are not fully understood. Questions such as under which conditions and for which types of data this hypothesis holds and how manifold curvature/connectivity/other geometric properties affect learning, remain unresolved:</p>
  <blockquote>
  <p>Machine learning models only have to fit relatively simple, low-dimensional, highly structured subspaces within their potential input space... The argument for reasoning about the information geometry on the latent space of distributions rests upon the existence and uniqueness of the Fisher information metric. In this general setting, we are trying to find a stochastic embedding of a statistical manifold.</p>
  </blockquote>
  <blockquote>
  <p>In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space.</p>
  </blockquote>
  <blockquote>
  <p>We find that the resultant information metric does not describe the full geometry of space but only its conformal geometry</p>
  </blockquote>
  <p>Manifolds such as those found in graph-based or geometric data (e.g., 3D point clouds) are not easy to represent with conventional neural network architectures, which are designed for Euclidean data. The development of deep learning techniques that can model data on these non-Euclidean manifolds, such as geometric deep learning, is still in its early stages, and many questions remain about how best to handle such data.</p>
  <p>Functional analysis is generally not a core skill for most machine learning roles outside of highly theoretical positions. For example if you want to fully understand and apply universal approximation theorem or various optimization methods you may need a good knowledge of functional analysis and approximation theory. A recent interesting <a href="https://ai.stackexchange.com/questions/47180/how-would-you-go-from-1-to-k-hidden-layers-in-cybenkos-result-that-neural-netwo">post</a> in this site requires this kind of knowledge. Since you described yourself as a math major with a burning passion for pure mathematics (to the point of obsession), maybe you can try to write a non-trivial answer for that post.</p>
  
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Answers.CreationDate: 2024-11-13T07:18:09.947Z
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Answers.UserDisplayName: cinch
Body: <p>I'm a math major with a burning passion for pure mathematics (to the point of obsession). However as I graduate I wanted to work in an industry that best utilise my math skills which brings me to my question.</p>
  <p>Does working in machine learning, deep learning and AI requires advanced math skills such as differential geometry and functional analysis? How does knowing such math topics helps in the work of a machine learning and AI practioner?</p>
  
Tags: <machine-learning><math><ai-field><education>
Title: Are pure math topics like functional analysis and differential geometry useful for machine learning and AI?
UserDisplayName: Wong Wen Hui
ThreadText: # Are pure math topics like functional analysis and differential geometry useful for machine learning and AI?
  
  (Asked by: username_0 on 2024-11-13)
  
  I'm a math major with a burning passion for pure mathematics (to the point of obsession). However as I graduate I wanted to work in an industry that best utilise my math skills which brings me to my question.
  
  Does working in machine learning, deep learning and AI requires advanced math skills such as differential geometry and functional analysis? How does knowing such math topics helps in the work of a machine learning and AI practioner?
  
  **Tags:** machine-learning, math, ai-field, education
  
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  ## 1 answers
  
  ### Answer by username_1 on 2024-11-13. Score: 0 | Accepted answer
  
  Apart from common math, any advanced math would be potentially helpful for an AI practitioner, especially for research positions related to deep learning which still has quite a few unproven theoretical conjectures albeit many seemingly convincing empirical results have been accumulated over the years. For example there's a famous [manifold hypothesis and information geometry of statistical manifolds](https://en.wikipedia.org/wiki/Manifold_hypothesis) which is related to differential geometry, and today's learned representations by deep learning are often assumed to be on a manifold, but the precise geometry and topology of these learned manifolds are not fully understood. Questions such as under which conditions and for which types of data this hypothesis holds and how manifold curvature/connectivity/other geometric properties affect learning, remain unresolved:
  
  > Machine learning models only have to fit relatively simple, low-dimensional, highly structured subspaces within their potential input space... The argument for reasoning about the information geometry on the latent space of distributions rests upon the existence and uniqueness of the Fisher information metric. In this general setting, we are trying to find a stochastic embedding of a statistical manifold.
  
  > In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space.
  
  > We find that the resultant information metric does not describe the full geometry of space but only its conformal geometry
  
  Manifolds such as those found in graph-based or geometric data (e.g., 3D point clouds) are not easy to represent with conventional neural network architectures, which are designed for Euclidean data. The development of deep learning techniques that can model data on these non-Euclidean manifolds, such as geometric deep learning, is still in its early stages, and many questions remain about how best to handle such data.
  
  Functional analysis is generally not a core skill for most machine learning roles outside of highly theoretical positions. For example if you want to fully understand and apply universal approximation theorem or various optimization methods you may need a good knowledge of functional analysis and approximation theory. A recent interesting [post](https://ai.stackexchange.com/questions/47180/how-would-you-go-from-1-to-k-hidden-layers-in-cybenkos-result-that-neural-netwo) in this site requires this kind of knowledge. Since you described yourself as a math major with a burning passion for pure mathematics (to the point of obsession), maybe you can try to write a non-trivial answer for that post.
  
  
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