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I would like to know what mathematics topics are the most important to learn before actually studying the theory on neural networks.

I ask that because I will start to learn about neural networks and machine learning on my own to help in the analysis I am doing on my PhD about patterns of genome evolution.

Thank you in advance.

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Marcos, machine learning is still very young, and in a state of great flux. What this means is that the techniques used are constantly developing, and if you looked across the whole spectrum, you'd see a dizzying variety of methods. What I recommend is to google around for courses on topics you are interested in, both at the school you will attend and at others you respect, and see what the textbooks, notes, and homework assignments look like. –  Matus Telgarsky Jan 15 '10 at 10:15
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For basic neural networks (i.e. if you just need to build and train one), I think basic calculus is sufficient, maybe things like gradient descent and more advanced optimization algorithms. For more advanced topics in NNs (convergence analysis, links between NNs and SVMs, etc.), somewhat more advanced calculus may be needed.

For machine learning, mostly you need to know probability/statistics, things like Bayes theorem, etc.

Since you are a biologist, I don't know whether you studied linear algebra. Some basic ideas from there are definitely extremely useful. Specifically, linear transformations, diagonalization, SVD (that's related to PCA, which is a pretty basic method for dimensionality reduction).

The book by Duda/Hart/Stork has several appendices which describe the basic math needed to understand the rest of the book.

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Take a look at the web page for Michael Steele's course Probability Inequalities and Machine Learning, and the various texts linked to there.

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That page seems to be more related to what's called "learning theory", i.e. theoretical (mathematical) analysis of various learning algorithms, their performance bounds (hence probability inequalities), etc. –  user3035 Jan 15 '10 at 0:36
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