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## Most Important Mathematics in Advanced AI/Robotics

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### #1DevLiquidKnight  Members

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Posted 24 January 2013 - 10:55 AM

What would constitute the very most important mathematics used in AI and Robotics? Not just game AI, all AI, I realize the obvious answer here is "everything" but I am looking for something more specific. Like, I can see probability being used in machine learning, linear algebra is a given. Anything else?

### #2ApochPiQ  Moderators

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Posted 24 January 2013 - 11:32 AM

Statistics.

Nobody knows enough statistics.
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### #3Álvaro  Members

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Posted 24 January 2013 - 11:42 AM

There is something called Control Theory that deals with many of the problems involved, especially in robotics. It does involve a lot of statistics, for sure, but some calculus is also required.

Nobody knows enough statistics.

Hmmm... Not to brag, but I think I know enough statistics, at least for what I have needed so far. I work with a couple of people that know a lot more statistics than I do, but I rarely feel the need to ask for their help.

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Posted 24 January 2013 - 12:06 PM

Impossibly broad question since there are completely disjoint circles in that Venn diagram.

However, I'm going to go out on a limb and say "addition".

Dave Mark - President and Lead Designer of Intrinsic Algorithm LLC

Professional consultant on game AI, mathematical modeling, simulation modeling
Co-advisor of the GDC AI Summit
Co-founder of the AI Game Programmers Guild
Author of the book, Behavioral Mathematics for Game AI

Blogs I write:
IA News - What's happening at IA | IA on AI - AI news and notes | Post-Play'em - Observations on AI of games I play

"Reducing the world to mathematical equations!"

### #5ApochPiQ  Moderators

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Posted 24 January 2013 - 12:19 PM

There is something called Control Theory that deals with many of the problems involved, especially in robotics. It does involve a lot of statistics, for sure, but some calculus is also required.

Nobody knows enough statistics.

Hmmm... Not to brag, but I think I know enough statistics, at least for what I have needed so far. I work with a couple of people that know a lot more statistics than I do, but I rarely feel the need to ask for their help.

Constructing a basic implementation of an AI may not require much stats, sure; in fact, for most game AI, you can implement everything with a little linear algebra and some simple formal logic.

If you want to get good results, though, you need stats: analyzing when and why things happen, how they correlate, and so forth. Stats are also crucial for building good models if you're trying to approximate an existing behavior set.

Plus, if you have to work with a game designer in any capacity whatsoever, knowing how to use Excel will make you a vastly more effective teammate.
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### #6Álvaro  Members

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Posted 24 January 2013 - 12:29 PM

I think we are not understanding each other. You said that nobody knows enough statistics, and I said that I think I do. It's not like I think statistics are not useful, but I studied them in college, I use them every day at my job and I think I know enough.

Gathering evidence, understanding correlations, testing hypotheses, evaluating with what certainty I know that a change to a complex system will have the desired results, etc. is what I do for a living.

My point is that you *can* learn enough about these things.

### #7ApochPiQ  Moderators

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Posted 24 January 2013 - 12:57 PM

It's hyperbole, man... not an attack on your character ;-)
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### #8Emergent  Members

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Posted 25 January 2013 - 10:59 PM

Short post:

Convex optimization.

In reality, there are about four kinds of problems in the world that you can solve, and everything consists of reducing other problems to them:

1.) Linear system of equations

2.) Eigenvector/eigenvalue problems

3.) Convex optimization problems (which really includes 1, and mostly also includes 2)

4.) Low-dimensional dynamic programming

I exaggerate slightly, but not by much.

Long post:

There are a couple of core ideas that a lot of different people study, from slightly different angles, and with slightly different tools.  These people include,

- control theoreticians

- roboticists

- classical AI (planning) people,

- machine learning researchers, and

- operations researchers.

The differences between these groups are as much social, cultural, and historical, as they are technical.

In my own view, there are a small number of core problems that they all aim to solve.  The two that really stand out are,

1.) "Markov decision problems" -- a term that I use broadly to include both discrete time and state spaces (what people normally mean when they say "Markov Decision Problem" (MDP)), and with continuous time and state spaces (which people normally call "optimal control").  Additionally, problems here may or may not contain randomness, and each of these variants have very slightly different corresponding theories, but the underlying ideas are essentially the same.  Do you want to find the shortest path through a maze?  Or how to steer a fighter jet to perform a barrel roll?  Or how to move a robot arm around an obstacle?  These are MDP/optimal-control problems.

2.) State estimation problems ("filtering," "smoothing," and trajectory estimation).  Whereas in my last bullet point the goal was to get a system to a particular state, here the goal is to figure out what state a system is in.  Depending again on whether you are interested in discrete- or continuous- time- or state-spaces, depending on what noise model you assume, and depending on which of these problems you want to solve (filtering, smoothing, trajectory estimation) you end up with e.g. the "forward algorithm" for hidden Markov models, the Kalman filter for linear systems, or the Viterbi algorithm, among other closely-related algorithms.

There are generalizations and combinations of these ideas in all sorts of directions.

E.g., an operations researcher might say that Markov decision problems
are "just" optimization problems with a particular structure and go off
to look at Lagrange multipliers and duality gaps.  He might also point
out all the problems that don't most naturally have this structure.  Likewise, some
machine learning people might say that the idea of different "states"
at different "times," again, "just" describes a particular structure of
Bayes net.  You can also combine #1 and #2 to arrive at POMDPs, which
are the most general (single-player) discrete--time-and-state problem.  But despite all this, I think the two problems I listed above capture the essence of most things.

As for "statistics:" I do not think that there is a single unified intellectual edifice with this name.  The only thing that makes any sense to me philosophically is personalist Bayesian statistics, and even that I'm still figuring out.

### #9Selenaut  Members

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Posted 29 January 2013 - 03:17 PM

.......Why has nobody included basic calculus...?

Selenaut

### #10Álvaro  Members

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Posted 29 January 2013 - 08:39 PM

.......Why has nobody included basic calculus...?

There is something called Control Theory that deals with many of the problems involved, especially in robotics. It does involve a lot of statistics, for sure, but some calculus is also required.

### #11wodinoneeye  Members

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Posted 29 January 2013 - 10:35 PM

Robotics probably means optical/sensory interpretation would be part of the maths.

Im not sure of the names of the areas of mathematics for the different cognitive/perception processing.

Would include mass data algorithms, filtering, pattern feature conversion and matching.

Triangulation for 3D vision...

-----

Isnt Fuzzy Mathematics part of another more general classificaltion ?

Edited by wodinoneeye, 29 January 2013 - 10:36 PM.

--------------------------------------------Ratings are Opinion, not Fact

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Posted 29 January 2013 - 11:07 PM

I'm still sticking with addition... because without it, the rest of y'all are screwed.

Dave Mark - President and Lead Designer of Intrinsic Algorithm LLC

Professional consultant on game AI, mathematical modeling, simulation modeling
Co-advisor of the GDC AI Summit
Co-founder of the AI Game Programmers Guild
Author of the book, Behavioral Mathematics for Game AI

Blogs I write:
IA News - What's happening at IA | IA on AI - AI news and notes | Post-Play'em - Observations on AI of games I play

"Reducing the world to mathematical equations!"

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