Ask HN: Ramping up math skills?
I went through a decent mathematical curriculum during college including calculus and linear algebra several years ago, and while I was nowhere near the best I had good skills. Recently I have been trying to implement machine learning and what I have found out is that I have lost most of the capacity for math, specifically following proofs etc.
I think this may have happened to other people as well, and while I am in no way averse to hard work or putting in long hours, a little direction to start off would be very helpful.
Keeping that in mind, are there books or tutorials that are useful when trying to refresh mathematical knowledge (refresh being the key here)? My area of interest is machine learning so the main topics I need to be good at are algebra and calculus. I have already ordered How To Prove It as it was recommended elsewhere to me.
Thanks a lot. Since you have done math previously, the best way to ramp up rapidly is not through fluff material geared towards beginners and to focus on stuff that will actually force you to learn. You might have noticed that the only way to acquire mathematical intuition is to solve lots and lots of problems. One possible path to follow: 1. Start off with Sheldon Axler's Linear Algebra done right. This is a more theoretical book (than Strang) but should help keep you challenged and motivated. Work through most of the problems. The best way to attack the proofs is to do them yourself. 2. Feller is the best probability book barring none. This is the kind of stuff that Persi Diaconis went through. Solve as many problems as possible but remember that trying to finish it all will take you years. 3. An excellent introductory stats book that doesn't assume you are an immature child is Freedman's book on statistics. This focuses less on the math and more on what statistics really means. Techniques in stats are fairly trivial but using them right is hard. 4. Calculus is useful stuff. As you go through your probability education, you will eventually hit the world of continuous probability which requires a good amount of calculus to go through. Spivak is an awesome book which should prepare you for that. 5. Learn some real analysis. Real analysis from the machine learning perspective is useful because a lot of measure theoretic arguments in research papers have underpinnings here. Thanks a lot, especially for the names of the books. After reading yours and other advice in this thread I am convinced that the way to get up to speed quickly would be to start with harder problems and backtrack as needed. Do you have any recommendations on books on analysis, the one I have with me is Walter Rudin. I studied in a school that uses the Moore method along with the Professor's notes. This meant that any analysis textbooks were kind of banned. Rudin however is supposed to be fine (a bit harder but that is what you probably need). Also, Spivak is so freaking good that once you are done with that, Rudin should become much much easier to grasp and follow. How to Solve It is a very good book, but it doesn't address the issue you want; it is mainly about problem solving and was intended for math teachers. For review, I suggest getting cheap, out of print textbooks and working the chapter review problems. When the review suggests you are having difficulties, only then reading the appropriate section. That lets you target your problem areas without wasting time, and getting bored, doing a lot of unnecessary reviewing. _How To Solve It_ [1] is indeed a good book, but I think fjellfras is referring to _How To Prove It_ [2], which prepares students "to make the transition from solving problems to proving theorems." [1] http://www.amazon.com/dp/4871878309/
[2] http://www.amazon.com/dp/0521675995/ Thanks, solving questions before reading sounds like an excellent strategy, especially given that I'm supposed to have read it all before. Now I will try to make a list of books to work on. You may want to try working through Spivak's _Calculus_ [1] textbook. It is a bit more involved than most calculus textbooks used in universities today, so it will likely not feel like you are simply repeating something that you have already done. I would recommend this book to anybody who wants to brush up on calculus after studying it before. I have Spivak's book actually, I got the hardcover from a used book store several years ago. It did seem a bit daunting to me from what I recall, I had used Tom Apostle's book when studying calculus back then. I'll give more time to working through Spivak, hopefully it feels more approachable this time around. As for studying proofs, _How To Prove It_ [1] is indeed a good book. You may also be interested in _Book of Proof_ [2], which is available under a Creative Commons license. (You can download the PDF for free, and you can order it from Amazon [3] if you want a hard copy.) [1] http://www.amazon.com/dp/0521675995/ [2] http://www.people.vcu.edu/~rhammack/BookOfProof/index.html I'm in a similar situation (took my last math course a decade ago, and now struggling with texts on neural networks). I'm reading this book to revive my dormant knowledge:
http://www.amazon.com/Guide-Essential-Math-Engineering-Compl... An even shorter book has also been suggested for those who need a refresher :
http://www.amazon.com/Essential-Mathematical-Skills-Engineer... I find that at a practical level, linear algebra and probability/stochastic processes are the most valuable and heavily used topics in machine learning. For Linear algebra, i'd recommend Gilbert Strang's book+his MIT OCW lectures. Check out Papoulis's text for probability. It is very dense, and packs in lots of insight per page. Thank you, fortunately I was able to find an economy edition of both books (Linear Algebra and its applications by Strang though) after reading your post, so I have lots of work to do now. Also I don't know how I missed probability in my original post. That has been one of my weak areas and definitely needs fixing. Thank you for the link, I used to subscribe to r/math at one point. However, my problem is a bit different. I am looking to re-learn the stuff I had gone through in college, as opposed to many people on r/math (as well as the audience for that post I presume) who want to cover new ground in math. I seem to have just forgotten a lot of stuff or maybe not studied it right the first time around. Still I'll look at the books there, I'm sure there should be a significant overlap in my goals and those of people who want to start learning math. When you write "algebra" do you mean the algebra people study in high school or abstract algebra? I meant linear algebra, mostly matrix related analysis and such. Try Khan Academy ? Thanks, I will look into it I was under the impression that Khan Academy is for people new to the subject? I will check out the calculus and algebra lectures there.