14

I completed a Bachelor's in Mathematics May 2018 with a 3.6 major GPA. I had trouble with real analysis, scoring B-, B, B, B+ in the four courses I took on the subject despite significant effort and paying for a PhD Candidate tutor.

My goal is to go to graduate school in Machine Learning. I want to learn as much theory as I can on my own while working to afford Graduate School. Over the next ~6 years years, I want to self study up to 12 graduate level topics related to Probability / Point Estimation / Optimization or Control / Dynamics and Statistic Learning. As much as I can finish in ~6 years.

I will also schedule time over those ~6 years to work on programming at least 6 non-trivial personal projects in Machine Learning and on replicating one peer-reviewed academic paper every 1 - 2 months. After that, I'll crack open the Deep Learning Book and Reinforcement Learning Book over another year to study them thoroughly using my experience and studied theory while applying to graduate schools.

The answer here (https://www.quora.com/How-can-I-self-study-functional-analysis) raises an important point:

"if you want to understand it [Functional Analysis] in depth, you have to solve problems, which usually means proving stuff (as opposed to calculating stuff), and that's pretty hard for anyone to self-critique.

You'll want someone to help you out of tight spots as you're reading the text, and look over your solutions to see if you're actually getting it. It's not too hard to delude yourself into thinking that you've proved something while in fact you did not. If you miss a subtlety or fail to understand a definition, you might be proving the wrong thing or nothing at all - and you may have no way of even realizing that."

Partial progress is still amazing. However, what proactive stategies avoid falling into these pitfalls? I'm not always an A student, and I want to avoid spending more than 8 months average per subject. Do I repeatedly post every problem I attempt to prove onto Stack Exchange for advice / correctness, and do I occasionally contact a professor from my Alma Mater when I am really stuck?

mrtaurho
  • 16,103
Kalkirin
  • 151
  • 3
    May I know why you want to learn functional analysis? I'm also on the same boat as you. I'm a third year undergrad student and I'm determined to improve my problem solving skills before I go for a PhD in data science and machine learning. But as far as I know, functional analysis doesn't really have much to do with machine learning. So, why do you want to learn it? It's more useful for people who want to study mathematical physics or PDEs, I think. But I'm not an expert. I'm just asking. – stressed out Jan 02 '19 at 00:13
  • Hey Stressed Out, sorry for the wait. I went on a walk. The functional analysis was for this book: Foundations of Modern Probability by Olav Kallenberg. It says you should have experience with functional analysis, complex variables and topology. The text covers stuff like measure theory, distributions, random sequence, characteristic function + CLT, conditioning, Martingales, Markov Processes, Random Walks, Ergodic Theory, Poisson Processes, Gaussian / Brownian Motion, Embeddings, Convergence Theorems, Stochastic Calculus. Func. Anal. also has ties with Control Theory which I want to study. – Kalkirin Jan 02 '19 at 01:10
  • 1
    Thanks for the answer. I know nothing about control theory and ergodic theory, but the other things you mentioned require mostly measure theory I think. I say that because I had a course in stochastic differential equations (which I passed with a C- :P). I think you might want to take a course in PDEs more than functional analysis. Read a PDE book like Evans. I hope one day I can read it. I suggest you to ask a separate question about what prerequisites you need to have to study machine learning before you study a difficult and time-consuming subject like functional analysis. – stressed out Jan 02 '19 at 01:19
  • Here are the complete list of topics I want to study (ideally): Measure Theory, Complex Analysis, Fourier Series, Topology for Analysis, Functional Analysis, Found. of Modern Probability, Nonlinear Dynamical Systems, Point Estimation, Hypothesis Testing, Statistical Learning, Pattern Recognition, Differential Geometry, Nonlinear Dimensional Reduction, Matrix Computations (Applied Lin Algebra), Linear/Nonlinear Programming, and Information Theory. After that it's the Deep Learning book and Reinforcement Learning. Then onto signal processing, control theory and feature engineering/Selection. – Kalkirin Jan 02 '19 at 01:20
  • Are you sure those topics are absolutely necessary? Many engineers work in deep learning and have never encountered any topological space other than $\mathbb{R}^n$ and $\mathcal{L}^p$ spaces. – stressed out Jan 02 '19 at 01:22
  • I will do that! PDEs instead of Functional Analysis it is. The other books I own are: Stochastic Calculus, Stochastic Differential Equations, Random FIelds and Geometry, Markov Random Fields, Graph Theory, Probabilistic Graphical Models, and Bayesian Decision Analysis. However, I feel that those are fields for very specific problems (Except Bayesian Decision Analysis probably. I may move that up the list). What do you think of this list of topics? It's definitely like a decade worth of studying! – Kalkirin Jan 02 '19 at 01:25
  • I guess I just wanted to be versatile, and see if I could find any fun connections to the material. From what I can tell via my research, each of these topics has some motivation to machine learning. For example, Differential Geometry is related to the Fisher Information Matrix and Gradient Descent while Information Theory is becoming more and more important to GANs. However, as you say, it might be completely impractical – Kalkirin Jan 02 '19 at 01:27
  • I will write an answer now. What I'm going to write is too long for a comment. Please wait. – stressed out Jan 02 '19 at 01:28
  • Thank you. I updated my last comment. – Kalkirin Jan 02 '19 at 01:32
  • 1
    Well, I hadn't read your comment when I wrote the answer. I have heard from people that information geometry is more or less theoretical. Many top engineers have never studied it. You can find Goodfellow's article about GANs on arxiv. It doesn't use information geometry. Does it? I mean, mathematicians can develop many impractical tools, but the ones that usually make it into the engineering world use little math. – stressed out Jan 02 '19 at 01:56

5 Answers5

17

You can post some (not all) proofs here with the proof-verification tag. It would be helpful if you flagged the few particular places where you were in doubt.

If an old professor is willing to spend occasional time, go for it.

One suggestion. Rather than learning the basics from the bottom up, start with something you really want to know for its own sake and work backwards through the prerequisites as necessary. You will probably discover that you need a lot more linear algebra than you thought, and a lot less functional analysis.

Finally, six years is a long time to study all alone. Good grad schools do support their students. Consider applying sooner.

Ethan Bolker
  • 95,224
  • 7
  • 108
  • 199
  • 2
    Your suggestion about working backwards through the prerequisites has helped me learn one or two things in physics. I usually try to find homework problems with their solutions online. It's very slow though. I think that it's safe to say that every great mathematician has had great teachers and professors. The Jacobis, Euler, Gauss, Lagrange, Dedekind, Hilbert, etc. – stressed out Jan 02 '19 at 00:22
  • @stressedout What about Galois? – John Douma Jan 02 '19 at 00:33
  • @JohnDouma Yeah, I already had Galois in mind, but then again, Galois never found widespread recognition for his work before his death. Probably because he had never learned mathematics formally. And as far as my memory from math history books tells me, his work was never understood well until Emil Artin rewrote his entire work in terms of vector spaces and modern algebra. Even the French committee of mathematicians didn't understand his submitted work if I'm not mistaken. Another example is Ramanujan before he left India for the UK. And I meant the Bernoullis. Not the Jacobis, for sure. :D – stressed out Jan 02 '19 at 00:44
8

To answer your first question, about how you can catch errors during self-study, I think that you need to have others check your proofs. There have been numerous alleged proofs in the history of mathematics by well-known mathematicians that were later demonstrated to be insufficient or wrong. So, I think you need to find a community of researchers, online or not, to exchange your ideas with them.

As a matter of fact, these days I talk to many people about my future B.Sc. thesis which is going to be about machine learning. What I'm going to write is something that has been said to me by my professors and students studying at higher levels, and I don't claim that it's the best possible approach. So, please keep that in mind.

I think the starting point is to get a copy of the book Elements of Statistical Learning by Hastie and Tibshirani. As a more advanced text to supplement it, you can use Pattern Recognition and Machine Learning by C. Bishop. I think you already know this or probably have even better suggestions for this part.

After reading these two books, you can read the book that Ian Goodfellow, Yoshua Bengio and Aaron Courville have written about deep learning with the same name: Deep Learning. Once you start reading the book, you will be surprised to see how little you need to know to read through the chapters.

You need to take a course in Stochastic Processes. Now, engineering students take this course too. If you can, take this course from the engineering department because they usually avoid measure theory and depending on the lecturers, you may learn some things about signals and systems during the course.

If you want to take the rigorous path, you will need to learn measure theory first. Then you'll be able to understand stochastic calculus rigorously. Last semester, I took a course in stochastic processes from the computer engineering department. You will be surprised to know that most computer engineers know little about the rigorous treatment of the stuff they work with everyday. A book that engineers use for a more or less mathematical treatment is Gallagher's Stochastic Processes which is a terrible book in my opinion. It doesn't satisfy mathematicians, neither does it explain the beautiful intuitions that sometimes engineering offers.

One advantage to the rigorous path is that you get to learn about some other fields like financial mathematics as well. The rigorous approach is helpful when you want to define things like conditional expectation and Radon-Nikodym derivative. But after all, I think it's not wise to spend too much time on 'abstractions'.

You need to spend a lot of time on programming. Learn Python or R, preferably. You need to learn about Markov chain Monte Carlo methods. You also may need to learn about calculus of variation at some point. Overall, the list of things that you can learn is endless. You may like to learn differential geometry to understand information geometry which is more theoretical than practical. Also, some knowledge from physics like thermodynamics can be helpful when you study things like the Boltzmann machine, etc. Again, I would like to emphasize that many of the recent advances in neural networks and deep learning do not really require advanced (abstract) mathematics. Just some linear algebra, a good understanding of probability theory, some experience with matrix calculations as in The Matrix Cookbook and some creativity that engineers have is enough to start your journey. Once you have started and you have chosen your final destination, you will acquire the knowledge you need along the way.

stressed out
  • 8,130
  • Thank you again. If I had to create a minimal list of topics to cover it would be: 1) Introduction to Statistical Learning, 2) Applied Predictive Models, 3) Some Statistical Inference, 4) Measure Theory, 5) Complex Analysis, 6) Topology, 7) Modern Probability Theory, 8) PDEs, 9) Stochastic Calculus, 10) Linear Algebra, 11) Bishop Pattern Recognition, 12) Bayesian Decision Analysis, 13) Elements of Statistical Learning, 14) Deep Learning Book, 15) Reinforcement Learning Book. Do you think this can be cut down even further? I might still be biased. – Kalkirin Jan 02 '19 at 02:13
  • Well, many of those topics you mentioned have enormous overlaps. I don't think you need topology. The amount of topology one learns when learning analysis is enough. I don't think learning about non-Hausdorff spaces would be useful in future, but I can be wrong. It's not easy to learn stochastic calculus because it's too broad. For example, there is Ito's calculus with many results like Feynman-Kac formula, etc. There's also Malliavin's calculus. I think a fair amount of measure theory, complex analysis, linear algebra and PDEs should provide you with a reasonable mathematical background. – stressed out Jan 02 '19 at 02:20
  • Thank you so much! I'll rework a budget of my available time per day with my job and a budget of my growing savings when I can comfortably go to graduate school. I'll use that to decide if I can add any other subjects onto this minimal list. I'll follow your recommendation to let Topology for Analysis go for now. However, I think it could be useful for certain projects in Machine Vision where you try to classify objects by similar topology through depth perception and 3D model prediction as one layer of an analysis. What do you think of that? – Kalkirin Jan 02 '19 at 02:31
  • @Kalkirin Well, I'm afraid most of the topics you mention are so deep that if you want to study them up to a level that helps you understand research stuff, you will never be able to study for a PhD. If you want to study topology, a natural thing to do is to study algebraic topology because Betti numbers and cohomology are computational while topology itself is not very computational and therefore, it's hard to implement it in computers. If you want to learn about computer vision, you need to learn about projective geometry and the camera model more than topology. Are you a CS student? – stressed out Jan 02 '19 at 02:39
  • Also, Topological Data Analysis using Algebraic Topology is still in its infancy, but have you heard of uses of it in machine learning yet? Here's a potentially good article showing how TDA and Machine Learning Together are greater than the sum of their parts (https://www.kdnuggets.com/2015/09/ayasdi-topological-analysis-machine-learning.html) – Kalkirin Jan 02 '19 at 02:39
  • I am taking a class in CS online every semester right now. – Kalkirin Jan 02 '19 at 02:41
  • @Kalkirin The reason I asked is because I have a CS friend who is very much like you. :D The thing is that the mathematical subjects you mentioned, when you study them like a mathematician, are not as fun as you think they are. They are very abstract. And it takes months or years to get acquainted with them. As I said, if you want to learn topology, you can't just stop there. You need to study simplicial complexes, CW structures, cohomology, etc. So, you'll need to study algebraic topology. And it's like a tree with an endless number of branches. You can't cover all of them in a life time. – stressed out Jan 02 '19 at 02:52
  • Here's a project I would like to work on, even if I can only dent the surface my entire life: Make an unsupervised multi-layer algorithm that identifies objects by classifying connectedness, topology, geometry, color, texture, etc by analyzing video. You determine how close one classification type is to another in the same layer by the extent of stochastic transformation needed to transform one type to the other. Then you have an overall distance calculated using a linear combination of these distances per layer. This allows the program to identify images without labeling. – Kalkirin Jan 02 '19 at 02:53
  • @Kalkirin That sounds cool. I think Ethan Bolker gave you a fantastic suggestion. I think you need to find a professor who has time and is willing to help you with your ideas. Or try to get in touch with researchers who work in these fields. – stressed out Jan 02 '19 at 03:01
  • Thank you again Stressed Out :D – Kalkirin Jan 02 '19 at 03:05
  • @Kalkirin You're welcome. Since you are interested in computer vision, I think you might also want to check 'Multiple View Geometry' by R. Hartley. It's not machine learning, it uses classical techniques in computer vision. But you might find it interesting. Anyway, I got to go now. Good luck with your future project. – stressed out Jan 02 '19 at 03:07
  • 1
    I also highly recommend Bishop's Pattern Recognition. Fantastic resource and easily understandable. – Alex L Jan 02 '19 at 06:47
  • 1
    This is an amazing answer. Thanks! – Higurashi Jan 02 '19 at 22:35
  • @ElieLouis Thank you for your comment. It always makes me glad to know that someone has found my answer helpful. :) – stressed out Jan 02 '19 at 22:45
1

By not stopping. If you keep thinking about the topic with a misconception in your head (even if it's a small untrue lemma that you never stated explicitly), you will eventually prove something you know to be untrue. Then you can have lots of fun* going over everything you thought you knew with a fine-toothed comb, and trying to find the bug.

*: Your mileage may vary.

Solveit
  • 838
  • A "proof" of an untrue proposition will, theoretically, eventually lead to a clearly false conclusion, but this requires that all further logic be perfect, it is hardly certain that this will occur within a practical period of time (where "practical" means not only "within a person's lifetime", but "before the heat death of the universe"), and it doesn't address the issue of invalid proofs of true propositions. – Acccumulation Jan 02 '19 at 17:29
  • I agree, but this seems to work very well in practice. – Solveit Jan 03 '19 at 00:12
0

Put it away for a while and come back to it later after you've forgotten most of it. Then either grade it yourself with a fresh brain, or try reproving it cold and compare the two proofs.

Learn the language and software for doing automated proof checking and have a computer check your proof for you. Previously: State of the progess of the automated proof checking

user3067860
  • 221
0

Find a study partner, or even better, group, and critique each other's proofs. If you go to professors' office hours and wait for them to not have any students there, they may be willing to go over your proofs.

If you can't find other people, user3067860 's suggestion of coming back to them is good. Also, try to look at what other conclusions you can derive from whatever claims you introduce in your proofs. If you have a proof that the number of primes is infinite, but your proof can also be used to show the number of primes less than 1000 is infinite, you know you messed up somewhere. Also, looking at the contrapositive can be helpful sometimes: if you have a proof $a\rightarrow b\rightarrow c \rightarrow d$, see what happens when you change it to $ \neg c \rightarrow \neg b \rightarrow \neg a$

Acccumulation
  • 12,210