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I am interested in continuing my education in math and I know that I'd eventually like to work on brain-computer interface (theory and application) like mind uploading but was curious if there is a discipline that merges computational neuroscience, biostatistics, AI, and cybersecurity: providing a rigourous curriculum that can be used to pursue these fields. Any input would be greatly appreciated! This is ultimately to maximize my chances of being employed, having a successful career long term.

If the opportunity exists, I would equally like to learn more about AI (neural net) and cybersecurity, and I currently enjoy the statistical, predictive modeling (machine learning) work that I do in genetics (similar to data science).

I have thoroughly looked through gradcafe, stackexchange, quora, reddit and amassed math topics important in each field. I have highlighted common topics and would like to get you guys' input on the accuracy of this list.

MATH TOPICS FOR EACH FIELD

  1. cybersecurity - applied number theory (abstract algebra), combinatorics (graph theory), algebraic geometry, information theory, asymptotic analysis, finite fields
  2. computational neuroscience - information theory, systems theory (nonlinear dynamics, dynamical systems), evolutionary algorithms (Monte Carlo), state space analysis, signal processing, probability theory
  3. AI/ML - neural networks, genetic algorithms, information geometry (Riemannian geometry, information theory, Fisher information), algebraic geometry, manifold geometry, learning theory (Fourier analysis), probability theory, game theory (topology, measure theory), graph theory, Model Free Methods

RECOMMENDATIONS

  • Some have recommended biostatistics programs because the curriculum offers a fair amount of 'theoretical' math work.
  • Others, however, have said that biostatistics is a bad choice - sticking to CS or EE would be better.
  • There is always the option to go into pure math but I am concerned about employability of a pure math PhD compared to an applied math PhD.
  • I have played with the idea of work towards becoming a fellow of actuarial science simultaneously instead to gain statistical training - although this would be more oriented towards business, not science

There is also the fact that I have a BS in biochemistry. I have done post-bacc work for CS fundamentals, calculus series, diff. eq., linear algebra, statistics, combinatorics, but there is a legitimate chance that I may not have sufficient background for fields (like statistics or applied math) other than biostatistics.

I have looked heavily into degrees for applied/computational mathematics, scientific computing (UPENN, Rice, JHU, MIT, Stanford, Maryland) but it seems that these fields are more broadly focused on application reseach for physics, chemistry, biology (like engineering). I've also looked into mathematical biology (aka biomathematics) but it seems not a lot of schools have such a department - it's commonly housed under computational/systems biology.

Thank you very much for your time and help!

Edited by inbrsuan

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For the topics you mention above, EECS (either CS or EE) is the appropriate venue. I think it's worth thinking about the one-line research summary of different fields (in context of areas you mentioned) to determine where you'll fit.

EECS looks at neural networks, learning, signal processing, control etc. from a computational viewpoint to endow machines with human-like capabilities. In other words, it's the study of information and decision systems. Computational neuroscience uses primarily biological and some computational tools to understand how the brain works. The difference between the two is that any theory of human brain coming out of neuroscience will be validated on how closely it mimics the human brain as opposed to how computationally sound it is. However, in EECS, as long as the algorithm provides the capabilities as intended, the biological plausibility takes a backseat.

Applied math studies mathematical principles to understand how the world works, and hence is primarily a service discipline to physics, chemistry, biology, engineering etc. They generally don't have close connections to the information sciences, which seems to be where your interest lies. Bioinformatics uses computational tools to adress some questions in biology. I wouldn't choose this discipline since it's overspecialization. You can always sit in EECS, and if you find your interests in biology taking over, do a PhD in bioinformatics. On the other hand, sitting in bioinformatics and doing ML for robotics is not possible.

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Thank you very much for your input. I am curious if, considering my background in biochemistry, getting into a PhD EECS program will be plausible.

Also are you familiar with computational science/engineering (Harvard, UT Austin)? At first glance it seems similar to applied math in that it focuses on application research but its a "nascent" field that commonly accepts people with life sciences background and may also study ML/AI from an interdesciplinary approach .

Lastly how much statistical training could I get by pursuing EECS? I know this varies by school but was curious to know generally what I should expect. Thank you again for your comment. It has been tremendously insightful.

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@inbrsuan 1. I don't think you require an undergrad degree in EECS to get into an EECS PhD program (I myself didn't have one). One option might be to get an MS in EECS, and then try for a PhD. This way, you'll have more exposure about the field of EECS and where your interests actually lie. 

2. Yes, I am familiar with many computational science, computational math, or applied math programs. I got into the ICES program at UTA and a similar program at UMD last year. After I spoke to the professors, it was clear that the presence of EECS in these programs is minimal. They may list a few people on their website, but in reality the program is mostly MechEs, AEs, and some physicists. The only exceptions I am aware of are Caltech CMS and possibly Stanford ICME, both of which are big on the interface of math, CS, EE, and statistics. Naturally, these are the most competitive programs, and I was rejected.

3. Depending on the area in EECS, PhD students know as much data science as statistics students. Obvious areas are machine learning and signal processing. Students working in control theory, information theory, and communications also take quite a few statistics courses and use them effectively for their work.

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@compscian, so I had thought that EECS was simply a notation difference from EE and CS programs but it seems that there is a difference in curriculum.

I had considered doing an MS first as well but I thought that it would be more cost effective (albeit more risky) to take a few classes (that are part of the core PhD curriculum) as a non-degree while working in a lab. I would then apply the following cycle.

I actually looked into Stanford, Caltech as well, but Caltech CMS did not have a MS option. MIT also has a similar program called CDO that looked promising.

May I ask how you transitioned from MechE to ML/AI - did you take classes as a post-bac? If so how much additional coursework/time did you take to be competitive?

Do you think the best point of contact to inquire about a dept's emphasis on EECS would be the program director or professors within the dept?

I'm trying to transition into a diff. field late (26 yr) and essentially navigating through this by myself. You've been a tremendous resource, @compscian.

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1 hour ago, inbrsuan said:

@compscian, so I had thought that EECS was simply a notation difference from EE and CS programs but it seems that there is a difference in curriculum.

When I say EECS programs, I mean either EE programs with CS focus (Berkeley, MIT, Washington, Michigan etc.) or CS programs with AI or AI+theory dominant research. There are of course EE programs with focus on devices or CS programs with systems-dominant research (eg Wisconsin Madison) - you shouldn't apply to them.

1 hour ago, inbrsuan said:

I actually looked into Stanford, Caltech as well, but Caltech CMS did not have a MS option. MIT also has a similar program called CDO that looked promising.

Stanford ICME has an MS option, but their stated interest is in "data science" which is different in both goal and spirit from AI. Caltech CMS or even CNS is probably the closest to what you are looking for. CDO is very very different, it is a traditional computational math program with ME, AE etc dominant research. You should probably be looking at some MS program in https://idss.mit.edu/

I realized the shift in my interest in senior year. So had 1 semester left, during which I completely overloaded myself with ML courses (normal load for final semester is 3 courses, I took 6 - nearly all in ML). I have since been working in an industrial research lab (think Microsoft, Google, IBM etc) related to ML and optimization. It was easier for me to transition to EE than CS-proper. So I applied to only those programs which are either merged with CS (ie EECS) or have close connections with CS and will let me work with a CS adviser. I am most likely to go to Washington which has an awesome ML group and shared resources among CS, EE, and Statistics. 

 

Edited by compscian

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@compscian, that is very admirable and awesome to hear that you successfully switched fields. If you took 6 ML courses (what courses specifically if I may ask), I think the barrier to entry for me would be higher. And I may not have the money/time to do additional post-bac work (although CMU/UWash seem to have affordable non-degree tracks).

What is your opinion on human-computer interaction programs (CMU, UMich, Stanford)? At first glance it seems to have a lesser emphasis on ML/AI but it seems much more accommodating to non-CS degree holders and it may serve as a good point to horizontally transition into ML dept.

UWash does seem like it has a really great interdesciplinary approach between EE/stat/CS. I also saw that CMU actually offers a joint PhD in statistics and machine learning and Princeton looks to do something similar in the near future. What do you think was the biggest factor for your success/admission? I'm assuming you applied only to top programs (Stanford, UWash, UT Austin, Columbia, Berkeley, Michigan) as well.

 

EDIT: It also seems that going into PhD statistics could be more viable/realistic as the pre-req is not as steep as that of EECS (like UChicago) and take a more mathematical (rather than CS) approach to machine learning (anecdote). It's said that a CS program is less rigourous on hard math training but I'm curious if students have the opportunity (in UWash, for example) to take advance pure math courses like algebraic geometry.

Edited by inbrsuan

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Hi @inbrsuan

I think you are just mixing up too many factors. First recognize what you want to do - AI, statistics, neuroscience etc etc. All of them are distinct fields, and though they share overlap, they are different in aims and scope. Once you have decided and are sure, you should just pick the appropriate program. For AI, it's EECS, plain and simple. It's better to drop a few years, get the relevant experience, and be in a department that supports your interest; as opposed to just joining something and feeling miserable about research fit mismatch. 

One way to make sure about research fit is to see what current students in the program are doing, whom they are working with, and whether the adviser is open to taking more students from the program. It's highly likely that you will have a career similar to the current alumni - are you satisfied. These are relevant and important questions, and I wouldn't cut corners like what you seem to be describing.

7 hours ago, inbrsuan said:

What is your opinion on human-computer interaction programs (CMU, UMich, Stanford)?

I am not familiar with them. if you are talking about "ischool" programs (ie PhD in information systems and such), they are quite related to ML and AI, but again the aim and scope are different. Also, I think ischool programs are not as strong as EECS, since most students who are admitted to both will prefer EECS. You should make sure that you can work with the adviser you want if you are admitted to such programs. For job in industry, EECS vs ischool isn't a very big difference I think. However, academia will definitely prefer EECS.

7 hours ago, inbrsuan said:

EDIT: It also seems that going into PhD statistics could be more viable/realistic as the pre-req is not as steep as that of EECS (like UChicago) and take a more mathematical (rather than CS) approach to machine learning (anecdote). It's said that a CS program is less rigourous on hard math training but I'm curious if students have the opportunity (in UWash, for example) to take advance pure math courses like algebraic geometry.

Again, EECS and Statistics have different aims and scope, different definitions for what constitutes a "successful project", and hence correspondingly slightly different training. Expected deliverable for theoretical statistics will be a strong theorem and proof; for applied statistics clear description of methodology and tabulated results. For EECS, it will be development, analysis, and numerical experiments for proposed algorithms. The nature of problems that interest ML/AI people and Statistics people are also very different. For example, much of recent ML/AI work (deep learning) was motivated by applications in computer vision, which has no presence in statistics. Unless you really care about vision, language, speech, or robotics - you can't hope to advance the state of the art in neural networks. However, statisticians don't work on any of these problems. IMHO, this is why statistics has lost ground to ML, even though it is basically the same thing, but done and conceived very differently.

Edited by compscian

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