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Can I start learning machine learning and deep learning without brushing up my maths knowledge first?
Originally Answered: What are the prerequisites in maths to master machine learning?
Are mathematics and machine learning related?
A couple of years ago, based on his experience, Bradford Cross gave a comprehensive list of the best resources on machine learning and the
How can I learn math?
prerequisites in his blog ("Measuring measures"). Unfortunately, it appears to be
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down right now.
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UPD:
In other languages
Here the blog post at WebArchive's mirror is: En español: ¿Cómo aprendo matemáticas para el aprendizaje de máquinas? Auf Deutsch: Wie kann ich Mathematik für Maschinenlernen erlernen? In italiano: Come posso imparare le leggi matematiche per l'apprendimento automatico?
http://web.archive.org/web/20101...
Bradford's lists at Amazon: Analysis [1] Linear Algebra [2] Probability [3] Statistics [4, 5] Optimization [6] Machine learning [7] Feature Selection [8]
I hope, Mr. Cross will be able to the discussion. [1] http://www.amazon.com/Analysis/l... [2] http://www.amazon.com/Matrix-Fu/...
[3] http://www.amazon.com/Probabilit... [4] http://www.amazon.com/Statistics...
Still have a question? Ask your own! [5] http://www.amazon.com/Nonparamet...
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[6] http://www.amazon.com/Heuristic-... [7] http://www.amazon.com/Machine-Le... [8] http://www.amazon.com/Feature-Se...
UPD 2: Here is the list of must-read books for theoretical machine learning [1], which is attributed to prof. Michael Jordan (UC Berkeley). The sources are [2] and [3]. [1] https://www.goodreads.com/review... [2] Learning About Statistical Learning
[3] AMA: Michael I Jordan • /r/MachineLearning 98.5k Views · View Upvoters Your response is private.
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What are the mathematical pre-requisites for studying machine learning? How did Roman Trusov learn machine learning? How did Vladimir Novakovski learn Machine Learning? How do I learn machine learning? How do I learn machine learning fast?
Ask New Question Calvin John, Autonomous vehicle researcher; ing UCLA in Fall 2017 Answered May 23 2017 · Author has 121 answers and 216.7k answer views Originally Answered: How can I become very good in applied mathematics for robotics and machine learning before graduate school and while studying alone?
Let me first caveat what I’m about to say with this: go to graduate school. [math]\dagger [/math] To show you just how super-serious I am about this, I’m even going to separate this caveat from the rest of the answer with one of the ultra-cool line breaks.
Alright, at this point, I’m assuming that you are still solely considering graduate school preparation without an undergraduate education. Let’s go.
Still have a question? Ask your own! My background consists of an undergraduate BS in mathematics, a minor in
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particle detectors (physics/EE) to autonomous vehicle system design for collision detection and evasion. Long story short: I’m far more qualified to answer your question when robotics is emphasized, so that’s what I’m going to do. Robotics is Multi-Disciplinary Robotics is a highly multi-disciplinary field. In fact, I’d argue that it could well be the academic field which encomes the largest quantity of distinct domains into its core structure. When we’re talking about robotics, we’re really talking about Computer science Mathematics Computer engineering Electrical engineering Control engineering Systems engineering Mechanical engineering Physics (mechanics, more specifically) What’s even more impressive about the above list than its size is the depth of each field. Aside from control and systems engineering, which are a bit more specialized and less fundamental than the others, each of the above domains are extremely broad—indicating that if you were to break down robotics concepts into a networked graph, it would resemble something like this: [1]
Needless to say, roboticist ultimately specialize in a much narrower range so that expertise in a topic can be attained. But that doesn’t change the fact that to pursue robotics, high breadth and versatility in engineering and math is a tool whose utility can’t be overstated.
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Now, regardless of whether you want to pursue a masters or a Ph.D., you will ultimately have to carve out a niche for yourself. As I mentioned above, mastery of all robotics is a hopelessly daunting task; it’s impossible. Therefore, it’s important that you expose yourself to the different areas of robotics, and gradually hone in on your desired path according to the topics with which you’re interested and at which you’re talented. Here’s my breakdown of robotics research, in increasing order of mathematical abstraction and decreasing order of hands-on engineering and building: 1. Sensors. About as applied and hands-on as you can get, the domain of sensors works on expanding the current technical constraints that robotics hardware faces. It’s because of these guys that the iphone magically gets smaller and smaller every year, while also increasing its technological capacities. An example of the importance of this domain which is even more specific to robotics is radar evading drones. when Osama Bin Laden got taken out because we flew a helicopter in Afghanistan that magically evades radars? Thanks sensors. 2. Nano-robotics. Focusing on developing robotic systems on the microlevel, nano-robotics explores how robotic agents can be built and implemented on a scale sufficiently small that they can be directly inserted into your body. Sound scary? It shouldn’t. Nano-robotics has a plethora of game-changing medical applications, some of which include legitimately curing cancer and preventing aging. 3. Machine vision. While the ability to process and interpret visual information comes very intuitively to humans, translating our abilities to an algorithmic environment in this matter has proven to be an intimidating process. In fact, I’d argue that the largest obstacle facing self-driving cars is machine vision. Just take a look at the self-driving car expert at Tesla who died because his car failed to distinguish between the bright sky and an incoming white truck. [2] 4. Robotic learning. When machine learning is applied in a robotic context, it basically becomes robotic learning. Robotic learning is the overlap between robotics and machine learning; it approaches the problem of developing tools for adaptation and learning in robotic systems. Very cool field, with a lot of promising application, and very well suited for someone interested in machine learning and robotics. 5. Robotic control. This is the area in which I’m currently nested. Control represents a mathematical approach to modeling the behavior and evolution of a Dynamical system - Wikipedia
in relation to inputs,
which can be used to affect the system’s output. The goal here is to mathematically demonstrate that a certain approach for input selection guarantees that the system’s output will quickly converge to a stabilized desired range, as illustrated in this kick-a** picture. [3]
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Because you have stated that robotics and machine learning are your interests, I’m going to assume your interests align with the #3–5 end of the spectrum. But even when your interests are honed in on these two areas, there is still a massive range of topics and skill sets spanned by these two very broad domains.
Developing Skills for Robotic Learning Again, I’m far from an expert in robotic learning and machine learning, but I’ll do my best to show some helpful tips for pursuing this domain. The fundamental fields from which machine learning constantly draws, as I understand it, are the following: Probability Statistics Algorithms Optimization Systems The last one is a bit more of a stretch in comparison to the others, but I’ve heard that a high portion of machine learning can actually be approached from a systems perspective, and that its inception actually arose from system theory modeling. For probability and statistics, both intuition and rigorous technicality will be important. I had a horrible textbook which provided very little conceptual basis for the theorems, and mostly included a bunch of isolated problems which were crudely connected in a very disted way. I recommend Introduction to Probability by Grinstead and Snell, [4] which provides a lot of clear, wellarticulated conceptual explanations which enhance both intuition and precise reasoning on the subject. It’s also free and available online, which ya’ know, is always a big plus.
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Becoming comfortable with algorithms is a task which can more easily be
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independently. Regarding a textbook to guide you through key concepts to algorithm theory, I recommend to look no further than the classic Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein. [5] Additionally, I would look to two additional sources to continually expand algorithmic skills: Project Euler Archived Problems - Project Euler
and
Topcoder Deliver Faster through Crowdsourcing . Project Euler encomes a diverse range of mathematical problems for algorithmic development which will strengthen your mathematical algorithmic thinking, and your “out of the box,” creativity. Topcoder provides challenges which will improve your technical programming skills, and diversify and expand your problem solving breadth. Of course, once you have a solid background in the above topics, you’ll want to receive a comprehensive introduction to robot learning, for which I’ve been told that Robot Learning by Connell and Mahadevan is a solid choice. [6] Although robotic learning and robotic control are distinct domains, robotic learning is intrinsically tied in to concepts from control theory. In fact, one of the most challenging problems facing the robotic learning community is that it lacks the rigorous analysis and descriptions that the control and systems theories possess. For example, a self-driving car that implements a series of clever robotic learning algorithms will never be implemented without tools from control systems. Why? Because without tools from control and systems theory, you will never get close to demonstrating rigorous, mathematically demanding qualities such as robustness, safety guarantees, stability, etc., without which, the government wouldn’t let your self-driving car see the light of day. Robotic Control I think that optimization, control, and systems are all presented and integrated very concisely in Design of Optimal Control Systems by Bini. [7] This book consists of more than a minimal amount of knowledge in any of these topics which is needed for machine learning. But a deep understanding of at least some of the ideas shown in this book will allow for insights to be drawn between these domains which most others will likely not be capable of seeing. Note that I recommend the above for someone interested in both machine learning and robotic control. If you’re primarily interested in robotic control, then your mathematical skills need to be more sophisticated than the vast majority of other engineers. This is likely the only engineering discipline in which highly abstract mathematical fields play a fundamental role. They include 1. Real analysis 2. Systems of Differential Equations 3. Dynamical Systems (similar to 2., but distinct from it) 4. Advanced Linear Algebra
5.Still have a question? Ask your own! Advanced Optimization What your question? 6. Basicis Topology https://www.quora.com/How-do-I-learn-mathematics-for-machine-learning
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7. Set Theory (more than the basics, but not quite “advanced” set theory) Clearly, your mathematical skills have to be beyond the more applied end of the spectrum in which things like formalities, proofs, theorems, and rigor are almost never relevant. For a comprehensive introduction to real analysis and topology that isn’t esoteric (difficult to find), I recommend Basic Analysis by Lebl. [8] While the book isn’t intended for studying topology specifically, it covers nearly all of the fundamentals which are relevant to control. Note that real analysis is the most important item in the above list. Advanced Linear algebra is the most difficult field for which to find an accessible, engaging textbook, I.M.O.. The majority of the texts are far too focused on minute, irrelevant details and burdensome proofs whose understanding gains little insight regarding the deeper concepts. More importantly, most textbooks totally fail to connect the ideas to deeper concepts which are both cool and incredibly useful. After a lot of searching, I found hope in an unexpected place: online lecture notes. [9] If you master this book, and its difficult problems, to the point where you can comfortably walk through the main concepts with a high school student, then you’ll be five steps ahead of me. As for dynamical systems, I’d say that Dynamical Systems by Sternberg does the trick. [10] Until you get to the more theoretical content like stability and invariance, you really want to focus more on the concepts; the details aren’t particularly important, surprisingly. You really just need to know what kind of assumptions you have to make about the system you’re modeling. Once you’re comfortable with most of the above, you can get your hands dirty with some actual control theory. For this, I recommend Mathematical Control Theory by Sontag. [11]
[math]\dagger: [/math] I have a hunch that’s not what you want to here, since you didn’t ask for advice regarding this matter. So I’m sorry if this caveat irks you in any way, but it’s the best advice I can give, and I think it’s important for you to hear. I’m a firm believer in pragmatic optimism, and while it’s optimistic to believe that ittance into graduate school—especially in a technical field—is feasible without an undergraduate degree, it is far from optimistic. Without an undergraduate degree, you are immediately excluded from consideration for all departments at the majority of universities. I can’t find any specific statistics on this matter, so you’ll have to choose whether or not to take my word for it. But trust me when I say that I can currently think of one graduate school that doesn’t necessitate an undergraduate degree as a strict requirement. Even putting the strict requirements aside, for deeply embedded multidisciplinary fieldsAsk likeyour robotics Still have a question? own! and machine learning, an undergraduate education is crucial. Although I do think that the ability to interact with
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designed by experts on which you are tested in a competitive environment are all vital assets for initiating the engineering experience in any field, they are especially true for robotics. Another important distinction regarding your question is are you planning for a masters or a Ph.D.? [1] Pawel Pralat: Graph Theory [2] Tesla driver killed while using autopilot was watching Harry Potter, witness says [3] Vehicle stability control systems: An overview of the integrated ... [4]https://www.dartmouth.edu/~chanc... [5] Introduction to Algorithms, 3rd Edition (MIT Press): Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein: 9780262033848: Amazon.com: Books [6] Robot Learning | J. H. Connell | Springer [7] http://retis.sssup.it/~bini/math... [8] http://www.jirka.org/ra/realanal... [9] https://www.math.uh.edu/~climenh... [10] Dynamical Systems (Dover Books on Mathematics): Shlomo Sternberg: 9780486477053: Amazon.com: Books [11]http://www.mit.edu/~esontag/FTPD... 15.5k Views · View Upvoters · Answer requested by Abdulmajeed Kabir
Abhinav Sharma, I've designed for and built ML systems Updated Dec 14 2013 · Upvoted by Tudor Achim, phd student working on machine learning · Author has 841 answers and 5.7m answer views
When going through my Machine Learning course last semester, I felt like I had the most catching up to do with Linear Algebra. I felt key ideas from LinAlg are harder to over time than Probability. I found myself to be mostly working with probability distributions, Bayes' rule, MLEs and MAPs, while the algebra side of it was mostly optimization in higher dimensions, was mostly Matrix calculus.
I discovered that the Matrix Cookbook was popular with most students for working with Matrix Calculus as it seems to have a never-ending list of matrix derivatives: http://www2.imm.dtu.dk/pubdb/vie...
As far as brushing up on the rest of your Linear Algebra knowledge is concerned, I highly recommend Strang's lectures/book: http://ocw.mit.edu/courses/mathe... Still have a question? Ask your own!
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Highly relevant topics include knowing about rank and inversion, SVD, and also https://www.quora.com/How-do-I-learn-mathematics-for-machine-learning
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make sure you're very comfortable with eigenvalues and eigenvectors, amongst other things.
Finally, with Analysis, I don't think ML requires a formal introduction to Analysis at all. Its important to know higher dimensional calculus well, especially parts related to optimization, such as Lagrange multipliers, the primal-dual form, and in general, the calculus of Matrices, and you should be good to go.
Overall, I think the case with Linear Algebra and Calculus is to work your way through an ML book/course, and stop and look at the relevant math when necessary, whereas you need a strong foundation in Probability right from the beginning, and most textbooks on ML tend to talk a lot about probability while skimming over the mathematical details of LinAlg and Calculus. 124.3k Views · View Upvoters Your response is private.
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Ansup Babu, Working on Uncertainity Quantification Answered Dec 15 2016 · Upvoted by Colby, MS Machine Learning & Computer Science, Fu Foundation School of Engineering and Applied Science, Co…
If you want to be a real Data Scientist Not the fake ones with skills of Analyst and not any mathematical intuition or point of view. Real Data Scientist Need to have very strong mathematical grounding. So to learn Mathematics for ML this should be the order :1. Start with probability ( Conditional Basic Marginal etc …) 2. Mathematical Series and Convergence , Numerical methods for Analysis 3. Matrix and Linear Algebra 4. Bayesian Statistics 5. Vectors ( Most Important) 6. Calculus 7. Markov Process and Chains 8. Basics of Optimization ( Linear/ Quadratic) 9. Advanced Matrix Algebras and Calculus ( Gradient , Divergence , Curls etc)
Still have a question? Ask your own! This much mathematics will enable the understanding behind the core ideas of
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You should pause now and start analysing certain Packages from Scratch in Python : 1. K-NN is great starting point learn it , and code it from scratch. 2. Logistic Regression with Gradient Descent. Till now you can see the parameters and numbers moving in a matrix form , and understand the mathematics of prediction, And if you feel this is enough. Hold your breath. There is more exciting stuff to come. This will enable you to be a beginner of being a “Real Data Scientist”. Next Start with :1. Stochastic Models and Time Series Analysis 2. Differential Equations 3. Dynamic Programming and Optimization Techniques 4. Fourier's and Wavelengths 5. Random Fields 6. Basic Knowledge of PDEs 7. Techniques to solve PDEs using Monte-Carlo , Polynomial Expansions. These mathematical techniques will help you visualize the model’s working and how to model and process raw data to create unique models whose functionality can be tuned. Parameters can be optimized for the problems and fine tuned with these techniques. For a Next Level Up:- ( Statistics of Higher Dimensions) 1. PDEs numerical solution with numerical input/ random input. ( fascinating subject to work on ) 2. Stochastic Differential Equations and Solutions 3. PCA etc 4. Dirichlet Processes, Markov Decision Process. 5. Uncertainty Quantification - Polynomial Chaos, Projections on vector space I think these are subject which one must learn to be a good Machine learning engineer in 21st century. with a knowledge base like this one can connect dots very rapidly and build systems and model of high accuracy. ( I am not a big fan of Neural nets,..so forgot to mention here) 5k Views · View Upvoters
Biswarup Bhattacharya Updated Jan 4 Originally Answered: How much should I go into mathematical concepts in machine learning?
I will try to keep this as concise as possible.
Still have a question? Ask your own! Edit: Somebody merged the original question to this question, so the premise
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To become a full stack AI/ML engineer, it is imperative that you have a complete grasp of the mathematical foundations of ML so that you can build upon concepts easily. The basic mathematical skills required are Linear Algebra, Matrix Algebra, Probability and some basic Calculus. Linear Algebra The best source to study Linear Algebra is Prof. Gilbert Strang’s Linear Algebra book/course. Video Lectures | Linear Algebra | Mathematics | MIT OpenCourseWare
(MIT OCW). There are 34 lectures and believe me, they are
completely worth it as after completing this, linear algebra should not pose any more problems for you. Solve some exercises/exams if you want to achieve mastery (recommended). Matrix Algebra Matrix algebra is an essential component of deep learning. I personally recommend this (Matrix Cookbook by Kaare Brandt Petersen & Michael Syskind Pedersen): http://www2.imm.dtu.dk/pubdb/vie...
(PDF). There are 66
pages of pure matrix operations and this is the absolute “go-to” in case you are stuck trying to understand certain matrix manipulations that a researcher might have done. Probability & Statistics Understanding probability is a very important aspect of understanding ML. Some of the key probability concepts that you must be aware of include Bayes’ Theorem, distributions, MLE, regression, inference and so on. The best resource for this is Think Stats (Exploratory Data Analysis in Python) by Allen Downey: http://greenteapress.com/thinkst...
(PDF). This absolute gem of a
book is 264 pages long and covers all the aspects of probability and statistics that you need to understand with relevant Python code. Optimization The go-to book for Convex Optimization is Convex Optimization by Stephen Boyd and Lieven Vandenberghe: https://web.stanford.edu/~boyd/c...
(PDF).
This is a 730 page book and you need not read it all in one go. Choose the concept which you need to learn depending on your requirements and interest and read that part. It is complete and extremely well written. This book is free as part of the CVX 101 MOOC on EdX. This 263 page book on metaheuristics, Essentials of Metaheuristics by Sean Luke (http://cs.gmu.edu/~sean/book/met...
(PDF)) talks about gradient based
optimization, policy optimization etc. and it is well written. One can choose to go through this also if interested. Data science concepts are covered in the above topics. Other topics can be learnt by googling for sources easily as and when you encounter them. But complete understanding of the above should suffice for 95% of all scenarios.
Still have a question? Ask your own! Achieving mastery of the above topics will surely make you a mathematically
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your feet into research papers. They are absolutely essential as these clearly show the standards of AI researchers/engineers. Firstly, find out the famous papers of AI like RNN, LSTM, SVM etc. and go through the technical content. Can you understand the jargon? Can you understand the mathematics? Can you implement the mathematics in code now without the help of overly sufficient libraries? These are the key questions to be answered. Once you can answer “Yes/Mostly Yes” to these 3 questions, you are good to go. After trying to read these papers dealing with the most popular concepts, try to read the not-so-famous papers. arXiv is a great site with hundreds of preprints being published everyday by top researchers and reading the papers from here is like drinking straight out of the fire-hose. Try to choose a paper which looks fairly well written and the abstract seems interesting. Then, read that paper and try to answer those 3 questions again. The same can be done with papers of top AI conferences like NIPS, AAAI, AAMAS, IJCAI, ICML etc. You may not be able to fully implement the papers due to data constraints and other issues, but if you are able to understand even 60% of the mathematical reasoning, then I can safely say you have completed your training. Do not concentrate on learning more and more “packages”. Concentrate on the concept. While implementing, you will automatically see that you require “this” package and then you will automatically learn to use it. Learning the various commands of random packages won’t help. If you start implementing and writing codes to solve problems or simulate results from a paper, you will automatically learn about packages and use them appropriately; they’ll be the least of your concerns. This is the correct way to maintain “balance” between math and coding. You can also participate in competitions (e.g. Kaggle or conference competitions) to improve speed, development and processing skills if you feel the need to do so. (All the links in this answer are working as of 6th July 2017) 8.3k Views · View Upvoters
Pankesh Bamotra, ML is recollecting what humans have learnt so far Answered Oct 14 2015 · Upvoted by Yuval Feinstein, Algorithmic Software Engineer in NLP,IR and Machine Learning · Author has 93 answers and 367.9k answer views
Mathematics is too vast a subject to be considered for this question. The breadth and depth of mathematical awareness you require for machine learning totally depends on what you are learning in the subject. Keeping this in mind, let's deal with what you need to know in "mathematics" for machine learning. 1. Probability and mathematical statistics This is a fundamental requirement for machine learning and so you need to know well. When I say probability it's more than what you studied in High school and almost everything you probably not paid attention to during your undergrad. You need Still have a question? your own! toknow about Random Ask variables, their distributions, probabilistic convergence,
and estimation theory. That covers a major part of what you need to know here. What is your question? Ask Two of my favourite resources are:https://www.quora.com/How-do-I-learn-mathematics-for-machine-learning
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1. Joseph Blitzstein - Harvard Stat 110 lectures 2. Larry Wasserman's book - All of statistics
2. Linear algebra Linear algebra will pop up every now and then in ML. PCA, SVD, LU decomposition, QR decomposition, symmetric matrices, othogonalization, projections, matrix operations are needed many a times. The good thing is that there are countless resources available on linear algebra. My all time favourite is Gilbert Strang's MIT lectures on linear algebra.
3. Optimisation Though only a few things from optimisation are needed most of the time, a strong foundational knowledge will help long way. You need to know Langrange multipliers, gradient descent, and primal-dual formulation. The best resource on this is Boyd and Vandenberghe's course on Convex optimisation from Stanford.
4. Calculus I wanted to put this on the top, but I'm putting it in the last just to emphasise on the fact that only a fundamental knowledge is needed in of calculus. Know about 3-D geometry, integration, and differentiation and you'll survive. It's the easiest to start with amongst the topic I've mentioned here. MIT has good lectures on calculus.
I think with these 4 tools you'll most likely find ML easy to understand. Other than these you may find real analysis and functional analysis relevant too, but they are just formal generalisations of the topics mentioned before. 13.7k Views · View Upvoters
Related Questions What does it take to do a PhD at Oxford in machine learning? Can I start learning machine learning and deep learning without brushing up my maths knowledge first? Are mathematics and machine learning related? How can I learn math? How can I learn machine learning at home? How should one start learning Algorithms? How does a total beginner start to learn machine learning if they have some knowledge of programming languages? Is Machine Learning easier than Mathematics? What math subject should I start to learn and what is the last one in order to start machine learning? Where can I find some online courses? What programming language is used for machine learning? Still have a question? Ask your own! Ask New Question
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Nikita Zhiltsov, Chief Scientist, Orb Intelligence Updated Sep 12 2014 · Upvoted by Yuval Feinstein, Algorithmic Software Engineer in NLP,IR and Machine Learning · Author has 66 answers and 346.1k answer views
Can I start learning machine learning and deep learning without brushing up my maths knowledge first?
Originally Answered: What are the prerequisites in maths to master machine learning?
Are mathematics and machine learning related?
A couple of years ago, based on his experience, Bradford Cross gave a comprehensive list of the best resources on machine learning and the
How can I learn math?
prerequisites in his blog ("Measuring measures"). Unfortunately, it appears to be
How can I learn machine learning at home?
down right now.
Ask New Question
UPD:
In other languages
Here the blog post at WebArchive's mirror is: En español: ¿Cómo aprendo matemáticas para el aprendizaje de máquinas? Auf Deutsch: Wie kann ich Mathematik für Maschinenlernen erlernen? In italiano: Come posso imparare le leggi matematiche per l'apprendimento automatico?
http://web.archive.org/web/20101...
Bradford's lists at Amazon: Analysis [1] Linear Algebra [2] Probability [3] Statistics [4, 5] Optimization [6] Machine learning [7] Feature Selection [8]
I hope, Mr. Cross will be able to the discussion. [1] http://www.amazon.com/Analysis/l... [2] http://www.amazon.com/Matrix-Fu/...
[3] http://www.amazon.com/Probabilit... [4] http://www.amazon.com/Statistics...
Still have a question? Ask your own! [5] http://www.amazon.com/Nonparamet...
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[6] http://www.amazon.com/Heuristic-... [7] http://www.amazon.com/Machine-Le... [8] http://www.amazon.com/Feature-Se...
UPD 2: Here is the list of must-read books for theoretical machine learning [1], which is attributed to prof. Michael Jordan (UC Berkeley). The sources are [2] and [3]. [1] https://www.goodreads.com/review... [2] Learning About Statistical Learning
[3] AMA: Michael I Jordan • /r/MachineLearning 98.5k Views · View Upvoters Your response is private.
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What are the mathematical pre-requisites for studying machine learning? How did Roman Trusov learn machine learning? How did Vladimir Novakovski learn Machine Learning? How do I learn machine learning? How do I learn machine learning fast?
Ask New Question Calvin John, Autonomous vehicle researcher; ing UCLA in Fall 2017 Answered May 23 2017 · Author has 121 answers and 216.7k answer views Originally Answered: How can I become very good in applied mathematics for robotics and machine learning before graduate school and while studying alone?
Let me first caveat what I’m about to say with this: go to graduate school. [math]\dagger [/math] To show you just how super-serious I am about this, I’m even going to separate this caveat from the rest of the answer with one of the ultra-cool line breaks.
Alright, at this point, I’m assuming that you are still solely considering graduate school preparation without an undergraduate education. Let’s go.
Still have a question? Ask your own! My background consists of an undergraduate BS in mathematics, a minor in
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particle detectors (physics/EE) to autonomous vehicle system design for collision detection and evasion. Long story short: I’m far more qualified to answer your question when robotics is emphasized, so that’s what I’m going to do. Robotics is Multi-Disciplinary Robotics is a highly multi-disciplinary field. In fact, I’d argue that it could well be the academic field which encomes the largest quantity of distinct domains into its core structure. When we’re talking about robotics, we’re really talking about Computer science Mathematics Computer engineering Electrical engineering Control engineering Systems engineering Mechanical engineering Physics (mechanics, more specifically) What’s even more impressive about the above list than its size is the depth of each field. Aside from control and systems engineering, which are a bit more specialized and less fundamental than the others, each of the above domains are extremely broad—indicating that if you were to break down robotics concepts into a networked graph, it would resemble something like this: [1]
Needless to say, roboticist ultimately specialize in a much narrower range so that expertise in a topic can be attained. But that doesn’t change the fact that to pursue robotics, high breadth and versatility in engineering and math is a tool whose utility can’t be overstated.
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Now, regardless of whether you want to pursue a masters or a Ph.D., you will ultimately have to carve out a niche for yourself. As I mentioned above, mastery of all robotics is a hopelessly daunting task; it’s impossible. Therefore, it’s important that you expose yourself to the different areas of robotics, and gradually hone in on your desired path according to the topics with which you’re interested and at which you’re talented. Here’s my breakdown of robotics research, in increasing order of mathematical abstraction and decreasing order of hands-on engineering and building: 1. Sensors. About as applied and hands-on as you can get, the domain of sensors works on expanding the current technical constraints that robotics hardware faces. It’s because of these guys that the iphone magically gets smaller and smaller every year, while also increasing its technological capacities. An example of the importance of this domain which is even more specific to robotics is radar evading drones. when Osama Bin Laden got taken out because we flew a helicopter in Afghanistan that magically evades radars? Thanks sensors. 2. Nano-robotics. Focusing on developing robotic systems on the microlevel, nano-robotics explores how robotic agents can be built and implemented on a scale sufficiently small that they can be directly inserted into your body. Sound scary? It shouldn’t. Nano-robotics has a plethora of game-changing medical applications, some of which include legitimately curing cancer and preventing aging. 3. Machine vision. While the ability to process and interpret visual information comes very intuitively to humans, translating our abilities to an algorithmic environment in this matter has proven to be an intimidating process. In fact, I’d argue that the largest obstacle facing self-driving cars is machine vision. Just take a look at the self-driving car expert at Tesla who died because his car failed to distinguish between the bright sky and an incoming white truck. [2] 4. Robotic learning. When machine learning is applied in a robotic context, it basically becomes robotic learning. Robotic learning is the overlap between robotics and machine learning; it approaches the problem of developing tools for adaptation and learning in robotic systems. Very cool field, with a lot of promising application, and very well suited for someone interested in machine learning and robotics. 5. Robotic control. This is the area in which I’m currently nested. Control represents a mathematical approach to modeling the behavior and evolution of a Dynamical system - Wikipedia
in relation to inputs,
which can be used to affect the system’s output. The goal here is to mathematically demonstrate that a certain approach for input selection guarantees that the system’s output will quickly converge to a stabilized desired range, as illustrated in this kick-a** picture. [3]
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Because you have stated that robotics and machine learning are your interests, I’m going to assume your interests align with the #3–5 end of the spectrum. But even when your interests are honed in on these two areas, there is still a massive range of topics and skill sets spanned by these two very broad domains.
Developing Skills for Robotic Learning Again, I’m far from an expert in robotic learning and machine learning, but I’ll do my best to show some helpful tips for pursuing this domain. The fundamental fields from which machine learning constantly draws, as I understand it, are the following: Probability Statistics Algorithms Optimization Systems The last one is a bit more of a stretch in comparison to the others, but I’ve heard that a high portion of machine learning can actually be approached from a systems perspective, and that its inception actually arose from system theory modeling. For probability and statistics, both intuition and rigorous technicality will be important. I had a horrible textbook which provided very little conceptual basis for the theorems, and mostly included a bunch of isolated problems which were crudely connected in a very disted way. I recommend Introduction to Probability by Grinstead and Snell, [4] which provides a lot of clear, wellarticulated conceptual explanations which enhance both intuition and precise reasoning on the subject. It’s also free and available online, which ya’ know, is always a big plus.
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Becoming comfortable with algorithms is a task which can more easily be
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independently. Regarding a textbook to guide you through key concepts to algorithm theory, I recommend to look no further than the classic Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein. [5] Additionally, I would look to two additional sources to continually expand algorithmic skills: Project Euler Archived Problems - Project Euler
and
Topcoder Deliver Faster through Crowdsourcing . Project Euler encomes a diverse range of mathematical problems for algorithmic development which will strengthen your mathematical algorithmic thinking, and your “out of the box,” creativity. Topcoder provides challenges which will improve your technical programming skills, and diversify and expand your problem solving breadth. Of course, once you have a solid background in the above topics, you’ll want to receive a comprehensive introduction to robot learning, for which I’ve been told that Robot Learning by Connell and Mahadevan is a solid choice. [6] Although robotic learning and robotic control are distinct domains, robotic learning is intrinsically tied in to concepts from control theory. In fact, one of the most challenging problems facing the robotic learning community is that it lacks the rigorous analysis and descriptions that the control and systems theories possess. For example, a self-driving car that implements a series of clever robotic learning algorithms will never be implemented without tools from control systems. Why? Because without tools from control and systems theory, you will never get close to demonstrating rigorous, mathematically demanding qualities such as robustness, safety guarantees, stability, etc., without which, the government wouldn’t let your self-driving car see the light of day. Robotic Control I think that optimization, control, and systems are all presented and integrated very concisely in Design of Optimal Control Systems by Bini. [7] This book consists of more than a minimal amount of knowledge in any of these topics which is needed for machine learning. But a deep understanding of at least some of the ideas shown in this book will allow for insights to be drawn between these domains which most others will likely not be capable of seeing. Note that I recommend the above for someone interested in both machine learning and robotic control. If you’re primarily interested in robotic control, then your mathematical skills need to be more sophisticated than the vast majority of other engineers. This is likely the only engineering discipline in which highly abstract mathematical fields play a fundamental role. They include 1. Real analysis 2. Systems of Differential Equations 3. Dynamical Systems (similar to 2., but distinct from it) 4. Advanced Linear Algebra
5.Still have a question? Ask your own! Advanced Optimization What your question? 6. Basicis Topology https://www.quora.com/How-do-I-learn-mathematics-for-machine-learning
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7. Set Theory (more than the basics, but not quite “advanced” set theory) Clearly, your mathematical skills have to be beyond the more applied end of the spectrum in which things like formalities, proofs, theorems, and rigor are almost never relevant. For a comprehensive introduction to real analysis and topology that isn’t esoteric (difficult to find), I recommend Basic Analysis by Lebl. [8] While the book isn’t intended for studying topology specifically, it covers nearly all of the fundamentals which are relevant to control. Note that real analysis is the most important item in the above list. Advanced Linear algebra is the most difficult field for which to find an accessible, engaging textbook, I.M.O.. The majority of the texts are far too focused on minute, irrelevant details and burdensome proofs whose understanding gains little insight regarding the deeper concepts. More importantly, most textbooks totally fail to connect the ideas to deeper concepts which are both cool and incredibly useful. After a lot of searching, I found hope in an unexpected place: online lecture notes. [9] If you master this book, and its difficult problems, to the point where you can comfortably walk through the main concepts with a high school student, then you’ll be five steps ahead of me. As for dynamical systems, I’d say that Dynamical Systems by Sternberg does the trick. [10] Until you get to the more theoretical content like stability and invariance, you really want to focus more on the concepts; the details aren’t particularly important, surprisingly. You really just need to know what kind of assumptions you have to make about the system you’re modeling. Once you’re comfortable with most of the above, you can get your hands dirty with some actual control theory. For this, I recommend Mathematical Control Theory by Sontag. [11]
[math]\dagger: [/math] I have a hunch that’s not what you want to here, since you didn’t ask for advice regarding this matter. So I’m sorry if this caveat irks you in any way, but it’s the best advice I can give, and I think it’s important for you to hear. I’m a firm believer in pragmatic optimism, and while it’s optimistic to believe that ittance into graduate school—especially in a technical field—is feasible without an undergraduate degree, it is far from optimistic. Without an undergraduate degree, you are immediately excluded from consideration for all departments at the majority of universities. I can’t find any specific statistics on this matter, so you’ll have to choose whether or not to take my word for it. But trust me when I say that I can currently think of one graduate school that doesn’t necessitate an undergraduate degree as a strict requirement. Even putting the strict requirements aside, for deeply embedded multidisciplinary fieldsAsk likeyour robotics Still have a question? own! and machine learning, an undergraduate education is crucial. Although I do think that the ability to interact with
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designed by experts on which you are tested in a competitive environment are all vital assets for initiating the engineering experience in any field, they are especially true for robotics. Another important distinction regarding your question is are you planning for a masters or a Ph.D.? [1] Pawel Pralat: Graph Theory [2] Tesla driver killed while using autopilot was watching Harry Potter, witness says [3] Vehicle stability control systems: An overview of the integrated ... [4]https://www.dartmouth.edu/~chanc... [5] Introduction to Algorithms, 3rd Edition (MIT Press): Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein: 9780262033848: Amazon.com: Books [6] Robot Learning | J. H. Connell | Springer [7] http://retis.sssup.it/~bini/math... [8] http://www.jirka.org/ra/realanal... [9] https://www.math.uh.edu/~climenh... [10] Dynamical Systems (Dover Books on Mathematics): Shlomo Sternberg: 9780486477053: Amazon.com: Books [11]http://www.mit.edu/~esontag/FTPD... 15.5k Views · View Upvoters · Answer requested by Abdulmajeed Kabir
Abhinav Sharma, I've designed for and built ML systems Updated Dec 14 2013 · Upvoted by Tudor Achim, phd student working on machine learning · Author has 841 answers and 5.7m answer views
When going through my Machine Learning course last semester, I felt like I had the most catching up to do with Linear Algebra. I felt key ideas from LinAlg are harder to over time than Probability. I found myself to be mostly working with probability distributions, Bayes' rule, MLEs and MAPs, while the algebra side of it was mostly optimization in higher dimensions, was mostly Matrix calculus.
I discovered that the Matrix Cookbook was popular with most students for working with Matrix Calculus as it seems to have a never-ending list of matrix derivatives: http://www2.imm.dtu.dk/pubdb/vie...
As far as brushing up on the rest of your Linear Algebra knowledge is concerned, I highly recommend Strang's lectures/book: http://ocw.mit.edu/courses/mathe... Still have a question? Ask your own!
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make sure you're very comfortable with eigenvalues and eigenvectors, amongst other things.
Finally, with Analysis, I don't think ML requires a formal introduction to Analysis at all. Its important to know higher dimensional calculus well, especially parts related to optimization, such as Lagrange multipliers, the primal-dual form, and in general, the calculus of Matrices, and you should be good to go.
Overall, I think the case with Linear Algebra and Calculus is to work your way through an ML book/course, and stop and look at the relevant math when necessary, whereas you need a strong foundation in Probability right from the beginning, and most textbooks on ML tend to talk a lot about probability while skimming over the mathematical details of LinAlg and Calculus. 124.3k Views · View Upvoters Your response is private.
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Ansup Babu, Working on Uncertainity Quantification Answered Dec 15 2016 · Upvoted by Colby, MS Machine Learning & Computer Science, Fu Foundation School of Engineering and Applied Science, Co…
If you want to be a real Data Scientist Not the fake ones with skills of Analyst and not any mathematical intuition or point of view. Real Data Scientist Need to have very strong mathematical grounding. So to learn Mathematics for ML this should be the order :1. Start with probability ( Conditional Basic Marginal etc …) 2. Mathematical Series and Convergence , Numerical methods for Analysis 3. Matrix and Linear Algebra 4. Bayesian Statistics 5. Vectors ( Most Important) 6. Calculus 7. Markov Process and Chains 8. Basics of Optimization ( Linear/ Quadratic) 9. Advanced Matrix Algebras and Calculus ( Gradient , Divergence , Curls etc)
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You should pause now and start analysing certain Packages from Scratch in Python : 1. K-NN is great starting point learn it , and code it from scratch. 2. Logistic Regression with Gradient Descent. Till now you can see the parameters and numbers moving in a matrix form , and understand the mathematics of prediction, And if you feel this is enough. Hold your breath. There is more exciting stuff to come. This will enable you to be a beginner of being a “Real Data Scientist”. Next Start with :1. Stochastic Models and Time Series Analysis 2. Differential Equations 3. Dynamic Programming and Optimization Techniques 4. Fourier's and Wavelengths 5. Random Fields 6. Basic Knowledge of PDEs 7. Techniques to solve PDEs using Monte-Carlo , Polynomial Expansions. These mathematical techniques will help you visualize the model’s working and how to model and process raw data to create unique models whose functionality can be tuned. Parameters can be optimized for the problems and fine tuned with these techniques. For a Next Level Up:- ( Statistics of Higher Dimensions) 1. PDEs numerical solution with numerical input/ random input. ( fascinating subject to work on ) 2. Stochastic Differential Equations and Solutions 3. PCA etc 4. Dirichlet Processes, Markov Decision Process. 5. Uncertainty Quantification - Polynomial Chaos, Projections on vector space I think these are subject which one must learn to be a good Machine learning engineer in 21st century. with a knowledge base like this one can connect dots very rapidly and build systems and model of high accuracy. ( I am not a big fan of Neural nets,..so forgot to mention here) 5k Views · View Upvoters
Biswarup Bhattacharya Updated Jan 4 Originally Answered: How much should I go into mathematical concepts in machine learning?
I will try to keep this as concise as possible.
Still have a question? Ask your own! Edit: Somebody merged the original question to this question, so the premise
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To become a full stack AI/ML engineer, it is imperative that you have a complete grasp of the mathematical foundations of ML so that you can build upon concepts easily. The basic mathematical skills required are Linear Algebra, Matrix Algebra, Probability and some basic Calculus. Linear Algebra The best source to study Linear Algebra is Prof. Gilbert Strang’s Linear Algebra book/course. Video Lectures | Linear Algebra | Mathematics | MIT OpenCourseWare
(MIT OCW). There are 34 lectures and believe me, they are
completely worth it as after completing this, linear algebra should not pose any more problems for you. Solve some exercises/exams if you want to achieve mastery (recommended). Matrix Algebra Matrix algebra is an essential component of deep learning. I personally recommend this (Matrix Cookbook by Kaare Brandt Petersen & Michael Syskind Pedersen): http://www2.imm.dtu.dk/pubdb/vie...
(PDF). There are 66
pages of pure matrix operations and this is the absolute “go-to” in case you are stuck trying to understand certain matrix manipulations that a researcher might have done. Probability & Statistics Understanding probability is a very important aspect of understanding ML. Some of the key probability concepts that you must be aware of include Bayes’ Theorem, distributions, MLE, regression, inference and so on. The best resource for this is Think Stats (Exploratory Data Analysis in Python) by Allen Downey: http://greenteapress.com/thinkst...
(PDF). This absolute gem of a
book is 264 pages long and covers all the aspects of probability and statistics that you need to understand with relevant Python code. Optimization The go-to book for Convex Optimization is Convex Optimization by Stephen Boyd and Lieven Vandenberghe: https://web.stanford.edu/~boyd/c...
(PDF).
This is a 730 page book and you need not read it all in one go. Choose the concept which you need to learn depending on your requirements and interest and read that part. It is complete and extremely well written. This book is free as part of the CVX 101 MOOC on EdX. This 263 page book on metaheuristics, Essentials of Metaheuristics by Sean Luke (http://cs.gmu.edu/~sean/book/met...
(PDF)) talks about gradient based
optimization, policy optimization etc. and it is well written. One can choose to go through this also if interested. Data science concepts are covered in the above topics. Other topics can be learnt by googling for sources easily as and when you encounter them. But complete understanding of the above should suffice for 95% of all scenarios.
Still have a question? Ask your own! Achieving mastery of the above topics will surely make you a mathematically
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your feet into research papers. They are absolutely essential as these clearly show the standards of AI researchers/engineers. Firstly, find out the famous papers of AI like RNN, LSTM, SVM etc. and go through the technical content. Can you understand the jargon? Can you understand the mathematics? Can you implement the mathematics in code now without the help of overly sufficient libraries? These are the key questions to be answered. Once you can answer “Yes/Mostly Yes” to these 3 questions, you are good to go. After trying to read these papers dealing with the most popular concepts, try to read the not-so-famous papers. arXiv is a great site with hundreds of preprints being published everyday by top researchers and reading the papers from here is like drinking straight out of the fire-hose. Try to choose a paper which looks fairly well written and the abstract seems interesting. Then, read that paper and try to answer those 3 questions again. The same can be done with papers of top AI conferences like NIPS, AAAI, AAMAS, IJCAI, ICML etc. You may not be able to fully implement the papers due to data constraints and other issues, but if you are able to understand even 60% of the mathematical reasoning, then I can safely say you have completed your training. Do not concentrate on learning more and more “packages”. Concentrate on the concept. While implementing, you will automatically see that you require “this” package and then you will automatically learn to use it. Learning the various commands of random packages won’t help. If you start implementing and writing codes to solve problems or simulate results from a paper, you will automatically learn about packages and use them appropriately; they’ll be the least of your concerns. This is the correct way to maintain “balance” between math and coding. You can also participate in competitions (e.g. Kaggle or conference competitions) to improve speed, development and processing skills if you feel the need to do so. (All the links in this answer are working as of 6th July 2017) 8.3k Views · View Upvoters
Pankesh Bamotra, ML is recollecting what humans have learnt so far Answered Oct 14 2015 · Upvoted by Yuval Feinstein, Algorithmic Software Engineer in NLP,IR and Machine Learning · Author has 93 answers and 367.9k answer views
Mathematics is too vast a subject to be considered for this question. The breadth and depth of mathematical awareness you require for machine learning totally depends on what you are learning in the subject. Keeping this in mind, let's deal with what you need to know in "mathematics" for machine learning. 1. Probability and mathematical statistics This is a fundamental requirement for machine learning and so you need to know well. When I say probability it's more than what you studied in High school and almost everything you probably not paid attention to during your undergrad. You need Still have a question? your own! toknow about Random Ask variables, their distributions, probabilistic convergence,
and estimation theory. That covers a major part of what you need to know here. What is your question? Ask Two of my favourite resources are:https://www.quora.com/How-do-I-learn-mathematics-for-machine-learning
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1. Joseph Blitzstein - Harvard Stat 110 lectures 2. Larry Wasserman's book - All of statistics
2. Linear algebra Linear algebra will pop up every now and then in ML. PCA, SVD, LU decomposition, QR decomposition, symmetric matrices, othogonalization, projections, matrix operations are needed many a times. The good thing is that there are countless resources available on linear algebra. My all time favourite is Gilbert Strang's MIT lectures on linear algebra.
3. Optimisation Though only a few things from optimisation are needed most of the time, a strong foundational knowledge will help long way. You need to know Langrange multipliers, gradient descent, and primal-dual formulation. The best resource on this is Boyd and Vandenberghe's course on Convex optimisation from Stanford.
4. Calculus I wanted to put this on the top, but I'm putting it in the last just to emphasise on the fact that only a fundamental knowledge is needed in of calculus. Know about 3-D geometry, integration, and differentiation and you'll survive. It's the easiest to start with amongst the topic I've mentioned here. MIT has good lectures on calculus.
I think with these 4 tools you'll most likely find ML easy to understand. Other than these you may find real analysis and functional analysis relevant too, but they are just formal generalisations of the topics mentioned before. 13.7k Views · View Upvoters
Related Questions What does it take to do a PhD at Oxford in machine learning? Can I start learning machine learning and deep learning without brushing up my maths knowledge first? Are mathematics and machine learning related? How can I learn math? How can I learn machine learning at home? How should one start learning Algorithms? How does a total beginner start to learn machine learning if they have some knowledge of programming languages? Is Machine Learning easier than Mathematics? What math subject should I start to learn and what is the last one in order to start machine learning? Where can I find some online courses? What programming language is used for machine learning? Still have a question? Ask your own! Ask New Question
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