Mathematics Of Artificial Intelligence

[Siiri]

Theworkshop focuses on the interplay between mathematics, artificial intelligence and machine learning. The aim of the programme is to encourage mathematical research in these areas, to promote the dissemination of results and to facilitate interaction with other disciplines. Proposals for contributed talks are reserved for junior researchers; the presence of senior researchers is welcome. Participating and contributing is not restricted to UMI members, but is open to all interested researchers.

Our finances, politics, media, opportunities, information, shopping and knowledge production are mediated through algorithms and their statistical approaches to knowledge; increasingly, these methods form the organizational backbone of contemporary capitalism. Revolutionary Mathematics traces the revolution in statistics and probability that has quietly underwritten the explosion of machine learning, big data and predictive algorithms that now decide many aspects of our lives. Exploring shifts in the philosophical understanding of probability in the late twentieth century, Joque shows how this was not merely a technical change but a wholesale philosophical transformation in the production of knowledge and the extraction of value. This book provides a new and unique perspective on the dangers of allowing artificial intelligence and big data to manage society. It is essential reading for those who want to understand the underlying ideological and philosophical changes that have fueled the rise of algorithms and convinced so many to blindly trust their outputs, reshaping our current political and economic situation.

Artificial intelligence problems constitute two general categories: Search problems and representation problems. Following them are interconnected models and tools like rules, frames, logics and nets. All of them are mathematical topics.

Consider self-driving cars. Their goal is to recognize objects and people in video images. Math powers these cars in the form of minimization procedures and backpropagation. Math helps AI scientists solve challenging, deep abstract problems using traditional methods and techniques that have been known for hundreds of years.

Behind all significant advances in technology, there is mathematics. The concepts of linear algebra, calculus, game theory, probability, statistics, advanced logistic regressions and gradient descent are all major underpinnings of data science.

In linear programming, vectors are used to deal with inequalities and systems of equations for notational conveniences. AI scientists use different techniques of vectors to solve problems of regression, clustering, speech recognition and machine translation. The concepts are also used to store the internal representations of AI models like linear classifiers and deep learning networks.

To find the convergence and convergence rates, the matrix P is adjusted. When the row Google matrix P reaches the sum to 1, then it is called a row stochastic matrix. The page rank iteration represents the evolution of a Markov chain in which the web directed graph is represented as a transition probability matrix P. It shows the probability of a random surfer at each of the three pages at any point in time.

Differential calculus, multivariate calculus, integral calculus, error minimization and optimization via gradient descent, limits and advanced logistic regressions are all the concepts used in mathematical modeling. A well-designed mathematical model is used in the biomedical sciences to simulate complex biological processes of human health and diseases with high fidelity.

In silico modeling, which is the application of AI approaches in biomedicine, is a fully automated model that does not require human samples, crude animal tests, clinical trials or laboratory equipment. A differential mathematical equation is used in the model to test new mechanistic hypotheses and evaluate novel therapeutic targets. It is the most inexpensive and convenient way to study human physiology, drug responses, and diseases way more accurately by manipulating mathematical model parameters.

There are a lot of abstract problems in the AI world. You may experience uncertainty and stochasticity in many forms. Probability theory offers tools to deal with uncertainty. The concepts of probability are used to analyze the frequency of an event.

The Center focuses on artificial intelligence, big data, control, optimization, anomalous (nonlocal) diffusion, nonlinear partial differential equations, with a broad range of applications. CMAI is funded by George Mason University, National Science Foundation, Department of Energy, Defense Threat Reduction Agency, Air Force Office of Scientific Research, Army Research Office, and Department of Navy.

I am currently finishing my master in pure mathematics. I've studied mainly logic, mathematical analysis a bit of probability and I'm writing my thesis about Riemannian Geometry. I've been thinking about doing a PhD, but I am not so sure if I really want (and if I am good enough) to stay in the Academia for all my life.

I just want to supplement other answers (I have a PhD in AI). If your background is mathematics, I do not agree with others that you need to code per-se. Aside from machine learning pointed above, there are some other sides you can look at that do not require - much - programming (if any).

Logic - Symbolic AI is broadly focused around temporal logics, including those specifically formalised for game-theory, and non-monotonic logics. You can go in many directions with logic in AI: philosophical (i.e., thinking of how to capture certain types of common-sense reasoning, normative reasoning, etc.) or more solution-oriented (i.e., we need a logic to represent a specific problem X and then we will prove some general results about that logic). In the case that you work on logics to solve specific problems, you may need some Computer Science understanding (in particular, proving complexity for this kind of research is required for publishing in many top venues). In terms of the more philosophical aspects, you don't really need so much of a Comp. Sci. background, non-monotonic logics, for example, can be generalised in a graph-theoretical framework.

Game theory - similar, to the problem-oriented logic research I discussed previously, many papers in AI capture some new game-theoretical concepts and aim to prove something. In this area are related topics, such as negotiation (and to some extent, argumentation, although that is more related with non-monotonic logic). For some game-theoretical papers that relax assumptions of heterogeneity in the participants involved in the game (for example), simulation is used to find support for properties that cannot feasibly be proven (often, properties with assumptions of homogonous participants are proven and simulation supplements these results with experiments on trickier cases).

In short, I see no problem with you moving to AI, we have plenty of mathematicians (and philosophers and computer scientists) in the field. In terms of logic or game-theory, you may want to pick the topic carefully if you want to avoid spending time learning 'theoretical' Comp. Sci and/or programming.

What's troubling about what I've seen of your background is your lack of computer science courses. Maybe that's not an issue, if you have the equivalent of a minor in computer science that wasn't featured in your question. Certainly, that would be an important "tool." And if you don't have it, you might consider going back to school for a five or six course minor or certificate program in CS.

The most interesting part of your question was, "I find the Artificial Intelligence field really fascinating." Why is that? And how does that connect with the rest of your background? Or are you starting from scratch, which would be difficult, though not impossible. In any event, these issues should be addressed in your statement of purpose.

Here's my experience in unfamiliar territory: In high school, I had four years of French, and none of Spanish (unless you count my one year of self-taught Spanish). In college, I bluffed my way into a second year Spanish class, and completed it successfully. Here's an example where my interest and natural aptitude compensated for my lack of formal training. Perhaps it will be the same for you and artificial intelligence.

The longer answer: assuming you can catch up with some core programming based on Matlab or numpy/scipy, with your background you should have very good chances in theoretically oriented Machine Learning groups. Riemannian geometry will put you in a very good position for instance to pursue Information Geometry, which is a branch of information theory/probability theory of high relevance to learning theory which makes heavy use of differential geometry.

I know of a student with with a pure math background who continued to do a PhD in AI, with great success and strong publications. He was a good programmer, though, without actually having studied CS formally.

I have recently become interested in Machine Learning and AI as a student of theoretical physics and mathematics, and have gone through some of the recommended resources dealing with statistical learning theory and deep learning in neural networks.

One of the main issues I find in my personal study of this discipline is that an overwhelming proportion of such resources focus on the practical side of ML, forfeiting rigour in favour of useful heuristics. This approach has its obvious merits, considering the great current interest in their applications both in science and technology, but I would like to go beyond what the average engineer might need and explore the more theoretical sides.

The elephant in the room is, of course, the fact that to date the inner workings of the main tools of AI, neural networks above all others, are not well understood. From what I can tell, there are a variety of approaches drawing from very diverse field, including a physical perspective (see Huang's Statistical Mechanics of Neural Networks, or Statistical Field Theory for Neural Networks by Helias and Dahmen).

3a8082e126