Summary

Computational Mathematics is a branch of mathematical science that involves advanced computations and algorithms for solving mathematical problems. It is instrumental in the success of modern AI and machine learning by providing the theory and tools for numerical simulation, optimization, and statistical analysis.

ELI5

Imagine you are trying to solve a puzzle, but this puzzle is so large and complex that you would not be able to solve it by hand and in a reasonable time. Computational Mathematics is like the set of special tools and methods you would use to speed up solving this puzzle, or even make it possible in the first place.

In-depth explanation

Computational Mathematics is a field of mathematics that utilizes advanced computational methods to solve complex mathematical problems, often ones that are challenging or impossible to address with traditional analytical methods. It crosses with several subdomains, including numerical analysis, computational geometry, optimization, statistics, and operations research, amongst others.

Traditionally, mathematical problems were solved analytically, i.e., exact solutions were derived using mathematical symbols. But, many real-world mathematical problems are either too complex or simply do not have exact symbolic solutions. This reality presents a need for numerical solutions - approximate solutions computed using numerical algorithms, and this is where Computational Mathematics shines.

In the context of artificial intelligence (AI) and machine learning (ML), Computational Mathematics is crucial. Numerous ML algorithms, including, for instance, gradient descent used in deep learning, are computational in nature. They involve iterative approximation, weighing trade-offs between precision and computational cost, which are themes central to Computational Mathematics. AI and ML tasks often involve huge datasets or complex models, which would not be possible to process with classical methods alone.

At its core, a neural network involves lots of matrix operations—multiplications and additions—and non-linear transformations. These kinds of computations fall within the scope of numerical linear algebra, a subfield of Computational Mathematics. Optimization methods, core to the training of machine learning models, are grounded in calculus and numerical optimization methods.

One area where Computational Mathematics is particularly apparent is in gradient-based optimization methods such as Gradient Descent or its many variants (Stochastic, Mini-batch, etc.). These methods, used to minimize the cost function and find the optimal model parameters, are rooted in the field.

Computing likelihoods, performing regression analysis, structuring decision trees, selecting features, handling skewed data, factorization, cluster analysis, and numerous other tasks in ML, all depend on the principles of Computational Mathematics.

Moreover, the concepts of Computational Mathematics are used in improving the efficiency of algorithms in terms of space and time complexity. High-dimensional data is pretty standard in machine learning, and performing operations in high-dimensional spaces requires sophisticated computational tools.

Hence, mastering the principles of Computational Mathematics is invaluable for anyone involved in the world of AI and ML, whether you’re designing new algorithms or implementing existing ones.

Numerical Analysis, Gradient Descent, Linear Algebra, Machine Learning (ML),, Artificial Intelligence, Feature Selection, Optimization Methods, Statistics, Matrix Factorization, Cluster Analysis, Operations Research, Computational Geometry