Motivation
We represent data as vectors. We then choose an appropriate model, and learn from available data using numerical optimisation methods with the aim that the model performs well on data not used for training.
But to go back - we represent numerical data as vectors and represent a table of such data as matrices. A study of vectors and matrices is called linear algebra.
Given two vectors representing two objects in the real world, if we want to make statements about their similarity, the idea is that vectors that are similar should be predicted to have similar outputs by our machine learning algorithm, our predictor. To formalize the idea of similarity, we need to take two vectors and perform some operation on them that returns a numerical value representing their similarity. The construction of similarity and distances is central to analytic geometry.
Later, we will be introduced to matrix decomposition and some operations on matrices that are extremely useful in machine learning as they allow for intuitive representation of data, and more efficient learning.
We often consider data to be noisy observations of some true underlying signal. This requires us to quantify what “noise” means, if we want to have an algorithm that separates then signal from the noise. We would also like to have predictors that allow us to express some sort of uncertainty, ex. quantify the confidence we have about the value of the prediction at a data point - quantification of uncertainty is the realm of probability theory.
To train ML models, we find parameters that maximize some performance measure. Such optimization techniques require the concept of a gradient, which is covered in vector calculus.