before we beginthe mathematical building blocks of neural networks let' consider new pointb [ ]which we'll add to the previous one this is done geometrically by chaining together the vector arrowswith the resulting location being the vector representing the sum of the previous two vectors (see figure + figure geometric interpret at ion of the sum of wo vect ors in generalelementary geometric operations such as affine transformationsrotationsscalingand so on can be expressed as tensor operations for instancea rotation of vector by an angle theta can be achieved via dot product with matrix [uv]where and are both vectors of the planeu [cos(theta)sin(theta)and [-sin(theta)cos(theta) geometric interpretation of deep learning you just learned that neural networks consist entirely of chains of tensor operations and that all of these tensor operations are just geometric transformations of the input data it follows that you can interpret neural network as very complex geometric transformation in high-dimensional spaceimplemented via long series of simple steps in dthe following mental image may prove useful imagine two sheets of colored paperone red and one blue put one on top of the other now crumple them together into small ball that crumpled paper ball is your input dataand each sheet of paper is class of data in classification problem what neural network (or any other machine-learning modelis meant to do is figure out transformation of the paper ball that would uncrumple itso as to make the two classes cleanly separable again with deep learningthis would be implemented as series of simple transformations of the spacesuch as those you could apply on the paper ball with your fingersone movement at time figure uncrumpling complicat ed manifold of data licensed to