 So, why is this high-dimensional integration also important for deep learning? Well, two random vectors are orthogonal, that is, why there's limited interference, where we can learn one thing and we can, at the same time, learn another thing, and they, by and large, don't interfere all that much with one another. It might be the reason why really big deep neural networks tend to learn better than smaller ones. We've seen that two random vectors are far away from one another. That is why high-dimensional systems can typically distinguish inputs. Now, arguably, most of our mistakes in deep learning are due to us having poor interactions about what can happen and cannot happen in high-dimensional spaces. Now, to wrap up for today, we talked about abstractions, which is one failure mode of us interacting with deep learning systems because the math, as we write it, isn't exactly what we implement. We talked about linear dynamics of learning, which generally has the key to understanding how learning really happens, how the components interact with one another. We've seen models of competition in learning, where we saw how different dimensions can kind of pop in one after the other. We talked about cost functions, which is, of course, one of the big components that we always use. We talked about high-dimensional spaces. All of these ideas matter for every deep learning system that you will ever design. Having strong interactions for these points is the key to be effective when you will be designing such systems for production systems.