[MUSIC] So now that you've all spent some time playing in the sandpit, we can introduce an additional concept, which relates to multivariate systems called the Hessian. In many ways, the Hessian can be thought of as a simple extension of the Jacobian vector. For the Jacobian, we collected together all of the first order derivatives of a function into a vector. Now, we're going to collect all of the second order derivatives together into a matrix, which for a function of n variables, would look like this. However, this is one of those scenarios where using an abbreviated notation style comes in really handy. We saw in the previous module that we can just keep differentiating a function using the same method to find higher and higher order derivatives. Similarly for a partial derivative, if you want to find the second derivative with respect to x1 then x2, it's as simple as just differentiating with respect to x1, assuming all the other variables are constant, and then differentiating with respect to x2, again, assuming all the other variables are constant. As you can see from this general form, our Hessian matrix will be an n by n square matrix, where n is the number of variables in our function f. Let's now take a look at a quick example. It often makes life easier to find the Jacobian first and then differentiate its terms again to find the Hessian. So for our function f(x,y,z) = x squared y z. So f(x,y,z) = x squared y z. We're going to first build the Jacobian for this thing, which, of course, is going to be J =, so we've got differentiate with respect to x, we get 2xyz, differentiate with respect to y, which is going to get x squared z, and differentiate with respect to z, x squared y. Now using this, we can then think about differentiating again with respect to each of the variables, which will then give us our Hessian matrix. So H is just going to equal, we'll put a big bracket here. So we want to take this thing and differentiate it with respect to x again. So we're going to get 2yz. So the next term along, we're going to differentiate this thing with respect to y, 2xz. And again with respect to z, 2xy. Okay, now we'll take the next row, and we'll say we're going to differentiate this thing with respect to x, then y, then z. So you get 2xz. We're going to differentiate with respect to y, so we're going to get nothing. We're going to differentiate this thing with respect to z, and we get x squared. Lastly, take this term, make the last row, differentiate with respect to x, we get 2xy. And then with respect to y, we get x squared. And with respect to z, we get nothing. So one thing to notice here is that our Hessian matrix is symmetrical across the leading diagonal. So actually, once I'd worked out the top right region, I could just have written these directly in for the bottom left region. This will always be true if the function is continuous, meaning that it has no sudden step changes. We can now simply pass our Hessian an xyz coordinate, and it will return a matrix of numbers, which hopefully tells us something about that point in the space. In order to visualize this, we're going to have to drop down to two dimensions again. Consider the simple function f(x,y) = x squared + y squared. Calculating the Jacobian and the Hessian are both fairly straightforward. And hopefully, you can visualize how this function would have looked in your head. However, if you hadn't known what function we were dealing with and calculated the value of the Jacobian at the point (0,0), you'd have seen that the gradient vector was also 0. But how would you know whether this thing was a maximum or a minimum at that point? You could, of course, go and check some other point and see if it was above or below, but this isn't very robust. Instead, we can look at the Hessian, which in this simple case is no longer even a function of x or y. Its determinant is clearly just 2 times 2 minus 0 times 0, which is 4. The power of the Hessian is, firstly, that if its determinant is positive, we know we are dealing with either a maximum or a minimum. Secondly, we then just look at the first term, which is sitting at the top left-hand corner of the Hessian. If this guy is also positive, we know we've got a minimum, as in this particular case. Whereas, if it's negative, we've got a maximum. Lastly, slightly modifying our function to include a minus sign and recalculating our Jacobian and our Hessian, and our Hessian determinant, we now see the third interesting case. This time, our Hessian determinant is negative. So we know that we're not dealing with a maximum or a minimum. But clearly at this point, (0,0), the gradient is flat. So what's going on? Well, if you look at the animation, what we've got here is a location with 0 gradient, but with slopes coming down towards it in one direction, but up towards it in the other. We call this kind of feature as saddle point, and they can also cause a lot of confusion when searching for a peak. In the last module of this course, you're also going to see another way that the Hessian can help us with optimisation. But for now, we've simply got another tool to help you navigate the sandpit. See you next time.
