In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we're aiming at data-driven applications, we'll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you'll write code blocks and encounter Jupyter notebooks in Python, but don't worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning.
Getting a handle on vectors
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來自 Imperial College London 的課程
Mathematics for Machine Learning: Linear Algebra
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從本節課中
Introduction to Linear Algebra and to Mathematics for Machine Learning
In this first module we look at how linear algebra is relevant to machine learning and data science. Then we'll wind up the module with an initial introduction to vectors. Throughout, we're focussing on developing your mathematical intuition, not of crunching through algebra or doing long pen-and-paper examples. For many of these operations, there are callable functions in Python that can do the adding up - the point is to appreciate what they do and how they work so that, when things go wrong or there are special cases, you can understand why and what to do.
與講師見面
David DyeProfessor of Metallurgy
Department of Materials
Samuel J. CooperLecturer
Dyson School of Design Engineering
A. Freddie PageStrategic Teaching Fellow
Dyson School of Design Engineering
[SOUND] So the first thing we need to do in this course on Linear Algebra is to
get a handle on vectors, which will turn out to be really useful for
us in solving those linear algebra problems.
That is the ones that are linear in their coefficients
such as most fitting parameters.
We're going to first step back and
look in some detail at the sort of things we're trying to do with data.
And why those vectors you first learned about in high school were even relevant?
And this will hopefully make all the work with vectors a lot more intuitive.
Let's go back to that simpler problem from the last video,
the histogram distribution of heights of people in the population.
There aren't many people above about 2 meters, say.
And there aren't really very many people below 1.5 meters.
Say we wanted to try fitting that distribution with an equation
describing the variation of height in the population.
And let's say that equation has just two parameters.
One describing the center of the distribution here, and we'll call that mu.
And one describing how wide it is, which we'll call sigma.
So we could fit it with some curve that
had two parameters, mu and sigma.
I would use an equation like this.
I'd call it f(x),
some function of x where x is the height,
= 1 over sigma root 2 pi times
the exponential of - (x - mu) squared
divided by 2 sigma squared.
So this equation only has two parameters, sigma here and mu, and it looks like this.
And it has an area here of 1,
because there's only 100% of people in the population as a whole.
Now don't worry too much about the form of the equation.
This is what happens, this is called the normal or Gaussian distribution.
And it's got a center of mu and a width of sigma.
And it's normalized, so that it's area is 1.
Now, how do we go about fitting this distribution?
That is, finding mu and sigma,
well, the best possible mu and sigma that fits the data as well as is possible.
Imagine that we had guessed that the width was wider than it really is, but
keeping the area of 1.
So if we guessed that it was a fatter and probably a bit shorter distribution,
something like that, say.
So this one has something like that.
This one has a wider sigma, but it's got the same mu.
It'd be too high at the edges here, and too low in the middle.
So then we could add up the differences between all of our measurements, and
all of our estimates.
We've got all of these places where we underestimate here, and
all of these places where we overestimate here.
And we could add up those differences or, in fact,
the squares of them to get a measure of the goodness or badness of the fit.
And we'll look in detail at how we do that once we've done all the vectors
work, and once we've done actually all the calculus work, then we could plot how
that goodness varied as we change the fitting parameters, sigma and
mu, and we get a plot like this.
So if we had a correct value, our best possible value for mu here,
and our best possible value for the width, sigma here.
We could then plot, for a given value of mu and sigma, what the difference was.
So if we were at the right value, we'd get a value
of goodness where the sum of the squares of the differences was nought.
And if mu was too far over, if we had mis-estimated mu and
we got the distribution shifted over, so the width was right, but
we had some wrong value of mu there, that we get some value of all the sums of
the squares of the differences of goodness being some value here that was higher.
And it might be the same if we went over the other side and
we had some value there.
And if we were too wide, we'd get something there or too thin,
we'd get something that was too thin like that, something like that, say.
So we'd get some other value of goodness.
We could imagine plotting out all of the values of where we have the same
value of goodness or badness for different values of mu and sigma.
And we could then do that for some other value of badness, and
we might get a contour that looked like this, and
another contour that looked like this, and so on and so forth.
Now, say we don't want to compute the value of this goodness parameter for
every possible mu and sigma.
We just want to do it a few times, and
then find our way to the best possible fit of all.
Say we started off here with some guess that was too big a mu and
too small a width.
We thought people were taller than they really are,
and that they were tighter packed in their heights than they really are.
But what we could do is we could say well,
if I do a little move in mu and sigma, then does it get better or worse?
And if it gets better, well, we'll keep moving that direction.
So we could imagine making a vector of a change in mu and a change in sigma.
And we could have our original mu and sigma there.
And we could have a new value, mu prime, sigma prime, and
ask if that gives us a better answer?
If it's better there or if mu prime sigma prime took us over here?
If we were better or worse there, something like that.
Now actually, if we could find what the steepest way down the hill was,
then we could go down this set of contours, this sort of landscape here
towards the minimum point, towards the point where get the best possible fit.
And what we're doing here,
these are vectors, these are little moves around space.
They're not moves around a physical space,
they're moves around a parameter space, but it's the same thing.
So if we understand vectors and we understand how to get down hills,
that sort of curviness of this value of goodness, that's calculus.
Then once we got calculus and
vectors, we'll be able to solve this sort of problem.
So we can see that vectors don't have to be just geometric objects in the physical
order of space.
They can describe directions along any sorts of axes.
So we can think of vectors as just being lists.
If we thought of the space of all possible cars, for example.
So here's a car.
There's its back, there's its window, there's the front, something like that.
There's a car, there's the window.
We could write down in a vector all of the things about the car.
We could write down its cost in euros.
We could write down its emissions performance in grams of CO2 per
100 kilometers.
We could write down its Nox performance,
how much it polluted our city and killed people due to air pollution.
We could write down its Euro NCAP star rating, how good it was in a crash.
We could write down its top speed.
And write those all down in a list that was a vector.
That'd be more of a computer science view of vectors,
whereas the spatial view is more familiar from physics.
In my field, metallurgy, I could think of any alloy as being described by a vector
that describes all of the possible components,
all the compositions of that alloy.
Einstein, when he conceived relativity,
conceived of time as just being another dimension.
So space-time is a four dimensional space, three dimension of metres, and
one of time in seconds.
And he wrote those down as a vector of space-time of x, y, z,
and time which he called space-time.
When we put it like that, it's not so
crazy to think of the space of all the fitting parameters of a function, and
then of vectors as being things that take us around that space.
And what we're trying to do then is find the location in that space,
where the badness is minimized, the goodness is maximized, and
the function fits the data best.
If the badness surface here was like a contour map of a landscape, we're trying
to find the bottom of the hill, the lowest possible point in the landscape.
So to do this well, we'll want to understand how to work with vectors and
then how to do calculus on those vectors in order to find
gradients in these contour maps and minima and all those sorts of things.
Then we'll be able to go and do optimizations, enabling us to go and
work with data and do machine learning and data science.
So, in this video, we've revisited the problem of fitting a function to some
data, in this case, the distribution of heights in the population.
What we've seen is the function we fit, whatever it is, has some parameters.
And we can plot how the quality of the fit, the goodness of the fit,
varies as we vary those parameters.
Moves around these fitting parameter space are then just vectors
in that space of fitting parameters.
And, therefore, we want to look at and revisit vector maths in order to be able
to build on that, and then do calculus, and then do machine learning.
[SOUND]
