Introduction: Solving data science challenges with mathematics

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來自 Imperial College London 的課程

Mathematics for Machine Learning: Linear Algebra

740 個評分

Imperial College London

Mathematics for Machine Learning: Linear Algebra

740 個評分

課程 1（共 3 門，Specialization Mathematics for Machine Learning）

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.

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.

Introduction: Solving data science challenges with mathematics5:54

與講師見面

David Dye
Professor of Metallurgy
Department of Materials
Samuel J. Cooper
Lecturer
Dyson School of Design Engineering
A. Freddie Page
Strategic Teaching Fellow
Dyson School of Design Engineering

Hi there. I'm David Dye,

and I'm going to introduce you to this course on linear algebra.

Later in the course Dr. Sam Cooper will take you through the last module,

and in his course on multivariate calculus,

I'll do a bit of a turn.

But in this video, I just want to take a moment to

introduce the course and what we're trying to do here and to set the scene.

The world is awash with huge amounts of data.

Most of us living in cities generate huge amounts of data as we go about our lives,

moving around, browsing on mobile phones,

using transit networks, and consuming energy.

If we just think about energy alone for a moment,

our society literally lights up our planet from space.

We can see where the people and economic activity are from the lights.

Zooming in to Europe and London here,

you can see how some of the world's great cities network together.

This takes a huge amount of energy.

And if we're going to figure out how we're going to generate

and use energy more sensibly in the future,

we're going to have to understand the data we have about energy much better.

If we don't want to damage our health with NOx and

particulates and if we want to combat global warming,

we're probably going to have to strongly cut back or even eliminate burning hydrocarbons,

as the leading approach to power in our society.

And we can. With wind,

with nuclear, with solar,

we can probably make huge strides towards reducing our need for fossil fuels.

But we're going to need storage of

batteries in order to have hybrid and battery electric vehicles.

We're going to need to insulate our buildings

much better and heat them much more efficiently.

But where should we start?

Where were the opportunities to have the biggest impact soonest?

If we install this energy saving measure,

how much will it cost and how much will it help?

Should we recommend that everyone buy

heat pumps or should people first insulate their homes better?

How much range do electric vehicles really need?

How big should their batteries be?

In improving the London underground so that more people use it and less people drive.

Are there stations that are so close together,

there's no need to stop at them most of the time?

All of these questions we could answer better,

if we could analyze the energy and usage data we had better.

If we had good models of data,

then we could also make predictions.

Practically speaking, we could build software that tells you

how to take the fastest journey across your city and put in a map.

Or which web pages you might want to see in response to a search query.

And we don't necessarily need a physical model in order to make such predictions.

Real Networks are very accurate in

many applications at modeling data and making predictions.

And the process by which we optimize those network is called Machine Learning.

Now, it turns out that linear algebra,

how to solve systems of equations,

like this where the variables obey the rules of vectors

like these and then taking those forward and turning,

recasting them as matrices like these objects here.

This is very important in machine learning and data science.

And also being able to do optimizations in machine learning and data science,

in turn depends on understanding how a thing we're trying to optimize,

changes in response to changes in the primer to describe it.

Which is multivariable calculus,

these sorts of symbols here.

The problem is, that machine learning and data science

have their origins in computer science and mathematics.

But more and more,

we're wanting to be used that maths by engineers and physical scientists,

and biologists and people working in medicine and social scientists and so on.

But one of the problems is that,

most courses on data science and machine learning presuppose

a level of familiarity with linear algebra and calculus that not everybody has.

So our aim in this course and in this specialization,

is to revisit the underpinnings in vectors and

matrices and calculus that you might have touched on in school,

but which you might never have applied.

Depending on how it was taught,

it may not be at all obvious to you why that maths would even apply to the world of data.

Usually high school maths is motivated,

if it is at all, through physics after all.

So this might not be that obvious.

To be clear, we're not really aiming to do a fundamental course,

where we do everything very rigorously.

As we would, if we are trying to set up for

graduate school class and apply mathematics later.

Nor are we aiming for the breadth of a linear algebra or

a calculus course that would apply everywhere across science and engineering.

We're just aiming to very quickly and in a way

strongly both motivated by machine learning problems,

give you enough underpinnings to get you going.

And more than anything else, we want to develop your mathematical intuition.

That is, we have computers to do all the mathematical operations for us now.

In Python or Matlab or R.

Some of the most complex exercises at the end of these courses,

we'll ask you to write some very short little mini blocks of code in Python,

to develop and demonstrate and use the mathematical insight you've developed.

But doing the operations and crunching through them with pen and paper,

plugging the numbers and chugging your way to the answer,

it's just not a very useful skill in the world anymore.

So our focus is on the ideas,

not on your endurance.

So, linear algebra is defined to be,

the study of vectors,

vector spaces, a mapping between vector spaces.

It emerged from the study of systems of linear equations and

the realization that these could be solved using matrices and vectors.

So the first thing we'll do,

is look at vectors and the operations we can do

with vectors and then we'll move on to look at matrices.

And then in the final module,

we'll put it altogether with an application to

Google's famous Page Rank algorithm for ranking web pages.

So that's what this course Linear Algebra is all about.

And how it relates to the other courses in this specialization are

multivariate calculus for machine learning. So welcome.

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