Introduction to the course

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Do curso por Imperial College London

Mathematics for Machine Learning: PCA

172 ratings

Imperial College London

Mathematics for Machine Learning: PCA

172 ratings

Curso 3 de 3 em Specialization Mathematics for Machine Learning

This intermediate-level course introduces the mathematical foundations to derive Principal Component Analysis (PCA), a fundamental dimensionality reduction technique. We'll cover some basic statistics of data sets, such as mean values and variances, we'll compute distances and angles between vectors using inner products and derive orthogonal projections of data onto lower-dimensional subspaces. Using all these tools, we'll then derive PCA as a method that minimizes the average squared reconstruction error between data points and their reconstruction. At the end of this course, you'll be familiar with important mathematical concepts and you can implement PCA all by yourself. If you’re struggling, you'll find a set of jupyter notebooks that will allow you to explore properties of the techniques and walk you through what you need to do to get on track. If you are already an expert, this course may refresh some of your knowledge. The lectures, examples and exercises require: 1. Some ability of abstract thinking 2. Good background in linear algebra (e.g., matrix and vector algebra, linear independence, basis) 3. Basic background in multivariate calculus (e.g., partial derivatives, basic optimization) 4. Basic knowledge in python programming and numpy Disclaimer: This course is substantially more abstract and requires more programming than the other two courses of the specialization. However, this type of abstract thinking, algebraic manipulation and programming is necessary if you want to understand machine learning algorithms.

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Statistics of Datasets

Principal Component Analysis (PCA) is one of the most important dimensionality reduction algorithms in machine learning. In this course, we lay the mathematical foundations to derive and understand PCA from a geometric point of view. In this module, we learn how to summarize datasets (e.g., images) using basic statistics, such as the mean and the variance. We also look at properties of the mean and the variance when we shift or scale the original data set. We will provide mathematical intuition as well as the skills to derive the results. We will also implement our results in code (jupyter notebooks), which will allow us to practice our mathematical understand to compute averages of image data sets.

Introduction to the course3:47

Conheça os instrutores

Marc P. Deisenroth
Lecturer in Statistical Machine Learning
Department of Computing

Hi, I'm Mark.

I'm a lecturer in statistical machine learning at Imperial College.

In this course, we are going through the mathematical foundations that we need

in order to do dimensionality reduction with principal component analysis.

Data in real life is often high dimensional.

For example, if you want to estimate the price of our house in a year's time,

we can use data that helps us to do this.

The type of the house, the size,

the number of bedrooms and bathrooms,

the value of houses in the neighborhood,

when they were bought,

the distance to the next train station and park,

the number of crimes committed in the neighborhood,

the economic climate and so on and so forth.

There are many things that influence the house price and we collect this information,

the data set that we can use to estimate the price of our house.

Another example is the 640 by 480 pixels

color image which is a data point in a one dimensional space,

where every pixel corresponds to three dimensions,

one for each color channel;

red, green and blue.

Working with high dimensional data comes with some difficulties.

It is hard to analyse, interpretation is difficult,

visualisation is nearly impossible and from a practical point of view,

storage can be quite expensive.

However, high dimensional data also has some nice properties.

For example, high dimensional data is often over complete that means

many dimensions are redundant and can be explained by a combination of other dimensions.

For example, if we took a color image with four channels, red, green,

blue and a gray-scale channel then the gray-scale channel can be explained by

a combination of the other three channels

and the image could equally be represented by red,

green and blue alone as we can reconstruct

the gray-scale channel just using that information.

Dimensionality reduction exploits structure and correlation and allows us to work with

a more compact representation of the data ideally without losing information.

We can think of dimensionality reduction as

a compression technique similar to JPEG or MP3,

which are compression algorithms for images and music.

Let's take a small binary image of the handwritten digit eight.

We're looking at 28 by 28 pixels,

each of which can be either black or white.

Therefore, this image can be represented as a 784 dimensional vector.

However, in this example,

the pixels are not randomly black or white, they are structured.

They often have the same value as the neighboring pixels.

In this data set,

there are many examples of an eight.

They differ a little bit but they look

sufficiently similar that we can identify them as eight.

We can use dimensionality reduction now to find a lower dimensional representation of

all eights that is easier to work with than a 784 dimensional vector.

The lower dimensional representation of

a high dimensional data point is often called a feature or a code.

In this course, we will look at

a classical algorithm for linear dimensionality reduction,

principle component analysis or PCA.

The purpose of this course is to go through

the necessary mathematical details to derive PCA.

In the first, module we will look at statistical representations of data,

for example, using means and variances.

And we also describe how means and variances

change if we linearly transform our data set.

In the second module,

we will look at the geometry in vector spaces and

define how to compute distances and angles between vectors.

In the third module,

we'll use these results to project data onto

lower dimensional sub-spaces and in the fourth module,

we'll derive PCA as a way to reduce dimensionality of data using orthogonal projections.

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