This course offers a brief introduction to the multivariate calculus required to build many common machine learning techniques. We start at the very beginning with a refresher on the “rise over run” formulation of a slope, before converting this to the formal definition of the gradient of a function. We then start to build up a set of tools for making calculus easier and faster. Next, we learn how to calculate vectors that point up hill on multidimensional surfaces and even put this into action using an interactive game. We take a look at how we can use calculus to build approximations to functions, as well as helping us to quantify how accurate we should expect those approximations to be. We also spend some time talking about where calculus comes up in the training of neural networks, before finally showing you how it is applied in linear regression models. This course is intended to offer an intuitive understanding of calculus, as well as the language necessary to look concepts up yourselves when you get stuck. Hopefully, without going into too much detail, you’ll still come away with the confidence to dive into some more focused machine learning courses in future.

本課程是 数学在机器学习领域的应用 專項課程 專項課程的一部分

# Mathematics for Machine Learning: Multivariate Calculus

提供方

## 課程信息

### 學生職業成果

## 35%

## 26%

### 您將獲得的技能

### 學生職業成果

## 35%

## 26%

#### 100% 在線

#### 第 2 門課程（共 3 門）

#### 可靈活調整截止日期

#### 初級

#### 完成時間大約為22 小時

#### 英語（English）

### 提供方

#### 伦敦帝国学院

Imperial College London is a world top ten university with an international reputation for excellence in science, engineering, medicine and business. located in the heart of London. Imperial is a multidisciplinary space for education, research, translation and commercialisation, harnessing science and innovation to tackle global challenges.

## 教學大綱 - 您將從這門課程中學到什麼

**完成時間為 4 小時**

## What is calculus?

Understanding calculus is central to understanding machine learning! You can think of calculus as simply a set of tools for analysing the relationship between functions and their inputs. Often, in machine learning, we are trying to find the inputs which enable a function to best match the data. We start this module from the basics, by recalling what a function is and where we might encounter one. Following this, we talk about the how, when sketching a function on a graph, the slope describes the rate of change of the output with respect to an input. Using this visual intuition we next derive a robust mathematical definition of a derivative, which we then use to differentiate some interesting functions. Finally, by studying a few examples, we develop four handy time saving rules that enable us to speed up differentiation for many common scenarios.

**完成時間為 4 小時**

**10 個視頻**

**4 個閱讀材料**

**6 個練習**

**完成時間為 3 小時**

## Multivariate calculus

Building on the foundations of the previous module, we now generalise our calculus tools to handle multivariable systems. This means we can take a function with multiple inputs and determine the influence of each of them separately. It would not be unusual for a machine learning method to require the analysis of a function with thousands of inputs, so we will also introduce the linear algebra structures necessary for storing the results of our multivariate calculus analysis in an orderly fashion.

**完成時間為 3 小時**

**9 個視頻**

**5 個練習**

**完成時間為 3 小時**

## Multivariate chain rule and its applications

Having seen that multivariate calculus is really no more complicated than the univariate case, we now focus on applications of the chain rule. Neural networks are one of the most popular and successful conceptual structures in machine learning. They are build up from a connected web of neurons and inspired by the structure of biological brains. The behaviour of each neuron is influenced by a set of control parameters, each of which needs to be optimised to best fit the data. The multivariate chain rule can be used to calculate the influence of each parameter of the networks, allow them to be updated during training.

**完成時間為 3 小時**

**6 個視頻**

**3 個練習**

**完成時間為 3 小時**

## Taylor series and linearisation

The Taylor series is a method for re-expressing functions as polynomial series. This approach is the rational behind the use of simple linear approximations to complicated functions. In this module, we will derive the formal expression for the univariate Taylor series and discuss some important consequences of this result relevant to machine learning. Finally, we will discuss the multivariate case and see how the Jacobian and the Hessian come in to play.

**完成時間為 3 小時**

**9 個視頻**

**5 個練習**

### 審閱

#### 4.7

##### 來自MATHEMATICS FOR MACHINE LEARNING: MULTIVARIATE CALCULUS的熱門評論

Great course to develop some understanding and intuition about the basic concepts used in optimization. Last 2 weeks were a bit on a lower level of quality then the rest in my opinion but still great.

Excellent course. I completed this course with no prior knowledge of multivariate calculus and was successful nonetheless. It was challenging and extremely interesting, informative, and well designed.

Very Well Explained. Good content and great explanation of content. Complex topics are also covered in very easy way. Very Helpful for learning much more complex topics for Machine Learning in future.

Very clear and concise course material. The inputs given during the videos and the subsequent practice quiz almost force the student to carry out extra/research studies which is ideal when learning.

Excellent course!\n\nI studied multivariate calculus during engineering. I hardly understood the concepts at that time, this course helped me understand and visualize what is going behind formulas.

i think some of concepts touched the surface and it was difficult to get a deep understanding .Probably the course could have provided some external links for those topics where people could read .

Superb quality. The way instructors teach is really innovative. The course is good in terms of the area it covers but lacks depth, but is a good starting point if you want to dwell more in detail.

I highly recommend this course.\n\nEvery Machine Learning student have to do it. Some concepts is so clearly explained that you will be able to perform better in following ML studies.

Just a great course for getting you ready to understand machine learning algorithms. The chapter on backpropagation is simply outstanding and the programming assignments are awesome!

I wish, Linear Regression was taught with a little more clarity. Seemed like too many things were happening. Otherwise, a very good course. Really enjoyed the back-propagation week.

As good as the first class in the Math for ML series. Instruction was interesting. Questions were not too confusing. Clearly a lot of time was spent producing this class. Thank you.

Nice course. Ppl with who don't have some experience with the content may find the instruction too sparse. But for someone with a decent background its a fucking fantastic course !

A wonderful course. I learnt a lot after struggling to finish it. Some foundations of calculus might be needed since the lecturer goes through differntiation in a tremendous speed.

the basic concepts are explained clearly, but the step of the lecture became more fast than the course of linear algebra. More detail proof and application of theory is expected.

The course is still a bit young, some errors appear here and there sometimes, and some parts of it are a bit steep.\n\nOtherwise, this is a good course, focused on derivatives.

Really good introduction for things like regression and gradient descent. An extremely good refresher for calculus and extension from what is taught in school (in UK at least).

Excellent course to understand what is behind the techniques and why not high-level functions that are used in machine learning programming. Thanks for your teaching Dave, Sam

Very practical and useful! I got an idea about what neural network is and what is inside of the regression algorithm. I enjoyed the course, although it was quite challenging.

Loved the course. Backpropogation section needs more elaborate explanation, where are we doing dot products, where are we doing matrix multiplications, things go confusing.

This course is really informative and builds intuition for the topics covered, I'd like to specially thank Sam for his amazing way of teaching and his visualizations :)

## 關於 数学在机器学习领域的应用 專項課程

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