*This article was posted by Ingrid Daubechies on Quanta Magazine. Ingrid is the James B. Duke Professor of Mathematics and Electrical and Computer Engineering at Duke University. She served as president of the International Mathematical Union from 2011 to 2014.*

Machine learning works spectacularly well, but mathematicians aren’t quite sure why.

At a dinner I attended some years ago, the distinguished differential geometer Eugenio Calabi volunteered to me his tongue-in-cheek distinction between pure and applied mathematicians. A pure mathematician, when stuck on the problem under study, often decides to narrow the problem further and so avoid the obstruction. An applied mathematician interprets being stuck as an indication that it is time to learn more mathematics and find better tools.

I have always loved this point of view; it explains how applied mathematicians will always need to make use of the new concepts and structures that are constantly being developed in more foundational mathematics. This is particularly evident today in the ongoing effort to understand "big data" — data sets that are too large or complex to be understood using traditional data-processing techniques.

Our current mathematical understanding of many techniques that are central to the ongoing big-data revolution is inadequate, at best. Consider the simplest case, that of supervised learning, which has been used by companies such as Google, Facebook and Apple to create voice- or image-recognition technologies with a near-human level of accuracy. These systems start with a massive corpus of training samples — millions or billions of images or voice recordings — which are used to train a deep neural network to spot statistical regularities. As in other areas of machine learning, the hope is that computers can churn through enough data to "learn" the task: Instead of being programmed with the detailed steps necessary for the decision process, the computers follow algorithms that gradually lead them to focus on the relevant patterns.

In mathematical terms, these supervised-learning systems are given a large set of inputs and the corresponding outputs; the goal is for a computer to learn the function that will reliably transform a new input into the correct output. To do this, the computer breaks down the mystery function into a number of layers of unknown functions called sigmoid functions. These S-shaped functions look like a street-to-curb transition: a smoothened step from one level to another, where the starting level, the height of the step and the width of the transition region are not determined ahead of time.

Inputs enter the first layer of sigmoid functions, which spits out results that can be combined before being fed into a second layer of sigmoid functions, and so on. This web of resulting functions constitutes the “network” in a neural network. A “deep” one has many layers.

To read more, click here.

**Top DSC Resources**

- Article: What is Data Science? 24 Fundamental Articles Answering This Question
- Article: Hitchhiker's Guide to Data Science, Machine Learning, R, Python
- Tutorial: Data Science Cheat Sheet
- Tutorial: How to Become a Data Scientist - On Your Own
- Categories: Data Science - Machine Learning - AI - IoT - Deep Learning
- Tools: Hadoop - DataViZ - Python - R - SQL - Excel
- Techniques: Clustering - Regression - SVM - Neural Nets - Ensembles - Decision Trees
- Links: Cheat Sheets - Books - Events - Webinars - Tutorials - Training - News - Jobs
- Links: Announcements - Salary Surveys - Data Sets - Certification - RSS Feeds - About Us
- Newsletter: Sign-up - Past Editions - Members-Only Section - Content Search - For Bloggers
- DSC on: Ning - Twitter - LinkedIn - Facebook - GooglePlus

Follow us on Twitter: @DataScienceCtrl | @AnalyticBridge

- Juniper adds Mist AIOps to its 128 Technology-based SD-WAN
- 10 microservices patterns all architects should know
- IBM extends Call for Code for Racial Justice program
- citizen development
- How to manage third-party risk in the supply chain
- Gartner predicts data storytelling will dominate BI by 2025
- AWS Data Exchange and the third-party cloud data marketplace
- Overcome common IoT edge computing architecture issues

Posted 1 March 2021

© 2021 TechTarget, Inc. Powered by

Badges | Report an Issue | Privacy Policy | Terms of Service

**Most Popular Content on DSC**

To not miss this type of content in the future, subscribe to our newsletter.

- Book: Applied Stochastic Processes
- Long-range Correlations in Time Series: Modeling, Testing, Case Study
- How to Automatically Determine the Number of Clusters in your Data
- New Machine Learning Cheat Sheet | Old one
- Confidence Intervals Without Pain - With Resampling
- Advanced Machine Learning with Basic Excel
- New Perspectives on Statistical Distributions and Deep Learning
- Fascinating New Results in the Theory of Randomness
- Fast Combinatorial Feature Selection

**Other popular resources**

- Comprehensive Repository of Data Science and ML Resources
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- 100 Data Science Interview Questions and Answers
- Cheat Sheets | Curated Articles | Search | Jobs | Courses
- Post a Blog | Forum Questions | Books | Salaries | News

**Archives:** 2008-2014 |
2015-2016 |
2017-2019 |
Book 1 |
Book 2 |
More

**Most popular articles**

- Free Book and Resources for DSC Members
- New Perspectives on Statistical Distributions and Deep Learning
- Time series, Growth Modeling and Data Science Wizardy
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- Comprehensive Repository of Data Science and ML Resources
- Advanced Machine Learning with Basic Excel
- Difference between ML, Data Science, AI, Deep Learning, and Statistics
- Selected Business Analytics, Data Science and ML articles
- How to Automatically Determine the Number of Clusters in your Data
- Fascinating New Results in the Theory of Randomness
- Hire a Data Scientist | Search DSC | Find a Job
- Post a Blog | Forum Questions

## You need to be a member of Data Science Central to add comments!

Join Data Science Central