Mathematical Tools for Data Science

Description

Instructor: Carlos Fernandez-Granda
Contributors: Brett Bernstein | Aakash Kaku | Sheng Liu | Sreyas Mohan

This course provides a rigorous introduction to mathematical tools for data science drawn from linear algebra, Fourier analysis, probability theory, and convex optimization. The main topics are covariance matrices, principal component analysis, linear regression, regularization, sparse regression, frequency representations, the short-time Fourier transform, wavelets, Wiener filtering, and convolutional neural networks. The material is illustrated by multiple applications to real data.

We gratefully acknowledge the support of the National Science Foundation through awards DMS-2009752, HDR-1940097 and NRT-HDR 1922658.

Week	Notes	Videos	Code
1	Covariance matrix	The covariance matrix	PCA of weather data
		Principal component analysis
		Gaussian random vectors
		Covariance Matrix Additional Slides
2	Linear regression	Mean squared error estimation	Linear regression
		Ordinary least squares
		Analysis of OLS coefficients
		Linear Regression Additional Slides
3	Regularization	Training and test error of OLS
		Ridge regression	Ridge Regression
		Regularization via early stopping	Gradient Descent
		Regularization Additional Slides
4	Sparse regression	The lasso	Sparse Regression
		Convexity
		Subgradients
		Theoretical analysis of the lasso
		Sparse Regression Additional Slides
5	The frequency domain (Sections 1–2)	Fourier series	Fourier analysis of an electrocardiogram
		The sampling theorem
		The Frequency Domain (1-2) Additional Slides
6	The frequency domain (Sections 3-6)	Discrete Fourier transform	Magnetic resonance imaging
		Fourier transformations in multiple dimensions
		The Frequency Domain (3-6) Additional Slides
7	Beyond Fourier (Section 1)	Time-frequency analysis via windowing	STFT of a speech signal
		Short-time Fourier transform
		Beyond Fourier (Section 1) Additional Slides
8	Beyond Fourier (Sections 2-3)	Wavelets
		Denoising via thresholding	Audio denoising
		Beyond Fourier (Sections 2-3) Additional Slides	Image denoising
9	Stationarity (Sections 1-4)	Linear translation-invariant models and convolution	PCA of images
		Stationarity and PCA	PCA of piecewise-constant functions
		Stationarity Additional Slides
10	Denoising (Sections 5-6)	Wiener filtering for denoising
		Convolutional neural networks for image denoising	Image denoising via bias-free CNNs
		Deep learning for denoising electron-microscope images	Deep denoising for electron microscopy

Contact

If you have any questions, please contact Carlos Fernandez-Granda at cfgranda@cims.nyu.edu.

Mathematical Tools for Data Science by New York University, Carlos Fernandez-Granda is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at https://cds.nyu.edu/math-tools/.