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nsanthan committed May 3, 2023
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Expand Up @@ -39,11 +39,11 @@ do. By design the course is open ended, where you have the option of
going "ahead of lectures", so to speak, and I am always available to help
you when you do so. Each module typically has:

* [Prerequisites](/prerequisites), describing skills you should have prior to starting the module.
* [Learning outcomes](/outcomes), describing the major goals for the module.
* [Readings and other online resources](/readings), providing background material.
* [Experiential learning activities](/experiences), providing a structured set of challenges enabling you to acquire mastery of material.
* [Assessments](/assessments), which help you determine if you have acquired mastery of the material.
* [Prerequisites](https://uhm-descartes.github.io/ee445/prerequisites), describing skills you should have prior to starting the module.
* [Learning outcomes](https://uhm-descartes.github.io/ee445/outcomes), describing the major goals for the module.
* [Readings and other online resources](https://uhm-descartes.github.io/ee445/readings), providing background material.
* [Experiential learning activities](https://uhm-descartes.github.io/ee445/experiences), providing a structured set of challenges enabling you to acquire mastery of material.
* [Assessments](https://uhm-descartes.github.io/ee445/assessments), which help you determine if you have acquired mastery of the material.

## About the instructor

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---
title: "Home"
morea_id: home
morea_type: home
published: true
---

## Welcome to EE 445, Spring 2023

<div class="alert alert-danger" role="alert" markdown="1">

<i class="fa-solid fa-circle-exclamation fa-xl"></i> **Warning: material has not been migrated, you may find demo fillers instead of EE 445 material.**
<hr/>

The course material is transitioning into the Morea Framework.

See the <a href="https://morea-framework.github.io">Morea Framework Project Site</a> for details.
</div>

EE 445 is an undergraduate level introduction to Machine Learning for
Electrical and Computer Engineering students. It augments your base in
probability and linear algebra (and to some extent related engineering
concepts), and leverages this foundation to provide a comprehensive
introduction to machine learning fundamentals.

## Who should take this course

This course is intended for undergraduates in ECE/Math, or graduate
students in ICS, Economics, Math, and Business with a basic working
knowledge of python, probability and linear algebra.

## Pedagogy

EE 445 is structured as a series of [modules](/modules), each taking
approximately 1-2 weeks to complete. Each module has material we will
cover in class (usually the harder or the more important parts), and
will often have supplementary material that you are encouraged to
do. By design the course is open ended, where you have the option of
going "ahead of lectures", so to speak, and I am always available to help
you when you do so. Each module typically has:

* [Prerequisites](/prerequisites), describing skills you should have prior to starting the module.
* [Learning outcomes](/outcomes), describing the major goals for the module.
* [Readings and other online resources](/readings), providing background material.
* [Experiential learning activities](/experiences), providing a structured set of challenges enabling you to acquire mastery of material.
* [Assessments](/assessments), which help you determine if you have acquired mastery of the material.

## About the instructor

[Narayana Prasad Santhanam](https://ee.hawaii.edu/faculty/profile?usr=63) is a Professor of Electrical and Computer Engineering at the University of Hawaii. My research interests are at the intersection of machine learning, information theory and statistics. A particular focus is on high dimensional and complex problems, that are not amenable to traditional statistical methods and guarantees.
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---

First review the section about eigenvalues and eigenspaces
[here]. Recall that \({\mathbb R}^p\) represents the linear space of
all vectors with \(p\) real coordinates. For a matrix \(n\times p\)
matrix \(X\), one can use spectral decomposition of \(X^TX\)
(respectively \(XX^T\)) to find an orthonormal basis for \({\mathbb
R}^p\) (respectively \({\mathbb R}^n\)) using eigenvectors of \(X^TX\)
(respectively \(XX^T\)), and therefore for the rows (respectively
columns) of \(X\). Assume that \(n \ge p\), and let the eigenvalues
of \(X^TX\) be \(\lambda_1\ge \lambda_2 \cdots \ge \lambda_p\), then
the highest \(p\) eigenvalues of \(XX^T\) are also
\(\lambda_1, \lambda_2 \upto \lambda_p\), while the remaining \(n-p\)
[here]. Recall that \\({\mathbb R}^p\\) represents the linear space of
all vectors with \\(p\\) real coordinates. For a matrix \\(n\times p\\)
matrix \\(X\\), one can use spectral decomposition of \\(X^TX\\)
(respectively \\(XX^T\\)) to find an orthonormal basis for \\({\mathbb
R}^p\\) (respectively \\({\mathbb R}^n\\)) using eigenvectors of \\(X^TX\\)
(respectively \\(XX^T\\)), and therefore for the rows (respectively
columns) of \\(X\\). Assume that \\(n \ge p\\), and let the eigenvalues
of \\(X^TX\\) be \\(\lambda_1\ge \lambda_2 \cdots \ge \lambda_p\\), then
the highest \\(p\\) eigenvalues of \\(XX^T\\) are also
\\(\lambda_1, \lambda_2 \upto \lambda_p\\), while the remaining \\(n-p\\)
eigenvalues are all 0.


Let \(V\) (respectively \(U\)) be the matrix formed
Let \\(V\\) (respectively \\(U\\)) be the matrix formed
by placing as columns the orthonormal basis obtained by the
eigendecomposition of \(X^TX\) (respectively \(XX^T\)).
Let \(\Sigma\)
be the \(n\times p\) matrix formed with the positive square roots of
the eigenvalues of \(X^TX\) in all the diagonal locations.
eigendecomposition of \\(X^TX\\) (respectively \\(XX^T\\)).
Let \\(\Sigma\\)
be the \\(n\times p\\) matrix formed with the positive square roots of
the eigenvalues of \\(X^TX\\) in all the diagonal locations.
The singular value decomposition observes that
$$ X = U \Sigma V^T. $$

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