cs534/dim_reduction/slides.html at master · yubin-park/cs534 · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
<!DOCTYPE html>
<html>
  <head>
    <title>DimReduce</title>
    <meta charset="utf-8">
    <link rel="stylesheet" href="https://unpkg.com/purecss@1.0.1/build/pure-min.css" integrity="sha384-oAOxQR6DkCoMliIh8yFnu25d7Eq/PHS21PClpwjOTeU2jRSq11vu66rf90/cZr47" crossorigin="anonymous">    <style>
      @import url(https://fonts.googleapis.com/css?family=Yanone+Kaffeesatz);
      @import url(https://fonts.googleapis.com/css?family=Droid+Serif:400,700,400italic);
      @import url(https://fonts.googleapis.com/css?family=Ubuntu+Mono:400,700,400italic);

      body { font-family: 'Droid Serif'; }
      h1, h2, h3 {
        font-family: 'Yanone Kaffeesatz';
        font-weight: normal;
      }
      .remark-code, .remark-inline-code { font-family: 'Ubuntu Mono'; }
      .reference{
      	font-size: 10px;
      }
      .smaller-font { font-size:14px }
      @page {
        size: 908px 681px;
        margin: 0;
      }

      @media print {
        .remark-slide-scaler {
          width: 100% !important;
          height: 100% !important;
          transform: scale(1) !important;
          top: 0 !important;
          left: 0 !important;
        }
      }

      .green {
        color: #45ADA8;
      }

      .figure img{
        height: 550px;
      }

      .figure-200 img{
        height: 200px;
      }

      .figure-250 img{
        height: 250px;
      }

      .figure-300 img{
        height: 300px;
      }

      .figure-350 img{
        height: 350px;
      }

      .figure-w500 img{
        width: 500px;
      }

      .figure-w600 img{
        width: 600px;
      }
    </style>
  </head>
  <body>
    <textarea id="source">

class: center, middle

# Dimensionality Reduction

CS534 - Machine Learning

Yubin Park, PhD

---

class: center, middle

"Less is more"

We want to reduce the dimension of data

$$ \mathbf{X} \rightarrow \mathbf{X}^\prime $$

where `\(\text{dim}(\mathbf{X}^\prime) \ll \text{dim}(\mathbf{X})\)`

---

class: center, middle

with **one condition**

$$ \mathcal{L}(\mathbf{y}, f(\mathbf{X})) \approx \mathcal{L}(\mathbf{y}, f(\mathbf{X}^\prime)) $$

i.e. without losing

much information

from the original data.

---

## Ways to Reduce Dimensions

Many approaches exist:

- Feature Selection
  - through manual analysis
  - through regularization techniques such as Lasso
- Clustering
  - such as [k-means](https://en.wikipedia.org/wiki/K-means_clustering)
- Linear Transformation
  - such as [Principal Component Analysis](https://en.wikipedia.org/wiki/Principal_component_analysis), [Singular Value Decomposition](https://en.wikipedia.org/wiki/Singular_value_decomposition)
- Non-linear Transformation
  - such as [Neural-Net Word Embeddings](https://en.wikipedia.org/wiki/Word_embedding)

We will cover the Linear Transformation approach in this lecture.

---

class: middle, center

.figure-350[![2D](img/2Ddata.png)]

The data seem like distrubuted in 2D, but...

---

class: middle, center

.figure-350[![1D](img/1Ddata.png)]

If we rotate the coordinates, they are distributed in 1D.

---

## Principal Component Analysis

The rotation in the example is a part of "[linear transformation](https://en.wikipedia.org/wiki/Linear_map)".

We want to find a linear transformation:

$$ f(\mathbf{x}) = \mathbf{\mu} + \mathbf{V}\mathbf{x} $$

that minimizes the Squared Loss function:

$$ \sum (\mathbf{x}_i - f(\mathbf{x}_i) )^T(\mathbf{x}_i - f(\mathbf{x}_i) ) $$

---

class: center, middle

## Questions?

</textarea>
    <script src="https://remarkjs.com/downloads/remark-latest.min.js">
    </script>
    <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-AMS_HTML&delayStartupUntil=configured" type="text/javascript">
    </script>
    <script type="text/javascript">
      var hljs = remark.highlighter.engine;
      /*
        Language: terminal console
        Author: Josh Bode <joshbode@gmail.com>
      */
      hljs.registerLanguage('terminal', function() {
        return {
          contains: [
            {
              className: 'string',
              begin: '^([\\w.]+)@([\\w.]+)'
            },
            {
              className: 'constant',
              begin: ' (.*) \\$ '
            },
            {
              className: 'ansi',
              begin: '<span style\\="([^"]+)">',
              end: '<\\/span>'
            }
          ]
        }
      });
      var slideshow = remark.create({
        highlightStyle: 'monokai'
      });

      // extract the embedded styling from ansi spans
      var highlighted = document.querySelectorAll("code.terminal span.hljs-ansi");
      Array.prototype.forEach.call(highlighted, function(next) {
        next.insertAdjacentHTML("beforebegin", next.textContent);
        next.parentNode.removeChild(next);
      });

      // Setup MathJax
      MathJax.Hub.Config({
          tex2jax: {
          skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
          }
      });

      MathJax.Hub.Configured();
    </script>
  </body>
</html>