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<br>

# .monash-blue[ &nbsp; Burning sage]

<h2 style="font-weight:900!important;">Reversing the curse of dimensionality in the visualization of high-dimensional data</h2>
.center[.monash-gray50[work in collaboration with Di Cook and Stuart Lee]]
.center[.monash-gray50[arXiv:2009.10979]]

.bottom_abs.width100[

## .monash-blue[Ursula Laa]

Monash University

Department of Econometrics and Business Statistics &
School of Physics and Astronomy

<span><i class="fas  fa-envelope faa-float animated "></i></span>  ursula.laa@monash.edu

6th November 2020

<br>
]

]]
]

---
## Open problem described in 2018 (Di Cook)

.center[

<div >
    <div style="width: 33%; float: left">
        <b> pD object </b>
        <a href="">
<img src="images/uncut_cake.png" width = "90%" />        </a>
    </div>
        <div style="width: 33%; float: left">
        <b>3D slice</b>
         <br><br> <br>
        <a href="">
<img src="images/inside_cake_original.png" width = "90%" />      </a>
    </div>
        <div style="width: 33%; float: left">
       <b>pD slice</b>
        <br><br> <br>
        <a href="">
  <img src="images/inside_cake_pinched.png" width = "90%" />        </a>
    </div>
    </div>
]

<br>

<small>
Picture sources: [Samantha Cooper](https://www.pinterest.com.au/pin/551831760586431358Samantha Cooper) and [CakesDekor](https://www.pinterest.com.au/pin/383368987005449767/)
</small>

---
## Curse of dimensionality paradox

- **Origin**: Bellman (1961) described difficulty of optimization in high dimensions given exponential growth in space
- **Consequence**: most points are far from the sample mean, near the edge of the sample space
- **Paradox**: using dimension reduction we instead get an excessive amount of observations near the center of the distribution, most projections are approximately Gaussian

.center[
  <img src="images/density-1.png" width = "75%" />
]

---
## Concepts of projected volume

- To understand the piling near the center of projections, we can think about the high-dimensional volume projected onto a 2D area
- To impose rotation invariance and avoid edge effects, we start from a hypersphere in p dimensions

<br>

.center[
  <img src="images/diagram.png" width = "55%" />
]

---
## Concepts of projected volume

<br>

.center[
  <img src="images/cdf-1.png" width = "75%" />
]

---
## Burning sage transformation

We can define a radial transformation that will redistribute the projected volume such that equal pD volume is projected onto equal 2D area

`$$r'_y = R \sqrt{1-\left(1-\left(\frac{r_y}{R}\right)^2\right)^{p/2}}$$`

.center[
  <img src="images/radii-1.png" width = "55%" />
]

---
## Burning sage transformation

We can define a radial transformation that will redistribute the projected volume such that equal pD volume is projected onto equal 2D area

`$$r'_y = R \sqrt{1-\left(1-\left(\frac{r_y}{R}\right)^2\right)^{p/2}}$$`

.center[
  <img src="images/circles-1.png" width = "85%" />
]

---
## Sage tour

The new transformation is especially useful combined with a tour display, showing sequences of low-dimensional projections: for each new view we project the data to 2D and then show the sage display of the projected data

<br>

---
## Needle in a haystack: Pollen data

---
## Clustering in high dimensions: Single Cell Mouse Retina Data

<br>

.center[
<img src="gifs/mouse_grand_2c.gif" width="400"/> <img src="gifs/mouse_sage_2c_gam3.gif" width="400"/>
]
---
## Clustering in high dimensions: Single Cell Mouse Retina Data

---
## Discussion & Outlook

<br>

- New display that reverses piling effects when visualizing high-dimensional data in low-dimensional projections
- This is especially useful in combination with a tour, implemented as the **sage tour** display
- Related approach: **slice tour** [Laa, Cook, Valencia (2020)](https://doi.org/10.1080/10618600.2020.1777140)
- These new displays are complementary to non-linear dimension reduction methods for visualization, e.g. t-SNE
- Displays should be implemented in an interactive interface, for efficient tuning
- Thinking about new transformations and slicing methods

---
# Acknowledgements

<br>

My slides are made using `RMarkdown`, `xaringan` and the `ninjutsu` theme, and based on a Monash themed template from **Emi Tanaka**.

<br>

The main `R` packages used are `tourr`, `tidyverse`, `plotly`, `geozoo`.

<br>

This is joint work done in collaboration with **Dianne Cook** and **Stuart Lee**.

---

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# Thanks!

<br> 
<a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png" /></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike 4.0 International License</a>.

.bottom_abs.width100[

<h2 style="font-weight:900!important;">Ursula Laa</h2>

Monash University

Department of Econometrics and Business Statistics &
School of Physics and Astronomy

<span><i class="fas  fa-envelope faa-float animated "></i></span>  ursula.laa@monash.edu
]

]]

Notes for current slide

Notes for next slide

Burning sage

Reversing the curse of dimensionality in the visualization of high-dimensional data

work in collaboration with Di Cook and Stuart Lee arXiv:2009.10979

Ursula Laa

Monash University

Department of Econometrics and Business Statistics & School of Physics and Astronomy

ursula.laa@monash.edu

6th November 2020

Open problem described in 2018 (Di Cook)

pD object

3D slice

pD slice

Picture sources: Samantha Cooper and CakesDekor

Curse of dimensionality paradox

Origin: Bellman (1961) described difficulty of optimization in high dimensions given exponential growth in space
Consequence: most points are far from the sample mean, near the edge of the sample space
Paradox: using dimension reduction we instead get an excessive amount of observations near the center of the distribution, most projections are approximately Gaussian

Concepts of projected volume

To understand the piling near the center of projections, we can think about the high-dimensional volume projected onto a 2D area
To impose rotation invariance and avoid edge effects, we start from a hypersphere in p dimensions

Concepts of projected volume

To understand the piling near the center of projections, we can think about the high-dimensional volume projected onto a 2D area
To impose rotation invariance and avoid edge effects, we start from a hypersphere in p dimensions

Concepts of projected volume

To understand the piling near the center of projections, we can think about the high-dimensional volume projected onto a 2D area
To impose rotation invariance and avoid edge effects, we start from a hypersphere in p dimensions

Burning sage transformation

We can define a radial transformation that will redistribute the projected volume such that equal pD volume is projected onto equal 2D area

$r'_y = R \sqrt{1-\left(1-\left(\frac{r_y}{R}\right)^2\right)^{p/2}}$

Burning sage transformation

We can define a radial transformation that will redistribute the projected volume such that equal pD volume is projected onto equal 2D area

$r'_y = R \sqrt{1-\left(1-\left(\frac{r_y}{R}\right)^2\right)^{p/2}}$

Burning sage transformation

We can define a radial transformation that will redistribute the projected volume such that equal pD volume is projected onto equal 2D area

$r'_y = R \sqrt{1-\left(1-\left(\frac{r_y}{R}\right)^2\right)^{p/2}}$

Sage tour

Needle in a haystack: Pollen data

Clustering in high dimensions: Single Cell Mouse Retina Data

Discussion & Outlook

New display that reverses piling effects when visualizing high-dimensional data in low-dimensional projections
This is especially useful in combination with a tour, implemented as the sage tour display
Related approach: slice tour Laa, Cook, Valencia (2020)
These new displays are complementary to non-linear dimension reduction methods for visualization, e.g. t-SNE
Displays should be implemented in an interactive interface, for efficient tuning
Thinking about new transformations and slicing methods

Acknowledgements

My slides are made using RMarkdown, xaringan and the ninjutsu theme, and based on a Monash themed template from Emi Tanaka.

The main R packages used are tourr, tidyverse, plotly, geozoo.

This is joint work done in collaboration with Dianne Cook and Stuart Lee.

Thanks!

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Ursula Laa

Monash University

Department of Econometrics and Business Statistics & School of Physics and Astronomy

ursula.laa@monash.edu

Open problem described in 2018 (Di Cook)

pD object

3D slice

pD slice

Picture sources: Samantha Cooper and CakesDekor

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