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<!DOCTYPE html>
<html>
<head>
<title>MGC</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<link rel="stylesheet" href="fonts/quadon/quadon.css">
<link rel="stylesheet" href="fonts/gentona/gentona.css">
<link rel="stylesheet" href="slides_style2.css">
<script type="text/javascript" src="assets/plotly/plotly-latest.min.js"></script>
</head>
<body>
<textarea id="source">
<!-- class: middle -->
name:opening
#### Discovering and deciphering relationships across disparate data modalities
Joshua T. Vogelstein | <https://neurodata.io/mgc>
<img src="images/neurodata_purple.png" style="height:350px; float:right;"/>
<br><br><br><br><br><br><br><br><br>
<!-- <img src="images/funding/jhu_bme_blue.png" STYLE="HEIGHT:95px;"/>
<img src="images/funding/KNDI.png" STYLE="HEIGHT:95px;"/> -->
.footnote[[jovo@jhu.edu](mailto:[email protected]) | <http://neurodata.io/talks/> | [@neuro_data](https://twitter.com/neuro_data)]
<br>
---
#### Motivation
<br>
--
- "Understand" the relationship between physical (brain) and mental properties
--
- Question 1: are the two related at all?
--
- Question 2: how are they related?
--
<br><br>
- Note: many efforts dedicated to Q1, fewer to Q2.
- We will formally address Q1, and address Q2
---
<!-- class: center, middle -->
class: left
#### One often desires to test for independence
<br>
<br>
| \\(X\\) | \\(Y\\) |
| :---: | :---: |
| clouds | grass wetness |
| brain connectivity | creativity |
| brain shape | health |
| CLARITY | condition |
| gene expression | cancer |
--
| anything | anything else |
---
#### Statistics Background
<br>
- $X$ is a random variable (some measurement).
--
<br>
- $F_X$ is the distribution of $X$.
--
<br>
- This means $F_X(a) = P(X \leq a)$.
- This is denoted:
<font size="10">$$X \sim F_X$$</font>
---
#### Statistics Background
<br>
- For two random variables $X$ and $Y$, $F_{XY}$ is called the joint distribution.
--
<br>
- This means that $F_{XY} (a, b) = P(X \leq a\ \mathrm{and}\ Y \leq b)$,
or,
<font size="10">$$(X, Y) \sim F_{XY}$$</font>
---
#### Independence
<br>
$X$ and $Y$ are independent if neither contains information about the other.
In other words,
$$F_{XY} = P(X \leq a\ \mathrm{and}\ Y \leq b) = P(X \leq a) \times P(Y \leq b) = F_X F_Y$$
<font size="10">$$ F_{XY} = F_X F_Y$$</font>
Note: These ideas and notation generalize for multivariate $X$ and $Y$.
---
#### Informal Definition of Hypothesis Testing
<br>
- **Null Hypothesis**: The conventional belief about a phenomenon of interest, written $H_0$.
- **Alternative Hypothesis**: An alternate belief about the same phenomenon, written $H_A$.
- **p-value**: The probability (under the null) of measurements more extreme than what was observed.
---
#### Formal Definition of Independence Testing
<br><br>
<!-- <img src="images/setup.png" alt="Drawing" style="width: 60%;"/> -->
$$(X\_i,Y\_i) \sim F\_{XY} = F\_{X|Y} F\_Y, \quad i \in \{1,\ldots,n\}$$
$$H\_0: F\_{XY} = F\_X F\_Y $$
$$H\_A: F\_{XY} \neq F\_X F\_Y $$
---
class: top, left
## Outline
<br>
- intuition
- simulations
- real data
- extensions
- theory
- discussion
---
class: middle
# .center[intuition]
---
class: top, left
#### Intuitive Desiderata of Testing Procedure
<!-- <br> -->
1. Performant under *any* joint distribution
- low- and high-dimensional
- Euclidean and structured data (eg, sequences, images, networks, shapes)
- linear and nonlinear relationships
6. Reveals the "geometry" of dependence
5. Is computational efficiency
Provides a tractable algorithm that addresses the two motivating questions:
- Question 1: are the two related at all?
- Question 2: how are they related?
---
class: center
### correlation coefficient
<img src="images/linear.png" alt="Drawing" style="position:absolute; TOP: 350px; LEFT: 300px; height: 300px;"/>
<!-- $$ r^2 = \frac{(\sum x\_i - \bar{x}) (y\_i - \bar{y})}{\sum (x\_i - \bar{x})^2 \sum (y\_i- \bar{y})^2 } -->
$$r\_{XY}^2 = \frac{(\sum\_{i=1}^n ( x\_i - \bar{x}) (y\_i - \bar{y}))^2}{\sum\_{i=1}^n (x\_i - \bar{x})^2 \sum\_{i=1}^n (y\_i- \bar{y})^2} $$
---
class: center
### **mantel** correlation coefficient
<img src="images/linear.png" alt="Drawing" style="position:absolute; TOP: 350px; LEFT: 300px; height: 300px;"/>
<!-- $$ r^2 = \frac{(\sum x\_i - \bar{x}) (y\_i - \bar{y})}{\sum (x\_i - \bar{x})^2 \sum (y\_i- \bar{y})^2 } -->
$$r\_{XY}^2 = \frac{(\sum\_{i=1}^n ( x\_i - \bar{x}) (y\_i - \bar{y}))^2}{\sum\_{i=1}^n (x\_i - \bar{x})^2 \sum\_{i=1}^n (y\_i- \bar{y})^2} $$
<br>
$$d\_{XY}^2 = \frac{(\sum\_{i,\color{red}{j}=1}^n ( x\_i - \color{red}{x\_j}) (y\_i - \color{red}{y\_j}))^2}{\sum\_{i,\color{red}{j}=1}^n (x\_i - \color{red}{x\_j})^2 \sum\_{i,\color{red}{j}=1}^n (y\_i- \color{red}{y\_j})^2} $$
---
class: center
### **generalized** correlation coefficient
<!-- $$ r^2 = \frac{(\sum x\_i - \bar{x}) (y\_i - \bar{y})}{\sum (x\_i - \bar{x})^2 \sum (y\_i- \bar{y})^2 } -->
$$r\_{XY}^2 = \frac{(\sum\_{i=1}^n ( x\_i - \bar{x}) (y\_i - \bar{y}))^2}{\sum\_{i=1}^n (x\_i - \bar{x})^2 \sum\_{i=1}^n (y\_i- \bar{y})^2} $$
<br>
$$d\_{XY}^2 = \frac{(\sum\_{i,\color{red}{j}=1}^n ( x\_i - \color{red}{x\_j}) (y\_i - \color{red}{y\_j}))^2}{\sum\_{i,\color{red}{j}=1}^n (x\_i - \color{red}{x\_j})^2 \sum\_{i,\color{red}{j}=1}^n (y\_i- \color{red}{y\_j})^2} $$
<br>
$$c\_{XY}^2 = \frac{(\sum\_{i,j=1}^n \color{red}{\sigma\_x}(x\_i,x\_j) \color{red}{\sigma\_y}(y\_i,y\_j))^2}{\sum\_{i,j=1}^n \color{red}{\sigma\_x}(x\_i,x\_j)^2 \sum\_{i,j=1}^n \color{red}{\sigma\_y}(y\_i,y\_j)^2} $$
---
### local distance correlation
<img src="images/linear.png" alt="Drawing" style="position:absolute; TOP: 150px; LEFT: 100px; height: 180px;"/>
--
<img src="images/Fig1B.png" alt="Drawing" style="position:absolute; TOP: 150px; LEFT: 300px; height: 180px;"/>
<!-- --
<img src="images/Fig_centered1Panel2.png" alt="Drawing" style="position:absolute; TOP: 150px; LEFT: 500px; height: 150px;"/> -->
--
<div style="position:absolute; TOP: 150px; LEFT: 560px; height: 200px;">
dcorr(X,Y)=0.15, p-val < 0.001 <br> MGC(X,Y)=0.15, p-val < 0.001
</div>
--
<img src="images/spiral.png" alt="Drawing" style="position:absolute; TOP: 400px; LEFT: 100px; height: 180px;"/>
--
<img src="images/Fig8B.png" alt="Drawing" style="position:absolute; TOP: 400px; LEFT: 300px; height: 180px;"/>
<!-- --
<img src="images/Fig_centered8Panel2.png" alt="Drawing" style="position:absolute; TOP: 400px; LEFT: 500px; height: 150px;"/> -->
--
<div style="position:absolute; TOP: 400px; LEFT: 560px; height: 200px;">
dcorr(X,Y)=0.01, p-val 0.3 <br> MGC(X,Y)= .r[0.13], p-val < .r[0.001]
</div>
---
### multiscale distance correlation
- compute local dcorr **at all scales**
- find scale with **max** smoothed test statistic
- permutation test to determine p-value
---
class: top, center
### Multiscale Generalized Correlation (MGC)
<!-- <img src="https://github.com/neurodata/MGC/raw/master/Figures/FigA.png" alt="Drawing" style="width: 900px;"/> -->
<img src="images/mgc-linear.png" style="width: 780px;"/>
--
<img src="images/mgc-spiral.png" style="width: 780px;"/>
--
<img src="images/mgc-flowchart.png" style="width: 780px;"/>
---
class: top, left
#### Intuitive Desiderata of Testing Procedure
<br>
1. Performant under *any* joint distribution
- low- and high-dimensional
- Euclidean and structured data (eg, sequences, images, networks, shapes)
- linear and nonlinear relationships
6. Reveals the "geometry" of dependence
5. Is computational efficiency
Provides a tractable algorithm that addresses the two motivating questions:
- Question 1: are the two related at all?
- Question 2: how are they related?
---
class: middle, center
# .center[simulations]
---
class: top, left
### 20 Different Functions (1D version)
<img src="images/FigSimVisual.png" alt="local" style="width: 100%;"/>
---
class: top, left
### Definitions
- **power** is the probability of rejecting the null when the alternative is true
- $\beta_n(t)$ = power of test statistic $t$ given $n$ samples
- **relative power** power of one approach minus power of another
- $\beta_n(mgc)- \beta_n(t)$
<!-- ### Empirical Desiderata -->
<!-- 1. statistical efficiency: $$N\_\beta(t') / N\_\beta( \text{MGC}) \geq 1 \text{ for all } -->
<!-- n, \, d, \, F\_{XY}, \text{ and } t' $$ -->
<!-- $$\beta\_n(\text{MGC}) > \beta\_n(t') \text{ for all } -->
<!-- n, \, t' \in \mathcal{T}, \text{ and } F\_{XY} $$ -->
<!-- 2. quantification of relationship -->
---
class: top, left
### 1D Relative Power (MGC nearly dominates)
<img src="images/mgc-1DPower.png" alt="local" style="width: 100%;"/>
---
class: top, left
### HD Relative Power (MGC nearly dominates)
<img src="images/mgc-HDPower.png" alt="local" style="width: 100%;"/>
---
class: top, left
### MGC Requires 1/2 Samples to Achieve Power
<br>
<img src="images/mgc-table.png" alt="local" style="width: 100%;"/>
---
class: top, left
#### MGC Reveals Geometry of Dependence
<img src="images/mgc-maps.png" alt="local" style="width: 100%;"/>
<!-- .center[
<img src="https://github.com/neurodata/MGC/raw/master/Figures/FigHDHeat.png" style="width: 900px;"/>
] -->
---
class: middle, center
# .center[real data]
---
class: top, left
### Real Data Desiderata
<br>
1. when we believe dependence, MGC obtains a small p-value
2. MGC provides insight into the geometry of real dependence
3. when there is no dependence, MGC correcly controls FPR
---
class: top, left
### MGC Discovers Relationships between Brain & Mental Properties
<img src="images/mgc-real-brain.png" alt="Drawing" style="width: 100%;"/>
<img src="images/mgc-brain-table.png" alt="Drawing" style="width: 100%;"/>
---
class: top, left
### MGC Discovers Pancreatic Cancer Biomarkers
<img src="images/mgc-real-cancer.png" alt="Drawing" style="width: 100%;"/>
---
### MGC Does Not Make Too Many False Discoveries
<img src="images/FigFPR.png" alt="Drawing" style="height: 500px;"/>
---
class: middle
# .center[extensions]
---
#### Categorical Features
- All testing depends on measuring similarity/distance between observations
- Let $Y=$ {normal, condition 1, condition 2}
- $d(y_i,y_j) = 0$ if $y_i = y_j$
- $d(y_i,y_j) = 1$ if $y_i \neq y_j$
- Then everything just works :)
#### Mixed Data
- Define similarity/distance appropriately for each feature, then combine.
- In theory, the normalization procedure will work appropriately, though not yet tested.
#### Missing Data & Outliers
- *Missing* Does not yet work, though see RerF talk later today for ideas
- *Outliers* Natively handled, probably best to include them, otherwise, the statistics get (more) complicated. But haven't tested carefully.
---
### Computational Considerations
<br>
- n is # of samples
- T is # of threads
| Method | Complexity |
| :---: | :---: |
| Dcorr | n<sup>2</sup> /T |
| HHG | n<sup>2</sup> log n / T |
| MGC | n<sup>2</sup> log n / T |
- Python, MATLAB and R code at [http://neurodata.io/mgc](http://neurodata.io/mgc)
- Note: none of the implementations are parallelized
---
### Fast MGC
<img src="images/mgc-time.png" style="height: 450px;"/>
---
### K-Sample Tests are Independence Tests
Recall this definition of independence testing
$$(X\_i,Y\_i) \sim F\_{XY} = F\_{X|Y} F\_Y, \quad i \in \{1,\ldots,n\}$$
$$H\_0: F\_{XY} = F\_X F\_Y $$
$$H\_A: F\_{XY} \neq F\_X F\_Y $$
--
Here is the formal definition of 2-sample testing
$$U\_i \sim F_U, \, i \in \{1,\ldots,n\} \quad V\_i \sim F\_V, \, i \in \{1,\ldots,m\}$$
$$H\_0: F\_{U} = F\_V $$
$$H\_A: F\_{U} \neq F\_V $$
--
Let $X = [U\_1, \ldots U\_n | V\_1,\ldots, V\_m]$ and $Y=[0 \ldots 0 | 1 \cdots 1 ]$.
Now, testing whether $X$ and $Y$ are independent is equivalent to testing whether $F_U$ and $F_V$ are the same.
---
class: top, left
### 20 Different 2-Sample Functions (2D version)
<img src="images/mgc-rotation-sims.png" alt="local" style="height: 550px;"/>
---
class: top, left
### Relative Power vs Sample Size for 2D Rotations
<img src="images/mgc-rotation-samplesize.png" alt="local" style="height: 550px;"/>
---
class: top, left
### Relative Power vs Angle for 2D Rotations
<img src="images/mgc-rotation-angles.png" alt="local" style="height: 550px;"/>
---
class: top, left
### Relative Power vs Dimension for Rotations
<img src="images/mgc-rotation-dimension.png" alt="local" style="height: 550px;"/>
---
### Graph Equality Testing
For example, are the left and right hemispheres of the larval Drosophila Mushroom Body sampled from the same distribution?
$G=(V_1,E_1) \sim F_X, \quad H = (V_2, E_2) \sim F_Y$
$H\_0: F\_{X} = F\_y , \quad H\_A: F\_{X} \neq F\_Y $
<img src="images/drosophila-left.png" alt="local" style="height: 300px;"/>
<img src="images/drosophila-right.png" alt="local" style="height: 300px;"/>
---
### Graph Equality Testing
For example, are the left and right hemispheres of the larval Drosophila Mushroom Body sampled from the same distribution?
$G=(V_1,E_1) \sim F_X, \quad H = (V_2, E_2) \sim F_Y$
$H\_0: F\_{X} = F\_Y , \quad H\_A: F\_{X} \neq F\_Y $
<img src="images/mgc-left-v-right-drosophila.png" alt="local" style="height: 350px;"/>
---
### Graph Independence Testing
For example, in the Hermaphrodite *C. elegans*, is the chemical synapse connectome independent of the gap junction connectome?
$G=(V,E_1, Z_v) \sim F_U, \quad H = (V, E_2, Z_v) \sim F_V$
$H\_0: F\_{XY|Z} = F\_{X|Z} F\_{Y|Z} , \quad H\_A: F\_{XY|Z} \neq F\_{X|Z} F\_{Y|Z} $
<img src="images/elegans_connectome.png" alt="local" style="height: 350px;"/>
---
### Graph Independence Testing
For example, in the Hermaphrodite *C. elegans*, is the chemical synapse connectome independent of the gap junction connectome?
$G=(V,E_1, Z_v) \sim F_U, \quad H = (V, E_2, Z_v) \sim F_V$
$H\_0: F\_{XY|Z} = F\_{X|Z} F\_{Y|Z} , \quad H\_A: F\_{XY|Z} \neq F\_{X|Z} F\_{Y|Z} $
<img src="images/celegans_unweighted_ts.png" alt="local" style="height: 350px;"/>
---
### Graph Topology vs Vertex Attribute Testing
For example, in the *C elegans*, each node/neuron has an attribute (e.g., location), and we desire to understand whether connectivity and cell type are independent.
$G=(V,E, Z\_v) \sim F\_{UZ}$
$H\_0: F\_{EZ} = F\_{E} F\_{Z} , \quad H\_A: F\_{EZ} \neq F\_{E} F\_{Z} $
<img src="images/celegans_connectome-v-location.png" style="height: 300px;"/>
<img src="images/mgc-celegans_v_location.png" style="height: 300px;"/>
---
### Time Series Testing
For example, is a given ROI independent of another ROI?
Consider a strictly stationary time series $\{(X\_t,Y\_t)\}_{t=0}^{n}$.
Choose some $M$, the "maximum lag" hyperparameter. We wish to test the following hypothesis.
$$ H\_0: F\_{X\_t,Y\_{t-j}} = F\_{X\_t} F\_{Y\_{t-j}} \text{ for each } j \in \{0, 1, ..., M\}, \ \forall t$$
$$ H\_A: F\_{X\_t,Y\_{t-j}} \neq F\_{X\_t} F\_{Y\_{t-j}} \text{ for some } j \in \{0, 1, ..., M\}, \ \forall t$$
<br>
<img src="images/time-series-process.png" style="height: 100px;"/>
<img src="images/power_curve_nonlinearly_dependent_at_lag_1.png" style="height: 250px;" align="right"/>
---
### Application: Functional Connectomics
<br><br>
<img src="images/neg_log_p_val.png" style="height: 350px;"/>
<img src="images/optimal_lag.png" style="height: 350px;" align="right"/>
---
class: middle, center
# .center[theory]
---
class: top, left
### Definitions
- $t$ is sample MGC
- $T$ is population MGC
- $t^\prime$ is sample Dcorr
- $t^*$ is sample oracle MGC
- $\beta_n(t)$ = power of test statistic $t$ given $n$ samples
---
### Theoretical Desiderata
<!-- <br> -->
| name | principle |
| --- | --- |
| boundedness | $0 \leq t,T \leq 1 $|
| symmetric | $T(X,Y) = T(Y,X)$ |
| 1-linear. | $ T= 1 \Leftrightarrow y = A x + b$ |
| 0-indep. | $T=0 \Leftrightarrow H(X \lvert Y) = H(X)$ |
| ortho. invar. | $T(X,Y) = T(a_1 + b_1 C_1 X, a_2 + b_2 C_2 Y)$ |
--
| univ. consist. | $\beta\_n(t) \to 1, \quad \forall \, F\_{XY}$ |
| dominance | $\beta\_n(t^\prime) \geq \beta\_n(t), \, \forall n,\, t^\prime \in \mathcal{T}, \forall F\_{XY}$ |
| convergence | $t \to T$ as $n \to \infty$ |
<!-- | efficiency | $\mathbb{E}\beta_n(t) \geq \mathbb{E}\beta_n(t'), \, \forall n, t' \in \mathcal{T}$ | -->
<!-- | | comp. efficiency | $\mathcal{O}(d n^2 \log n)$ space and time | -->
<!-- 9. "reveal the "geometry/topology" of the relationship" -->
---
class: top, left
#### MGC is a Reasonable Dependence Statistic
<br>
Thm: Oracle MGC has the following properties:
- 0 ≤ MGC ≤ 1
- MGC = 0 only under independence
- MGC is symmetric
- MGC = 1 only under linear relationship
- MGC is invariant to rotation, translation, and scale of X and/or Y
---
class: left
### MGC has power 1 for all F<sub>XY</sub>
<br><br>
Lemma: \\( \beta_n (t) \to 1 \\) as \\( n \to \infty \\) whenever \\( E [X], E [Y] < \infty. \\)
<!-- <img src="images/thm1.png" style="position:absolute; LEFT: 200px; TOP: 300px; height: 40px;"/> -->
---
class: left
### Linear: Local = Global
<!-- Thm 2: If x is linearly dependent on y, then for any n it always holds that $\beta_n(c_t^*) = \beta_n(c_t)$ -->
<br><br>
Lemma: If $X$ is linearly dependent on $Y$, then it always holds that
$$ \beta_n(t^*) = \beta_n(t^\prime) $$
for any $n$.
<!-- <img src="images/thm2.png" style="position:absolute; LEFT: 60px; TOP: 300px; height: 80px;"/> -->
---
class: left
### (Certain) Nonlinear: Local > Global
<br><br>
Lemma: There exists certain nonlinear relationships between $X$ and $Y$, and $k$ and $l$ such that,
$$\beta_n( t^{k,l} ) > \beta_n(t^\prime). $$
<!-- <img src="images/thm3.png" style="width: 800px;"/> -->
---
class: left
### Oracle MGC > Dcorr
<br><br>
Thm: Oracle MGC statistically dominates dcorr, that is,
$$\beta_n( t^* ) \geq \beta_n(t^\prime). $$
<!-- <img src="images/thm4.png" style="width: 800px;"/> -->
---
class: left
### Sample MGC Converges
<br>
Thm: For any $F_{XY}$ with finite 1st moment,
$\beta_n(t) \to 1$ as $n \to \infty$
if and only if distance functions $\sigma_X$ and $\sigma_Y$ are of strong negative type.
---
class: top, left
### Theoretical Desiderata
<br>
| name | principle |
| --- | --- |
| boundedness | ✅|
| symmetric | ✅|
| linear. | ✅ |
| 0-indep. | ✅ |
| ortho. invar. | L$_p$ |
| univ. consist. | ✅ |
| dominance | ✅ |
| convergence | ✅ |
---
### Overall Summary
- Oracle MGC theoretically dominates, even in finite samples
- MGC empirically nearly dominates on extensive simulations
- Visual quantitative characterization of arbitrary relationships
- MGC reveals geometry of dependence in real data
- MGC mitigate "post selection inference" problems
---
### References
<br>
- MGC for scientists [[1]](https://elifesciences.org/articles/41690)
- Foundational theory for MGC [[2]](https://www.tandfonline.com/doi/full/10.1080/01621459.2018.1543125)
- MGC for independence between graph topology & attributes [[3]](https://arxiv.org/abs/1703.10136)
- MGC for signal subgraph detection [[4]](https://arxiv.org/abs/1801.07683)
- MGC for clustering (ish) [[5]](https://arxiv.org/abs/1710.09859)
- Energy and kernel tests are equivalent [[6]](https://arxiv.org/abs/1806.05514)
- MGC + RF [[7]](https://arxiv.org/abs/1812.00029)
- mgcpy [[8]](https://arxiv.org/abs/1907.02088)
- graph independence testing [[9]](https://arxiv.org/abs/1906.03661)
- MGCX [10]
<br>
### Forthcoming (drafts available upon request)
- fast mgc [11]
- graph two sample testing [12]
- multivariate nonpar k-sample tests [13]
---
### Next Steps
<br><br>
- Get you going with it ([download here](https://neurodata.io/mgc))
- Apply it to your data
---
### Acknowledgements
[[email protected]](mailto:[email protected]) | [neurodata.io](https://neurodata.io) | [@neuro_data](https://twitter.com/neuro_data)
<div class="small-container">
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<div class="centered">Carey Priebe</div>
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<div class="centered">Benjamin Falk</div>
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<div class="centered">Kwame Kutten</div>
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<div class="centered">Vince Lyzinski</div>
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<div class="centered">Daniel Sussman</div>
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<div class="centered">Youngser Park</div>
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<div class="centered">Cencheng Shen</div>
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<div class="centered">Shangsi Wang</div>