Multiscale Topic Modeling with Alto

class: title

# Multiscale Topic Modeling with Alto

<div id="subtitle_left">
Slides: <a href="https://go.wisc.edu/54f161">go.wisc.edu/54f161</a><br/>
Paper: <a href="https://go.wisc.edu/tify36">go.wisc.edu/tify36</a><br/>
Lab: <a href="https://measurement-and-microbes.org">measurement-and-microbes.org</a> <br/>
</div>
<div id="subtitle_right">
Kris Sankaran <br/>
CGSI Virtual Class<br/>
18 | April | 2025 <br/>
</div>

---

### Estimating Weights

The weights `\(W\)` can be estimated by solving the optimal transport problem,
`\begin{align*}
&\min_{W \in \mathcal{U}\left(p, q\right)} \left<C,W\right>
\end{align*}`
<span style="font-size: 20px;">
`\begin{align*}
\mathcal{U}\left(p, q\right) := &\{W\in \mathbb{R}^{\left|V_{p}\right| \times \left|V_{q}\right|}_{+} : W 1_{\left|V_{q}\right|} = p \text{ and } W^{T} 1_{\left|V_{p}\right|} = q\}.
\end{align*}`
</span>

---

### Indication of Consistency

The diagnostics become more reliable as the sample size increases.

---

### Extensions

The `alto` package is focused on topic models. However, it is possible to
generalize to nonnegative matrix factorization.

* We can fit `\(X \approx W\left(m\right)H\left(m\right)^\top\)` for a sequence of `\(m\)`. The scores `\(w_{k}\left(m\right)\)` are highly correlated across adjacent `\(m\)`.

* To build an alignment diagram, we only require that the alignment scores satisfy constraints on row and column totals. This can be achieved using iterative proportional fitting.

---

---

### Code Demo

Follow along: https://go.wisc.edu/d26lb4