# Amortized LDA#

LDA [1] (Latent Dirichlet Allocation) posits a generative model where a set of latent topics generates collections of elements. In the case of single-cell RNA sequencing, we can think of these topics as gene modules and each cell as a collection of UMI counts. Other features that can be ascribed to these topics include surface proteins and accessible chromatin regions, all of which have discrete count values. This implementation (Python class AmortizedLDA) of LDA amortizes the cost of performing variational inference for each cell by training a common encoder. Note: this is not an exact implementation of the model described in the original LDA paper.

The advantages of amortized LDA are:

• Can learn underlying topics without a reference.

• Scalable to very large datasets (>1 million cells).

The limitations of amortized LDA include:

• Optimal selection of the number of topics is unclear.

• Amortization gap in optimizing variational parameters.

## Preliminaries#

Amortized LDA takes as input a cell-by-feature matrix $$X$$ with $$C$$ cells and $$F$$ features. Because the LDA model assumes the input is ordered, we refer to this format as the bag-of-words (BoW) representation of the feature counts. Additionally, the number of topics to model must be manually set by the user prior to fitting the model.

## Generative process#

Amortized LDA posits that the $$N$$ observed feature counts for cell $$c$$ are treated as ordered. For all $$n \in [N]$$ feature counts for cell $$c \in [C]$$, the observed feature counts $$x_{cn}$$ are produced according to the following generative process:

\begin{align} \beta_k &\sim \mathrm{Dir}(\eta) &\forall k \in [K]\\ \theta_c &\sim \mathrm{Dir}(\alpha) &\\ x_{cn} &\sim \mathrm{Cat}(\theta_c \beta) &\forall n \in [N] \\ \end{align}

where $$\eta$$ denotes the prior on the Dirichlet distribution for the topic feature distribution $$\beta$$, and $$\alpha$$ denotes the prior on the Dirichlet distribution for the cell topic distribution $$\theta_c$$. In order to compute reparametrization gradients stably, we approximate the Dirichlet distribution with a logistic-Normal distribution, followed by a softmax operation. Specifically, we use the Laplace approximation which has a diagonal covariance matrix [2]:

\begin{align} \mu_k &= \log\alpha_k - \frac{1}{K}\sum_i \log\alpha_i \\ \Sigma_{kk} &= \frac{1}{\alpha_k} \left(1 - \frac{2}{K}\right) + \frac{1}{K^2} \sum_i \frac{1}{\alpha_k} \end{align}

for Dirichlet parameter $$\alpha \in \mathbb{R}^K$$ where $$K$$ denotes the number of topics.

The latent variables, along with their description are summarized in the following table:

Latent variable

Description

Code variable (if different)

$$\alpha \in (0, \infty)^K$$

Parameter for the Dirichlet prior on the cell topic distribution, $$\theta_c$$. Approximated by a logistic-Normal distribution.

cell_topic_prior

$$\eta \in (0, \infty)^K$$

Parameter for the Dirichlet prior on the topic feature distribution, $$\beta_k$$. Approximated by a logistic-Normal distribution.

topic_feature_prior

$$\theta_c \in \Delta^{K-1}$$

Cell topic distribution for a given cell $$c$$.

cell_topic_dist

$$\beta_k \in \Delta^{F-1}$$

Topic feature distribution for a given topic $$k$$.

topic_feature_dist

## Inference#

Amortized LDA uses variational inference and specifically auto-encoding variational bayes (see Variational Inference) to learn both the model parameters (the neural network params, topic feature distributions, etc.) and an approximate posterior distribution. Like scvi.model.SCVI, the underlying class used as the encoder for Amortized LDA is Encoder.

### Topic-based dimensionality reduction#

Users can retrieve the estimated topic proportions in each cell with the following code:

>>> topic_prop = model.get_latent_representation()


Due to the logistic-Normal distribution not having an analytic solution to the mean, we compute a Monte Carlo estimate of the expectation. The number of samples used for the estimate can be configured with the argument n_samples.

Additionally, once can estimate topic proportions on held-out data by passing in an AnnData object with the same format as the dataset used to train the model:

>>> test_topic_prop = model.get_latent_representation(test_adata)


If the learned topics generalize well to other datasets, this can serve as a dimensionality reduction method to the learned topic latent space.

### Feautre module discovery#

Once the model has been fitted, one can retrieve the estimated feature-by-topic distribution:

>>> feature_by_topic = model.get_feature_by_topic()


Like the get_latent_representation() method, this returns a Monte Carlo estimate of the logistic-Normal expectation. Similarly, we can configure the number of samples with n_samples.