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Title: Correcting Nuisance Variation using Wasserstein Distance

Paper content:
Profiling cellular phenotypes from microscopic imaging can provide meaningful biological information resulting from various factors affecting the cells.
One motivating application is drug development: morphological cell features can be captured from images, from which similarities between different drugs applied at different dosages can be quantified.
The general approach is to find a function mapping the images to an embedding space of manageable dimensionality whose geometry captures relevant features of the input images.
An important known issue for such methods is separating relevant biological signal from nuisance variation.
For example, the embedding vectors tend to be more correlated for cells that were cultured and imaged during the same week than for cells from a different week, despite having identical drug compounds applied in both cases.
In this case, the particular batch a set of experiments were conducted in constitutes the domain of the data; an ideal set of image embeddings should contain only the relevant biological information (e.g. drug effects).
We develop a general framework for adjusting the image embeddings in order to `forget' domain-specific information while preserving relevant biological information.
To do this, we minimize a loss function based on distances between marginal distributions (such as the Wasserstein distance) of embeddings across domains for each replicated treatment.
For the dataset presented, the replicated treatment is the negative control.
We find that for our transformed embeddings (1) the underlying geometric structure is not only preserved but the embeddings also carry improved biological signal (2) less domain-specific information is present.
In the framework where our approach is applicable, there are some inputs (e.g. images) and a map F sending the inputs to vectors in a low-dimensional space which summarizes information about the inputs.
F could either be engineered using specific image features, or learned (e.g. using deep neural networks).
We will call these vectors 'embeddings' and the space to which they belong the 'embedding space'.
Each input may also have corresponding semantic labels and domains, and for inputs with each label and domain pair, F produces some distribution of embeddings.
Semantically meaningful similarities between pairs of inputs can then be assessed by the distance between their corresponding embeddings, using some chosen distance metric.
Ideally, the embedding distribution of a group of inputs depends only on their label, but often the domain can influence the embedding distribution as well.
We wish to find an additional map to adjust the embeddings produced by F so that the distribution of adjusted embeddings for a given label is independent of the domain, while still preserving semantically meaningful distances between distributions of inputs with different labels.The map F can be used for phenotypic profiling of cells.
In this application, images of biological cells perturbed by one of several possible biological stimuli (e.g. various drug compounds at different doses, some of which may have unknown effects) are mapped to embeddings, which are used to reveal similarities among the applied perturbations.There are a number of ways to extract embeddings from images of cells.
One class of methods such as that used by BID10 relies on extracting specifically engineered features.
In the recent work by BID1 , a Deep Metric Network pre-trained on consumer photographic images (not microscope images of cells) described in BID16 was used to generate embedding vectors from cellular images, and it was shown that these clustered drug compounds by their mechanisms of action (MOA) more effectively.
See Figure 1 for example images of the different MOAs.Currently one of the most important issues with using image embeddings to discriminate the effects of each treatment (i.e. a particular dose of a drug, the 'label' in the general problem described above) on morphological cell features is nuisance factors related to slight uncontrollable variations in each biological experiment.
Many cell imaging experiments are organized into a number of batches of experiments occurring over time, each of which contains a number of sample plates (typically 3-6), each of which contains individual wells in which thousands of cells are grown and treatments are applied (typically around 96 wells per plate).
For this application, the 'domain' is an instance of one of these hierarchical levels, and embeddings for cells with a given treatment tend to be closer to each other within the same domain than from a different one.
For example, the experimentalist may apply slightly different concentrations or amounts of a drug compound in two wells in which the same treatment was anticipated.
Another example is the location of a particular well within a plate or the order of the plate within a batch, which may influence the rate of evaporation, and hence, the appearance of the cells.
Finally, 'batch' effects may result from differences in experiment conditions (temperature, humidity) from week to week; they are various instances of this hierarchical level that we will consider as 'domains' in this work.Our approach addresses the issue of nuisance variation in embeddings by transforming the embedding space in a possibly domain-specific way in order to minimize the variation across domains for a given treatment.
We remark that our main goal is to introduce a general flexible framework to address this problem.
In this framework, we use a metric function measuring the distances among pairs of probability distributions to construct an optimization problem whose solution yields appropriate transformations on each domain.
In our present implementation, the Wasserstein distance is used as a demonstration of a specific choice of the metric that can yield substantial improvements.
The Wasserstein distance makes few assumptions about the probability distributions of the embedding vectors.Our approach is fundamentally different than those which explicitly identify a fixed 'target' and 'source' distributions.
Instead, we incorporate information from all domains on an equal footing, transforming all the embeddings.
This potentially allows our method to incorporate several replicates of a treatment across different domains to learn the transformations, and not only the controls.
We highlight that other distances may be used in our framework, such as the Cramer distance.
This may be preferable since the Cramer distance has unbiased sample gradients BID3 .
This could reduce the number of steps required to adjust the Wasserstein distance approximation for each step of training the embedding transformation.
Additionally we propose several other extensions and variations in Section 4.1.
We have shown how a neural network can be used to transform embedding vectors to 'forget' specifically chosen domain information as indicated by our proposed domain classification metric.
The transformed embeddings still preserve the underlying geometry of the space and improve the k-NN MOA metrics.
Our approach uses the Wasserstein distance and can in principle handle fairly general distributions of embeddings (as long as the neural network used to approximate the Wasserstein function is general enough).
Importantly, we do not have to assume that the distributions are Gaussian.
The framework itself is quite general and extendible (see Section 4.1).
Unlike methods that use only the controls for adjusting the embeddings, our method can also utilize information from replicates of a treatment across different domains.
However, the dataset used did not have treatment replicates across batches, so we only relied on aligning based on the controls.
Thus we implicitly assume that the transformation for the controls matches that of the various compounds.
We expect our method to be more useful in the context of experiments where many replicates are present, so that they can all be aligned simultaneously.
We expect transformations learned for such experiments to have better generalizability since it would be using available knowledge from a greater portion of the embedding space.Our approach requires a choice of free parameters, either for regularization or early stopping, which we address by cross validation across compounds.
We discuss potential future directions below, as well as other limiting issues.
63.6 ± 1% 39.8 ± 0.6% 66.4 ± 0.7% 28.0 ± 0.8% 46.8 ± 0.9% 56.2 ± 0.9% 16.6% RF 45.9 ± 0.2% 34.4 ± 0.7% 46.8 ± 0.6% 26.7 ± 0.7% 33.3 ± 0.7% 39.5 ± 0.1% 16.6% Table 3 : We show the silhouette index for TVN only, TVN + WDN, and TVN + CORAL, as discussed in Section 3.2.2.
Here WDN refers to the the result using early stopping, and λ = 40, 80, 160 refers to the result when using a regularization with λ = λ M = λ b .
Both WDN and CORAL appear to increase the cohesion, as measured by this index.
The estimated error denoted by ± was determined by the bootstrapping procedure described in Section 3.

assistant: text: We correct nuisance variation for image embeddings across different domains, preserving only relevant information.