# Feature-wise transformations

A simple and surprisingly effective family of conditioning mechanisms.

July 9, 2018

### DOI

10.23915/distill.00011

Many real-world problems require integrating multiple sources of information. Sometimes these problems involve multiple, distinct modalities of information — vision, language, audio, etc. — as is required to understand a scene in a movie or answer a question about an image. Other times, these problems involve multiple sources of the same kind of input, i.e. when summarizing several documents or drawing one image in the style of another.

When approaching such problems, it often makes sense to process one source of information in the context of another; for instance, in the right example above, one can extract meaning from the image in the context of the question. In machine learning, we often refer to this context-based processing as conditioning: the computation carried out by a model is conditioned or modulated by information extracted from an auxiliary input.

Finding an effective way to condition on or fuse sources of information is an open research problem, and in this article, we concentrate on a specific family of approaches we call feature-wise transformations. We will examine the use of feature-wise transformations in many neural network architectures to solve a surprisingly large and diverse set of problems; their success, we will argue, is due to being flexible enough to learn an effective representation of the conditioning input in varied settings. In the language of multi-task learning, where the conditioning signal is taken to be a task description, feature-wise transformations learn a task representation which allows them to capture and leverage the relationship between multiple sources of information, even in remarkably different problem settings.

## Feature-wise transformations

To motivate feature-wise transformations, we start with a basic example, where the two inputs are images and category labels, respectively. For the purpose of this example, we are interested in building a generative model of images of various classes (puppy, boat, airplane, etc.). The model takes as input a class and a source of random noise (e.g., a vector sampled from a normal distribution) and outputs an image sample for the requested class.

Our first instinct might be to build a separate model for each class. For a small number of classes this approach is not too bad a solution, but for thousands of classes, we quickly run into scaling issues, as the number of parameters to store and train grows with the number of classes. We are also missing out on the opportunity to leverage commonalities between classes; for instance, different types of dogs (puppy, terrier, dalmatian, etc.) share visual traits and are likely to share computation when mapping from the abstract noise vector to the output image.

Now let’s imagine that, in addition to the various classes, we also need to model attributes like size or color. In this case, we can’t reasonably expect to train a separate network for each possible conditioning combination! Let’s examine a few simple options.

A quick fix would be to concatenate a representation of the conditioning information to the noise vector and treat the result as the model’s input. This solution is quite parameter-efficient, as we only need to increase the size of the first layer’s weight matrix. However, this approach makes the implicit assumption that the input is where the model needs to use the conditioning information. Maybe this assumption is correct, or maybe it’s not; perhaps the model does not need to incorporate the conditioning information until late into the generation process (e.g., right before generating the final pixel output when conditioning on texture). In this case, we would be forcing the model to carry this information around unaltered for many layers.

Because this operation is cheap, we might as well avoid making any such assumptions and concatenate the conditioning representation to the input of all layers in the network. Let’s call this approach concatenation-based conditioning.

Another efficient way to integrate conditioning information into the network is via conditional biasing, namely, by adding a bias to the hidden layers based on the conditioning representation.

Interestingly, conditional biasing can be thought of as another way to implement concatenation-based conditioning. Consider a fully-connected linear layer applied to the concatenation of an input $\mathbf{x}$ and a conditioning representation $\mathbf{z}$: The same argument applies to convolutional networks, provided we ignore the border effects due to zero-padding.

Yet another efficient way to integrate class information into the network is via conditional scaling, i.e., scaling hidden layers based on the conditioning representation.

A special instance of conditional scaling is feature-wise sigmoidal gating: we scale each feature by a value between $0$ and $1$ (enforced by applying the logistic function), as a function of the conditioning representation. Intuitively, this gating allows the conditioning information to select which features are passed forward and which are zeroed out.

Given that both additive and multiplicative interactions seem natural and intuitive, which approach should we pick? One argument in favor of multiplicative interactions is that they are useful in learning relationships between inputs, as these interactions naturally identify “matches”: multiplying elements that agree in sign yields larger values than multiplying elements that disagree. This property is why dot products are often used to determine how similar two vectors are. Multiplicative interactions alone have had a history of success in various domains — see Bibliographic Notes. One argument in favor of additive interactions is that they are more natural for applications that are less strongly dependent on the joint values of two inputs, like feature aggregation or feature detection (i.e., checking if a feature is present in either of two inputs).

In the spirit of making as few assumptions about the problem as possible, we may as well combine both into a conditional affine transformation. An affine transformation is a transformation of the form $y = m * x + b$.

All methods outlined above share the common trait that they act at the feature level; in other words, they leverage feature-wise interactions between the conditioning representation and the conditioned network. It is certainly possible to use more complex interactions, but feature-wise interactions often strike a happy compromise between effectiveness and efficiency: the number of scaling and/or shifting coefficients to predict scales linearly with the number of features in the network. Also, in practice, feature-wise transformations (often compounded across multiple layers) frequently have enough capacity to model complex phenomenon in various settings.

Lastly, these transformations only enforce a limited inductive bias and remain domain-agnostic. This quality can be a downside, as some problems may be easier to solve with a stronger inductive bias. However, it is this characteristic which also enables these transformations to be so widely effective across problem domains, as we will later review.

### Nomenclature

To continue the discussion on feature-wise transformations we need to abstract away the distinction between multiplicative and additive interactions. Without losing generality, let’s focus on feature-wise affine transformations, and let’s adopt the nomenclature of Perez et al. , which formalizes conditional affine transformations under the acronym FiLM, for Feature-wise Linear Modulation. Strictly speaking, linear is a misnomer, as we allow biasing, but we hope the more rigorous-minded reader will forgive us for the sake of a better-sounding acronym.

We say that a neural network is modulated using FiLM, or FiLM-ed, after inserting FiLM layers into its architecture. These layers are parametrized by some form of conditioning information, and the mapping from conditioning information to FiLM parameters (i.e., the shifting and scaling coefficients) is called the FiLM generator. In other words, the FiLM generator predicts the parameters of the FiLM layers based on some auxiliary input. Note that the FiLM parameters are parameters in one network but predictions from another network, so they aren’t learnable parameters with fixed weights as in the fully traditional sense. For simplicity, you can assume that the FiLM generator outputs the concatenation of all FiLM parameters for the network architecture.

As the name implies, a FiLM layer applies a feature-wise affine transformation to its input. By feature-wise, we mean that scaling and shifting are applied element-wise, or in the case of convolutional networks, feature map -wise. To expand a little more on the convolutional case, feature maps can be thought of as the same feature detector being evaluated at different spatial locations, in which case it makes sense to apply the same affine transformation to all spatial locations. In other words, assuming $\mathbf{x}$ is a FiLM layer’s input, $\mathbf{z}$ is a conditioning input, and $\gamma$ and $\beta$ are $\mathbf{z}$-dependent scaling and shifting vectors, $\textrm{FiLM}(\mathbf{x}) = \gamma(\mathbf{z}) \odot \mathbf{x} + \beta(\mathbf{z}).$ You can interact with the following fully-connected and convolutional FiLM layers to get an intuition of the sort of modulation they allow:

In addition to being a good abstraction of conditional feature-wise transformations, the FiLM nomenclature lends itself well to the notion of a task representation. From the perspective of multi-task learning, we can view the conditioning signal as the task description. More specifically, we can view the concatenation of all FiLM scaling and shifting coefﬁcients as both an instruction on how to modulate the conditioned network and a representation of the task at hand. We will explore and illustrate this idea later on.

## Feature-wise transformations in the literature

Feature-wise transformations find their way into methods applied to many problem settings, but because of their simplicity, their effectiveness is seldom highlighted in lieu of other novel research contributions. Below are a few notable examples of feature-wise transformations in the literature, grouped by application domain. The diversity of these applications underscores the flexible, general-purpose ability of feature-wise interactions to learn effective task representations.

Perez et al. use FiLM layers to build a visual reasoning model trained on the CLEVR dataset to answer multi-step, compositional questions about synthetic images.

The model’s linguistic pipeline is a FiLM generator which extracts a question representation that is linearly mapped to FiLM parameter values. Using these values, FiLM layers inserted within each residual block condition the visual pipeline. The model is trained end-to-end on image-question-answer triples. Strub et al. later on improved on the model by using an attention mechanism to alternate between attending to the language input and generating FiLM parameters layer by layer. This approach was better able to scale to settings with longer input sequences such as dialogue and was evaluated on the GuessWhat?! and ReferIt datasets.

de Vries et al. leverage FiLM to condition a pre-trained network. Their model’s linguistic pipeline modulates the visual pipeline via conditional batch normalization, which can be viewed as a special case of FiLM. The model learns to answer natural language questions about real-world images on the GuessWhat?! and VQAv1 datasets.

The visual pipeline consists of a pre-trained residual network that is fixed throughout training. The linguistic pipeline manipulates the visual pipeline by perturbing the residual network’s batch normalization parameters, which re-scale and re-shift feature maps after activations have been normalized to have zero mean and unit variance. As hinted earlier, conditional batch normalization can be viewed as an instance of FiLM where the post-normalization feature-wise affine transformation is replaced with a FiLM layer.

Dumoulin et al. use feature-wise affine transformations — in the form of conditional instance normalization layers — to condition a style transfer network on a chosen style image. Like conditional batch normalization discussed in the previous subsection, conditional instance normalization can be seen as an instance of FiLM where a FiLM layer replaces the post-normalization feature-wise affine transformation. For style transfer, the network models each style as a separate set of instance normalization parameters, and it applies normalization with these style-specific parameters.

Dumoulin et al. use a simple embedding lookup to produce instance normalization parameters, while Ghiasi et al. further introduce a style prediction network, trained jointly with the style transfer network to predict the conditioning parameters directly from a given style image. In this article we opt to use the FiLM nomenclature because it is decoupled from normalization operations, but the FiLM layers used by Perez et al. were themselves heavily inspired by the conditional normalization layers used by Dumoulin et al. .

Yang et al. use a related architecture for video object segmentation — the task of segmenting a particular object throughout a video given that object’s segmentation in the first frame. Their model conditions an image segmentation network over a video frame on the provided first frame segmentation using feature-wise scaling factors, as well as on the previous frame using position-wise biases.

So far, the models we covered have two sub-networks: a primary network in which feature-wise transformations are applied and a secondary network which outputs parameters for these transformations. However, this distinction between FiLM-ed network and FiLM generator is not strictly necessary. As an example, Huang and Belongie propose an alternative style transfer network that uses adaptive instance normalization layers, which compute normalization parameters using a simple heuristic.

Adaptive instance normalization can be interpreted as inserting a FiLM layer midway through the model. However, rather than relying on a secondary network to predict the FiLM parameters from the style image, the main network itself is used to extract the style features used to compute FiLM parameters. Therefore, the model can be seen as both the FiLM-ed network and the FiLM generator.

As discussed in previous subsections, there is nothing preventing us from considering a neural network’s activations themselves as conditioning information. This idea gives rise to self-conditioned models.

Highway Networks are a prime example of applying this self-conditioning principle. They take inspiration from the LSTMs’ heavy use of feature-wise sigmoidal gating in their input, forget, and output gates to regulate information flow:

The ImageNet 2017 winning model also employs feature-wise sigmoidal gating in a self-conditioning manner, as a way to “recalibrate” a layer’s activations conditioned on themselves.

For statistical language modeling (i.e., predicting the next word in a sentence), the LSTM constitutes a popular class of recurrent network architectures. The LSTM relies heavily on feature-wise sigmoidal gating to control the information flow in and out of the memory or context cell $\mathbf{c}$, based on the hidden states $\mathbf{h}$ and inputs $\mathbf{x}$ at every timestep $\mathbf{t}$.

Also in the domain of language modeling, Dauphin et al. use sigmoidal gating in their proposed gated linear unit, which uses half of the input features to apply feature-wise sigmoidal gating to the other half. Gehring et al. adopt this architectural feature, introducing a fast, parallelizable model for machine translation in the form of a fully convolutional network.

The Gated-Attention Reader uses feature-wise scaling, extracting information from text by conditioning a document-reading network on a query. Its architecture consists of multiple Gated-Attention modules, which involve element-wise multiplications between document representation tokens and token-specific query representations extracted via soft attention on the query representation tokens.

The Gated-Attention architecture uses feature-wise sigmoidal gating to fuse linguistic and visual information in an agent trained to follow simple “go-to” language instructions in the VizDoom 3D environment.

Bahdanau et al. use FiLM layers to condition Neural Module Network and LSTM -based policies to follow basic, compositional language instructions (arranging objects and going to particular locations) in a 2D grid world. They train this policy in an adversarial manner using rewards from another FiLM-based network, trained to discriminate between ground-truth examples of achieved instruction states and failed policy trajectories states.

Outside instruction-following, Kirkpatrick et al. also use game-specific scaling and biasing to condition a shared policy network trained to play 10 different Atari games.

The conditional variant of DCGAN , a well-recognized network architecture for generative adversarial networks , uses concatenation-based conditioning. The class label is broadcasted as a feature map and then concatenated to the input of convolutional and transposed convolutional layers in the discriminator and generator networks.

For convolutional layers, concatenation-based conditioning requires the network to learn redundant convolutional parameters to interpret each constant, conditioning feature map; as a result, directly applying a conditional bias is more parameter efficient, but the two approaches are still mathematically equivalent.

PixelCNN and WaveNet  — two recent advances in autoregressive, generative modeling of images and audio, respectively — use conditional biasing. The simplest form of conditioning in PixelCNN adds feature-wise biases to all convolutional layer outputs. In FiLM parlance, this operation is equivalent to inserting FiLM layers after each convolutional layer and setting the scaling coefficients to a constant value of 1. The authors also describe a location-dependent biasing scheme which cannot be expressed in terms of FiLM layers due to the absence of the feature-wise property.

WaveNet describes two ways in which conditional biasing allows external information to modulate the audio or speech generation process based on conditioning input:

1. Global conditioning applies the same conditional bias to the whole generated sequence and is used e.g. to condition on speaker identity.
2. Local conditioning applies a conditional bias which varies across time steps of the generated sequence and is used e.g. to let linguistic features in a text-to-speech model influence which sounds are produced.

As in PixelCNN, conditioning in WaveNet can be viewed as inserting FiLM layers after each convolutional layer. The main difference lies in how the FiLM-generating network is defined: global conditioning expresses the FiLM-generating network as an embedding lookup which is broadcasted to the whole time series, whereas local conditioning expresses it as a mapping from an input sequence of conditioning information to an output sequence of FiLM parameters.

Kim et al. modulate a deep bidirectional LSTM using a form of conditional normalization. As discussed in the Visual question-answering and Style transfer subsections, conditional normalization can be seen as an instance of FiLM where the post-normalization feature-wise affine transformation is replaced with a FiLM layer.

The key difference here is that the conditioning signal does not come from an external source but rather from utterance summarization feature vectors extracted in each layer to adapt the model.

For domain adaptation, Li et al. find it effective to update the per-channel batch normalization statistics (mean and variance) of a network trained on one domain with that network’s statistics in a new, target domain. As discussed in the Style transfer subsection, this operation is akin to using the network as both the FiLM generator and the FiLM-ed network. Notably, this approach, along with Adaptive Instance Normalization, has the particular advantage of not requiring any extra trainable parameters.

For few-shot learning, Oreshkin et al. explore the use of FiLM layers to provide more robustness to variations in the input distribution across few-shot learning episodes. The training set for a given episode is used to produce FiLM parameters which modulate the feature extractor used in a Prototypical Networks meta-training procedure.

Aside from methods which make direct use of feature-wise transformations, the FiLM framework connects more broadly with the following methods and concepts.

The idea of learning a task representation shares a strong connection with zero-shot learning approaches. In zero-shot learning, semantic task embeddings may be learned from external information and then leveraged to make predictions about classes without training examples. For instance, to generalize to unseen object categories for image classification, one may construct semantic task embeddings from text-only descriptions and exploit objects’ text-based relationships to make predictions for unseen image categories. Frome et al. , Socher et al. , and Norouzi et al. are a few notable exemplars of this idea.

The notion of a secondary network predicting the parameters of a primary network is also well exemplified by HyperNetworks , which predict weights for entire layers (e.g., a recurrent neural network layer). From this perspective, the FiLM generator is a specialized HyperNetwork that predicts the FiLM parameters of the FiLM-ed network. The main distinction between the two resides in the number and specificity of predicted parameters: FiLM requires predicting far fewer parameters than Hypernetworks, but also has less modulation potential. The ideal trade-off between a conditioning mechanism’s capacity, regularization, and computational complexity is still an ongoing area of investigation, and many proposed approaches lie on the spectrum between FiLM and HyperNetworks (see Bibliographic Notes).

Some parallels can be drawn between attention and FiLM, but the two operate in different ways which are important to disambiguate.

This difference stems from distinct intuitions underlying attention and FiLM: the former assumes that specific spatial locations or time steps contain the most useful information, whereas the latter assumes that specific features or feature maps contain the most useful information.

With a little bit of stretching, FiLM can be seen as a special case of a bilinear transformation with low-rank weight matrices. A bilinear transformation defines the relationship between two inputs $\mathbf{x}$ and $\mathbf{z}$ and the $k^{th}$ output feature $y_k$ as $y_k = \mathbf{x}^T W_k \mathbf{z}.$ Note that for each output feature $y_k$ we have a separate matrix $W_k$, so the full set of weights forms a multi-dimensional array.

If we view $\mathbf{z}$ as the concatenation of the scaling and shifting vectors $\gamma$ and $\beta$ and if we augment the input $\mathbf{x}$ with a 1-valued feature, As is commonly done to turn a linear transformation into an affine transformation. we can represent FiLM using a bilinear transformation by zeroing out the appropriate weight matrix entries:

For some applications of bilinear transformations, see the Bibliographic Notes.

## Properties of the learned task representation

As hinted earlier, in adopting the FiLM perspective we implicitly introduce a notion of task representation: each task — be it a question about an image or a painting style to imitate — elicits a different set of FiLM parameters via the FiLM generator which can be understood as its representation in terms of how to modulate the FiLM-ed network. To help better understand the properties of this representation, let’s focus on two FiLM-ed models used in fairly different problem settings:

• The visual reasoning model of Perez et al. , which uses FiLM to modulate a visual processing pipeline based off an input question.
• The artistic style transfer model of Ghiasi et al. , which uses FiLM to modulate a feed-forward style transfer network based off an input style image.

As a starting point, can we discern any pattern in the FiLM parameters as a function of the task description? One way to visualize the FiLM parameter space is to plot $\gamma$ against $\beta$, with each point corresponding to a specific task description and a specific feature map. If we color-code each point according to the feature map it belongs to we observe the following:

The plots above allow us to make several interesting observations. First, FiLM parameters cluster by feature map in parameter space, and the cluster locations are not uniform across feature maps. The orientation of these clusters is also not uniform across feature maps: the main axis of variation can be $\gamma$-aligned, $\beta$-aligned, or diagonal at varying angles. These findings suggest that the affine transformation in FiLM layers is not modulated in a single, consistent way, i.e., using $\gamma$ only, $\beta$ only, or $\gamma$ and $\beta$ together in some specific way. Maybe this is due to the affine transformation being overspecified, or maybe this shows that FiLM layers can be used to perform modulation operations in several distinct ways.

Nevertheless, the fact that these parameter clusters are often somewhat “dense” may help explain why the style transfer model of Ghiasi et al. is able to perform style interpolations: any convex combination of FiLM parameters is likely to correspond to a meaningful parametrization of the FiLM-ed network.

To some extent, the notion of interpolating between tasks using FiLM parameters can be applied even in the visual question-answering setting. Using the model trained in Perez et al. , we interpolated between the model’s FiLM parameters for two pairs of CLEVR questions. Here we visualize the input locations responsible for the globally max-pooled features fed to the visual pipeline’s output classifier:

The network seems to be softly switching where in the image it is looking, based on the task description. It is quite interesting that these semantically meaningful interpolation behaviors emerge, as the network has not been trained to act this way.

Despite these similarities across problem settings, we also observe qualitative differences in the way in which FiLM parameters cluster as a function of the task description. Unlike the style transfer model, the visual reasoning model sometimes exhibits several FiLM parameter sub-clusters for a given feature map.

At the very least, this may indicate that FiLM learns to operate in ways that are problem-specific, and that we should not expect to find a unified and problem-independent explanation for FiLM’s success in modulating FiLM-ed networks. Perhaps the compositional or discrete nature of visual reasoning requires the model to implement several well-defined modes of operation which are less necessary for style transfer.

Focusing on individual feature maps which exhibit sub-clusters, we can try to infer how questions regroup by color-coding the scatter plots by question type.

Sometimes a clear pattern emerges, as in the right plot, where color-related questions concentrate in the top-right cluster — we observe that questions either are of type Query color or Equal color, or contain concepts related to color. Sometimes it is harder to draw a conclusion, as in the left plot, where question types are scattered across the three clusters.

In cases where question types alone cannot explain the clustering of the FiLM parameters, we can turn to the conditioning content itself to gain an understanding of the mechanism at play. Let’s take a look at two more plots: one for feature map 26 as in the previous figure, and another for a different feature map, also exhibiting several subclusters. This time we regroup points by the words which appear in their associated question.

In the left plot, the left subcluster corresponds to questions involving objects positioned in front of other objects, while the right subcluster corresponds to questions involving objects positioned behind other objects. In the right plot we see some evidence of separation based on object material: the left subcluster corresponds to questions involving matte and rubber objects, while the right subcluster contains questions about shiny or metallic objects.

The presence of sub-clusters in the visual reasoning model also suggests that question interpolations may not always work reliably, but these sub-clusters don’t preclude one from performing arithmetic on the question representations, as Perez et al. report.

Perez et al. report that this sort of task analogy is not always successful in correcting the model’s answer, but it does point to an interesting fact about FiLM-ed networks: sometimes the model makes a mistake not because it is incapable of computing the correct output, but because it fails to produce the correct FiLM parameters for a given task description. The reverse can also be true: if the set of tasks the model was trained on is insufficiently rich, the computational primitives learned by the FiLM-ed network may be insufficient to ensure good generalization. For instance, a style transfer model may lack the ability to produce zebra-like patterns if there are no stripes in the styles it was trained on. This could explain why Ghiasi et al. report that their style transfer model’s ability to produce pastiches for new styles degrades if it has been trained on an insufficiently large number of styles. Note however that in that case the FiLM generator’s failure to generalize could also play a role, and further analysis would be needed to draw a definitive conclusion.

This points to a separation between the various computational primitives learned by the FiLM-ed network and the “numerical recipes” learned by the FiLM generator: the model’s ability to generalize depends both on its ability to parse new forms of task descriptions and on it having learned the required computational primitives to solve those tasks. We note that this multi-faceted notion of generalization is inherited directly from the multi-task point of view adopted by the FiLM framework.

Let’s now turn our attention back to the overal structural properties of FiLM parameters observed thus far. The existence of this structure has already been explored, albeit more indirectly, by Ghiasi et al. as well as Perez et al. , who applied t-SNE on the FiLM parameter values.

The projection on the left is inspired by a similar projection done by Perez et al. for their visual reasoning model trained on CLEVR and shows how questions group by question type. The projection on the right is inspired by a similar projection done by Ghiasi et al. for their style transfer network. The projection does not cluster artists as neatly as the projection on the left, but this is to be expected, given that an artist’s style may vary widely over time. However, we can still detect interesting patterns in the projection: note for instance the isolated cluster (circled in the figure) in which paintings by Ivan Shishkin and Rembrandt are aggregated. While these two painters exhibit fairly different styles, the cluster is a grouping of their sketches.

To summarize, the way neural networks learn to use FiLM layers seems to vary from problem to problem, input to input, and even from feature to feature; there does not seem to be a single mechanism by which the network uses FiLM to condition computation. This flexibility may explain why FiLM-related methods have been successful across such a wide variety of domains.

## Discussion

Looking forward, there are still many unanswered questions. Do these experimental observations on FiLM-based architectures generalize to other related conditioning mechanisms, such as conditional biasing, sigmoidal gating, HyperNetworks, and bilinear transformations? When do feature-wise transformations outperform methods with stronger inductive biases and vice versa? Recent work combines feature-wise transformations with stronger inductive bias methods , which could be an optimal middle ground. Also, to what extent are FiLM’s task representation properties inherent to FiLM, and to what extent do they emerge from other features of neural networks (i.e. non-linearities, FiLM generator depth, etc.)? If you are interested in exploring these or other questions about FiLM, we recommend looking into the code bases for FiLM models for visual reasoning and style transfer which we used as a starting point for our experiments here.

Finally, the fact that changes on the feature level alone are able to compound into large and meaningful modulations of the FiLM-ed network is still very surprising to us, and hopefully future work will uncover deeper explanations. For now, though, it is a question that evokes the even grander mystery of how neural networks in general compound simple operations like matrix multiplications and element-wise non-linearities into semantically meaningful transformations.

### Bibliographic Notes

Multiplicative interactions have succeeded on various tasks, ever since they were introduced in vision as “mapping units” and “dynamic mappings” around 40 years ago. These tasks include Character-level Language Modeling, Image Denoising, Pose Estimation, Tracking, Action Recognition, and, more generally, tasks involving relating or matching inputs, such as from different modalities or points in time .

Many models lie on the spectrum between FiLM and Hypernetworks:

• Adaptive CNN predicts the value of several of the model’s convolution filters as a function of auxiliary inputs like camera perspective, level of noise, etc. The resulting convolution filters turn out to be very effective in difficult vision tasks such as crowd counting or image deblurring.
• Residual Adapters also propose to predict entire convolutional filters conditioned on the visual recognition domain they are operating in.
• In zero-shot/one-shot learning, Ba et al. propose a model that predicts convolutional filters and classifiers weights based on textual descriptions of object classes.
• In reinforcement learning, Oh et al. propose a model that computes the parameters of a convolutional policy network conditioned on the task description.

Tenenbaum and Freeman first introduced bilinear models in the vision community to better disentangle latent perceptual factors. The authors wanted to separate an image’s style from its content, arguing that classic linear models were not rich enough to extract such complex interaction. They demonstrate the effectiveness of their approach by applying it to spoken vowel identification or zero-shot font classification. Notable applications include:

• Chuang et al. perform facial animation using bilinear transformations by separating key facial features (the style) from visual emotions (the content). Their method can modify a speaking subject’s expression in recorded sequence from happy to angry or neutral.
• Chu and Park and Yang et al. apply bilinear models to recommendation systems by extracting user and item information in various settings. More generally, recommendation systems rely heavily on matrix factorization methods , which can be viewed as a bilinear model where one of the latent vectors is fixed.
• More recently, bilinear models have inspired new neural architectures in visual recognition , video action recognition , and visual question-answering.

### Acknowledgements

This article would be nowhere near where it is today without the honest and constructive feedback we received from various people across several organizations. We would like to thank Chris Olah and Shan Carter from the Distill editorial team as well as Ludwig Schubert from the Google Brain team for being so generous with their time and advice. We would also like to thank Archy de Berker, Xavier Snelgrove, Pedro Oliveira Pinheiro, Alexei Nordell-Markovits, Masha Krol, and Minh Dao from Element AI; Roland Memisevic from TwentyBN; Dzmitry Bahdanau from MILA; Ameesh Shah and Will Levine from Rice University; Dhanush Radhakrishnan from Roivant Sciences; Raymond Cano from Plaid; Eleni Triantafillou from Toronto University; Olivier Pietquin and Jon Shlens from Google Brain; and Jérémie Mary from Criteo.

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### Citation

Dumoulin, et al., "Feature-wise transformations", Distill, 2018.
@article{dumoulin2018feature-wise,
}