The vast majority of deep learning is performed on Euclidean data. This includes datatypes in the 1-dimensional and 2-dimensional domain. But we don’t exist in a 1D or 2D world. All that we can observe exists in 3D, and our data should reflect that. It’s about time machine learning gets to our level.
Images, text, audio, and many others are all euclidean data.
Non-euclidean data can represent more complex items and concepts with more accuracy than 1D or 2D representation:
When we represent things in a non-euclidean way, we are giving it an inductive bias. This is based on the intuition that, given data of an arbitrary type, format, and size, one can prioritize the model to learn certain patterns by changing the structure of that data. In the majority of current research pursuits and literature, the inductive bias that is used is relational.
Building on this intuition, Geometric Deep Learning (GDL) is the niche field under the umbrella of deep learning that aims to build neural networks that can learn from non-euclidean data.
The prime example of a non-euclidean datatype is a graph. Graphs are a type of data structure that consists of nodes (entities) that are connected with edges (relationships). This abstract data structure can be used to model almost anything.
We want to be able to learn from graphs because:
Graphs allow us to represent individual features, while also providing information regarding relationships and structure.
There are various types of graphs, each with a set of rules, properties, and possible actions. Graph theory is the study of graphs and what we can learn from them. This will be covered in the next part of this series.
Examples of Geometric Deep Learning
These are the two more popular applications and research focuses in literature. They are often used as (unofficial) benchmarks.
Molecular Modeling and learning
For a concrete example of how Graph Learning can improve existing machine learning tasks we can look at the computational sciences.
One of the bottlenecks in computational chemistry, biology, and physics is the representation concepts, entities, and interactions. The nature of science is empirical and is therefore the result of many external factors and relationships. Here are some examples of where this is most obvious:
- Protein interaction networks
- Neural networks
- Feynman diagrams
- Cosmological maps
Our current methods of representing these concepts computationally can be considered “lossy”, since we lose a lot of valuable information. For example, using a simplified-molecular-input-line-entry-system (SMILE) string to represent molecules is easy to compute, but at the expense of structural information of the molecule.
By treating atoms as nodes, and bonds as edges, we can save structural information that can be used downstream in prediction or classification.
So instead of using a string that represents a molecule as input to a Recurrent Neural Network (RNN), we can use molecular graph as input to its geometric equivalent.
3D Modeling and Learning
As an example of how Geometric Deep Learning lets us learn from datatypes never used before, consider a person posing for a camera:
This image is 2D, although in our minds, we are aware that it represents a 3D person. Our current algorithms, namely Convolutional Neural Networks (CNN), are pretty good at predicting labels like the person posing and/or the kinds of poses given only a 2D image. The difficulty arises when poses become extreme and the angle is no longer fixed. Often times there may be clothing or objects in an image that obstruct the view of an algorithm, making it difficult to predict the pose.
Now imagine a 3D model of this same person making poses:
The CNN can now be run on the 3D object itself rather than a 2D image of the object.
Instead of learning from a 2D representation, which restricts the data to a single perspective angle, imagine if we could run a convolution directly on the object itself. Analogously to traditional CNNs, the kernel would pass through every “pixel” represented as a node in a point-cloud (basically a graph that wraps around the 3D object). Every corner and crevice on the 3D model would be covered and the information will be considered. In short, the difference between vanilla CNNs versus it’s Geometric equivalent is predicting the label of n object given a picture of it, versus predicting the label of an object given a 3D model of it.
As our 3D modelling, design, and printing technology improves, one could imagine how this would yield far more accurate and precise results.
The case of Dimensionality
Dimensions in the traditional sense
The notion of dimensionality is already commonly used in data science and machine learning, where the number of “dimensions” correlates to the number of features/attributes per example/datapoint in a dataset.
While at first, the performance of machine learning algorithms spikes, after a certain number of features (dimensions), the performance levels off. This is known as the curse of dimensionality.
Geometric Deep Learning doesn’t solve this problem. Rather, algorithms like graph convolutions reduce the performance penalties incurred when using datatypes that have alot of features, since relational data is considered via inductive bias and not as an additional feature.
What we talk about when we talk about dimensionality
Dimensionality in Geometric Deep learning is just a question of data being used in training a neural network. Euclidean data obeys the rules of euclidean geometry, while non-euclidean data is loyal to non-euclidean geometry.
As explained by this awesome StackExchange A.I stream post, Non-Euclidean geometry can be summed up with the phrase:
“the shortest path between 2 points isn’t necessarily a straight line”.
Other strange rules include:
- Interior angles of triangles always add up to more than 180 degrees
- Parallel lines can meet, either infinitely or never
- Quadrilateral shapes can have curved lines as sides
There is an entire field of non-euclidean geometry which is another topic on its own. For a bit of an intuition-boost, take an image, one of the most popular euclidean datatypes.
An image that is made up of pixels have a notion of left, right, up, and down. One can traverse the image by translating a function over the image recursively. This is exactly what a CNN does.
On a graph however, there is no notion of left, right, up, or down. There is just a node that is connected to an arbitrary number of nodes. A node can even be connected to itself.
Dimensions in the traditional sense of machine and deep learning still exist in the use of non-euclidean data for training neural networks. It is entirely possible to have many node features for example, where each feature is another “dimension”. But the term is rarely used in literature to represent this.
The standard vs the new
Machine learning has centered around Deep learning, which itself revolved around a handful of popular algorithms. Each algorithm roughly specializes in a specific datatype. Just as RNNs were built for time-dependent data and CNNs for image-type data, Graph neural networks (GNNs) are a type of Geometric Deep Learning algorithm built for graphs and networks. As said by Graham Ganssle, Head of Data Science at Expero:
Graph convolutional networks are the best thing since sliced bread because they allow algos to analyze information in its native form rather than requiring an arbitrary representation of that same information in lower dimensional space which destroys the relationship between the data samples thus negating your conclusions. — Graham Ganssle (in a Tweet)
Graph convolutional networks, or GCNs is to a Graph neural networks what CNNs are to Vanilla neural networks.
The implication of this new method makes a big difference; we are no longer forced to leave behind important information in a dataset. Information like structure, relationships, and connections, which are integral to some of the most important data-giving tasks and industries like transportation, social media, and protein networks.
In short, the field of Geometric Deep Learning has 3 main contributions:
- We can make use of non-euclidean data
- We can maximize on the information from the data we collect
- We can use this data to teach machine learning algorithms
In a paper where it was demonstrated that graph learning algorithms can be generalized and made modular for a various applications and augmentations, it was said that:
We argue for making generalization a top priority for AI, and advocate for embracing integrative approaches which draw on ideas from human cognition, traditional computer science, standard engineering practice, and modern deep learning. — DeepMind, Google Brain, MIT, and the University of Edinburgh
In other words, we have much to expect from Geometric Deep Learning.
So what is Geometric Deep Learning?
Geometric Deep Learning is a niche in Deep Learning that aims to generalize neural network models to non-Euclidean domains such as graphs and manifolds.
The notion of relationships, connections, and shared properties is a concept that is naturally occurring in humans and nature. Understanding and learning from these connections is something we take for granted. Geometric Deep Learning is significant because it allows us to take advantage of data with inherent relationships, connections, and shared properties.
- The Euclidean domain and non-euclidean domain have different rules that are followed; data in each domain specializes in certain formats (image, text vs graphs, manifolds) and convey differing amounts of information
- Geometric Deep Learning is the class of Deep Learning that can operate on the non-euclidean domain with the goal of teaching models how to perform predictions and classifications on relational datatypes
- The difference between traditional Deep Learning and Geometric Deep Learning can be illustrated by imagining the accuracy between scanning an image of a person versus scanning the surface of the person themselves.
- In traditional Deep Learning, dimensionality is directly correlated with the number of features in the data whereas in Geometric Deep Learning, it refers to the type of the data itself, not the number of features it has.
One of the reasons I set out to write about Geometric Deep Learning because there is hardly any entry-level resources, tutorial, or guides for this relatively new niche. With that ultimate goal in mind, I am writing a series of articles all on Graph Learning. All articles can be accessed here:
UPDATE: Nov 20th, 2020
The field has changed and grown a lot since this article was written, and I’ve learned a lot over the past year.
Geometric Deep Learning can now be found being used in the front lines of companies like Pinterest, Twitter, Uber, Accenture, and the list goes on to cover just about every company with a finger in the field of ML. In academia, Graph Representation Learning and related terms constantly top the list of topics in discussion for conferences from NeurIPS to ICML (1, 2). We’re beginning to see cross-domain comparisons and novel models with novel applications.
It was already difficult to summarize GDL merely a year ago and it’s futile to do it now. But I feel compelled to try again anyway and obligated to write a more accurate (and updated) summary for an ever-increasing audience because I really love what I do, and I appreciate the people that appreciate content like this. I will link to the fruits of that endeavor when it’s published.
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