Machine Studying in a Non-Euclidean Area | by Mastafa Foufa | Jun, 2023

Representing knowledge in a non-Euclidean house

It is tough understanding how we will symbolize knowledge in every other approach than with vectors in Rn. Plus, how can we transfer away from the Euclidean distance we all know so effectively to match two vector representations?

An answer is described by manifolds, in Riemannian geometry. Manifolds are objects that appear to be Rn however solely domestically. Meaning we will domestically use vectors for representing our knowledge factors. However solely domestically!

Useful resource: The tangent house [Light grey, TxM] at a degree x from the manifol M [Dark grey] and its tangent vector v. The vector x from the Manifold could be represented domestically within the Euclidean tangent house. From Wikipedia.

The notion of similarity or distances is vital in Machine Studying. If we’re constructing say an NLP mannequin, we wish to protect the notion of similarity in semantics throughout the embedding house that represents textual enter. In different phrases, we would like two phrases which might be comparable in which means to even be comparable within the Euclidean house, i.e. with a low Euclidean distance. Equally, two phrases which might be dissimilar in which means must be far-off within the Euclidean house, i.e. with a excessive Euclidean distance.

Therefore, there must be an equal method when escaping Euclidean geometry. This method is described by a Riemannian metric. The Riemannian metric permits us to match two entities within the non-Euclidean house and protect this intuitive notion of distance.

👀 I bear in mind.

Now, we have to keep in mind that on this non-Euclidean framework we will carry out operations domestically on our knowledge representations and we now have a metric to measure distances. Thus, we’re geared up to do ML in non-Euclidean areas.

🙌🏻 Why ought to I be taught extra about ML in a non-Euclidean house?

To this point, we all know that ML with out the genius Euclid is definitely one thing. There are precise tasks that exist and method our conventional machine studying issues with a special geometry framework.

Now, a query naturally emerges: is it price our time studying greater than the existence of this subject?

It’s a reasonably scary house that entails non-trivial arithmetic. However my good friend, Aniss Medbouhi, ML Doctoral Researcher at KTH, will assist us get previous the inherent complexity of this house.

The opposite purpose I wasn’t satisfied about this house is that I learn that it was largely match for hierarchical knowledge that may be described by timber. At a primary look, it doesn’t contain the information I work with every day.

Nevertheless, the abstracts beneath give us an thought of related datasets of curiosity:

“Nevertheless, latest work has proven that the suitable isometric house for embedding advanced networks shouldn’t be the flat Euclidean house, however negatively curved, hyperbolic house. We current a brand new idea that exploits these latest insights and suggest studying neural embeddings of graphs in hyperbolic house. We offer experimental proof that embedding graphs of their pure geometry signicantly improves efficiency on downstream duties for a number of real-world public datasets.” Chamberlain et al.

“Nevertheless, whereas advanced symbolic datasets usually exhibit a latent hierarchical construction, state-of-the-art strategies usually be taught embeddings in Euclidean vector areas, which don’t account for this property. For this function, we introduce a brand new method for studying hierarchical representations of symbolic knowledge by embedding them into hyperbolic house — or extra exactly into an n-dimensional Poincaré ball.” Nickel and Kiela

The datasets talked about above are listed as follows, by Chamberlain et al. :

(1) Karate: Zachary’s karate membership accommodates 34 vertices divided into two factions. [4]

(2) Polbooks: A community of books about US politics printed across the time of the 2004 presidential election and bought by the net bookseller Edges between books symbolize frequent co-purchasing of books by the identical patrons.

(3) Soccer: A community of American soccer video games between Division IA faculties throughout common season Fall 2000. [2]

(4) Adjnoun: Adjacency community of widespread adjectives and nouns within the novel David Coppereld by Charles Dickens. [3]

(5) Polblogs: A community of hyperlinks between weblogs on US politics, recorded in 2005. [1]

Moreover, in biology, we discover this reference dataset:

  • Biology: Evolutionary knowledge like proteins. [5]
Useful resource: A community illustration of social relationships among the many 34 people within the karate membership studied by Zachary. The inhabitants is split into two fractions based mostly on an occasion [4]. Tailored from Wikipedia.

Lastly, NLP knowledge, i.e. textual knowledge, is one other kind of hierarchical knowledge. In consequence, plenty of domains could profit from understanding the developments in non-Euclidean Machine Studying.

Now that we all know find out how to higher symbolize sure datasets, it’s key to tie it again to Machine Studying. Any downstream ML duties require ingesting knowledge first. Loads of time is spent in cleansing our underlying knowledge and representing it precisely. The standard of the information illustration is important because it immediately impacts the efficiency of our fashions. For instance, in NLP, I counsel my college students to concentrate on architectures that present good embeddings, e.g. contextual embeddings. There was intensive analysis in bettering embeddings, transferring from shallow neural networks (fasttext, word2vec) to deep neural networks and transformers (sentence-transformers, BERT, RoBERTa, XLM). Nevertheless, additionally it is price noting that knowledge illustration could be very a lot linked to the duty at hand, and analysis exhibits that sure shallow neural networks present higher outcomes than deep neural networks, for sure duties.


In this text, we noticed that we will leverage non-Euclidean geometry to deal with present issues particular to spherical knowledge and hierarchical datasets like graphs. When embedding such datasets right into a Euclidean house, the worth to pay is a distortion that doesn’t allow preserving distances from the unique house to the embedding house. Such distortion is intuitive in our earth illustration the place we now have some ways to symbolize our globe, a few of which don’t protect core properties anticipated akin to space preserving. Equally for graphs, core properties have to be preserved and distorting the underlying house could end in poorer efficiency for downstream Machine Studying duties.

Within the subsequent chapter, we’ll be taught extra about spherical and hyperbolic geometries. We are going to focus extra on the latter and provides instinct on how fashions in such house can higher embed hierarchical knowledge.

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