Annular similarity

When similarity is used for clustering, then obviously the most similar objects need to be placed in the same cluster.

But when similarity is being used for human consumption, a different dynamic is in play — humans usually already know what the most similar objects are, and are interested in those that are (just) beyond those.

This can be seen most clearly in recommender systems. Purchase an item or watch a Netlflix show, and your recommendation list will fill up with new objects that are very similar to the thing you just bought/watched.

From a strictly algorithm point of view, this is a success — the algorithm found objects similar to the starting object. But from a human point of view this is a total fail because it’s very likely that you, the human, already know about all of these recommended objects. If you bought something, you probably compared the thing you bought with many or all of the objects that are now being recommended to you. If you watched something, the recommendations are still likely to be things you already knew about.

The misconception about what similarity needs to mean to be useful to humans is at the heart of the failure of recommender systems, and even the ad serving systems that many of the online businesses make their money from. Everyone has had the experience of buying something, only to have their ad feed (should they still see it) fill up with ads for similar products (“I see you just bought a new car — here are some other new cars you might like”).

What’s needed is annular similarity — a region that is centred at the initial object, but excludes new objects that are too similar, and focuses instead on objects that are a bit similar.

Amazon tries to do this via “People who bought this also bought” which can show useful add-on products. (They also use “People who viewed this also viewed” but this is much less effective because motivations are so variable.) But this mechanism also fails because buying things together doesn’t necessarily mean that they belong together — it’s common to see recommendations based on the fact that two objects were on special on the same day, and so more likely to be bought together because of the opportunity, rather than any commonality.

Annular similarity is also important in applications that help humans to learn new things: web search, online courses, intelligence analysis. That’s why we built the ATHENS divergent web search engine (refs below) — give it some search terms and it returns (clusters of) web pages that contain information that is just over the horizon from the search terms. We found that this required two annuli — we first constructed the information implicit in the search terms, then an annulus around that of information that we assumed would be known to someone who knew the core derived from the search terms, and only then did we generate another annulus which contains the results returned.

We don’t know many algorithmic ways to find annular similarity. In any distance-based clustering it’s possible, of course, to define an annulus around any point. But it’s tricky to decide on what the inner and outer radii should be, the calculations have to happen in high-dimensional space where the points are very sparse, and it’s not usually clear whether the space is isotropic.

Annular similarity doesn’t work (at least straightforwardly) in density-based (e.g. DBScan) or distribution-based clustering (e.g. EM) because the semantics of ‘cluster’ doesn’t allow for an annulus.

One way that does work (and was used extensively in the ATHENS system) is based on singular vallue decomposition (SVD). An SVD projects a high-dimensional space into a low-dimensional one in such a way as to preserve as much of the variation as possible. One of its useful side-effects is that a point that is similar to many other points tends to be projected close to the origin; and a point that is dissimilar to most other points also tends to be projected close to the origin because the dimension(s) it inhabits have little variation and tend to be projected away. In the resulting low-dimensional projection, points far from the origin tend to be interestingly dissimilar to those at the centre of the structure — and so an annulus imposed on the embedding tends to find an interesting set of objects.

Unfortunately this doesn’t solve the recommender system problem because recommenders need to find similar points that have more non-zeroes than the initial target point — and the projection doesn’t preserve this ordering well. That means that the entire region around the target point has to be searched, which becomes expensive.

There’s an opportunity here to come up with better algorithms to find annular structures. Success would lead to advances in several diverse areas.

(A related problem is the Sound of Music problem, the tendency for a common/popular object to muddle the similarity structure of all of the other objects because of its weak similarity to all of them. The Sound of Music plays this role in movie recommendation systems, but think of wrapping paper as a similar object in the context of Amazon. I’ve written about this in a previous post.)

 

Tracy A. Jenkin, Yolande E. Chan, David B. Skillicorn, Keith W. Rogers:
Individual Exploration, Sensemaking, and Innovation: A Design for the Discovery of Novel Information. Decision Sciences 44(6): 1021-1057 (2013)
Tracy A. Jenkin, David B. Skillicorn, Yolande E. Chan:
Novel Idea Generation, Collaborative Filtering, and Group Innovation Processes. ICIS 2011
David B. Skillicorn, Nikhil Vats:
Novel information discovery for intelligence and counterterrorism. Decision Support Systems 43(4): 1375-1382 (2007)
Nikhil Vats, David B. Skillicorn:
Information discovery within organizations using the Athens system. CASCON 2004: 282-292

 

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