IEEE Spectrum October, 2012 - 58

recommendation odyssey: An
Amazon user interested in 2001: A Space
Odyssey sees suggestions from three
different collaborative recommenders.

braised short ribs four and a half
stars, fried chicken wings two stars,
cold tofu rolls one star, roasted portobello mushroom five stars, steamed
edamame with sea salt four stars,
and so forth.
A recommender using the matrix
wouldn't really care about your
particular rating of a particular food,
however. Instead, it wants to understand your preferences in general terms, so that it can apply
this knowledge to a wide variety of foods. For instance, given
the above, the recommender might conclude that you like beef,
salty things, and grilled dishes, dislike chicken and anything
fried, are neutral on vegetables, and so on. The number of
such taste attributes or dimensions would be much smaller
than the number of possible foods-there might be 50 or 100
dimensions in all. And by looking at those dimensions, a recommender could quickly determine whether you'd like a new
food-say, salt-crusted prime rib-by comparing its dimensions (salty, beef, not chicken, not fried, not vegetable, not
grilled) against your profile. This more general representation allows the recommender to spot users who prefer similar
yet distinct items. And it substantially compresses the matrix,
making the recommender more efficient.

It's a pretty cool solution.
But how do you find those taste
dimensions? Not by asking a
chef. Instead, these systems use
a mathematical technique called
singular value decomposition
to compute the dimensions.
The technique involves factoring the original giant matrix
into two "taste matrices"-one
that includes all the users and
the 100 taste dimensions and
another that includes all the
foods and the 100 taste dimensions-
plus a third matrix that, when multiplied by either of the other two, re-creates the original matrix.
Unlike the food example above, the dimensions that get
computed are neither describable nor intuitive; they are pure
abstract values, and try as you might, you'll never identify
one that represents, say, "salty." And that's okay, as long as
those values ultimately yield accurate recommendations.
The main drawback to this approach is that the time it takes
to factor the matrix grows quickly with the number of customers and products-a matrix of 250 million customers and
10 million products would take 1 billion times as long to factor as a matrix of 250 000 customers and 10 000 products.
And the process needs to be repeated frequently. The matrix
starts to grow stale as soon as new ratings are received;
at a company like Amazon, that happens every second.
Fortunately, even a slightly stale matrix works reasonably

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Table of Contents for the Digital Edition of IEEE Spectrum October, 2012