What is the meaning of sentence E | x?

Question

How to read this math notation?

Let closest(j) be the set of all points that are closest to centroid cj. The algorithm proceeds during several passes, during each of which it updates each centroid based on the data elements currently assigned to that (Chakrabarti et. al):

"The algorithm proceeds during several passes, during each of which it updates each centroid based on the data elements currently assigned to that centroid:" $$ c_j = \mathrm{E} _{x \epsilon closest(j ) } | (x) $$

Chakrabarti, Deepayan, Ravi Kumar, and Andrew Tomkins. "Evolutionary clustering." Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 2006.

Answer 1

OK. I looked at the paper. I searched for a capital "E" prior to the passage you quoted, and saw equation 4, where the paper says

Then, the distance between $T$ and $T'$ is defined to be the average distance between all pairs of objects $$ hc(T', T) = E_{i, j \in leaf(m_\ell)\\i \ne j} (d_{T', T}(i, j)).$$

From this, I infer that $E$ is actually being used to refer to average, which the authors seem to regard as "expectation" with respect to some probability measure -- probably the uniform probability measure.

After that ... it's reasonably clear from the context what's meant in the phrase you quoted: you should average the $x$s that are the nearest neighbors of the centroid. This is a pretty common step in $k$-means types of clustering algorithms. The vertical bar is almost certainly either a typo or merely being used to separate the expectation operator from its argument (which is weird, because it hasn't been used elsewhere --- perhaps the authors thought that just writing $$ E_{x \in closest(j)} x $$ would confuse folks, although I cannot fathom why. Anyhow, if you want the math to match the surrounding words, $E$ must mean "average", and the bar is just noise.

I suppose, if you're really uncertain about this, you could try emailing one of the authors.