Another Word For It Patrick Durusau on Topic Maps and Semantic Diversity

May 16, 2015

The tensor renaissance in data science

Filed under: Data Science,Mathematics,Tensors — Patrick Durusau @ 8:02 pm

The tensor renaissance in data science by Ben Lorica.

From the post:

After sitting in on UC Irvine Professor Anima Anandkumar’s Strata + Hadoop World 2015 in San Jose presentation, I wrote a post urging the data community to build tensor decomposition libraries for data science. The feedback I’ve gotten from readers has been extremely positive. During the latest episode of the O’Reilly Data Show Podcast, I sat down with Anandkumar to talk about tensor decomposition, machine learning, and the data science program at UC Irvine.

Modeling higher-order relationships

The natural question is: why use tensors when (large) matrices can already be challenging to work with? Proponents are quick to point out that tensors can model more complex relationships. Anandkumar explains:

Tensors are higher order generalizations of matrices. While matrices are two-dimensional arrays consisting of rows and columns, tensors are now multi-dimensional arrays. … For instance, you can picture tensors as a three-dimensional cube. In fact, I have here on my desk a Rubik’s Cube, and sometimes I use it to get a better understanding when I think about tensors. … One of the biggest use of tensors is for representing higher order relationships. … If you want to only represent pair-wise relationships, say co-occurrence of every pair of words in a set of documents, then a matrix suffices. On the other hand, if you want to learn the probability of a range of triplets of words, then we need a tensor to record such relationships. These kinds of higher order relationships are not only important for text, but also, say, for social network analysis. You want to learn not only about who is immediate friends with whom, but, say, who is friends of friends of friends of someone, and so on. Tensors, as a whole, can represent much richer data structures than matrices.

The passage:

…who is friends of friends of friends of someone, and so on. Tensors, as a whole, can represent much richer data structures than matrices.

caught my attention.

The same could be said about other data structures, such as graphs.

I mention graphs because data representations carry assumptions and limitations that aren’t labeled for casual users. Such as directed acyclic graphs not supporting the representation of husband-wife relationships.

BTW, the Wikipedia entry on tensors has this introduction to defining tensor:

There are several approaches to defining tensors. Although seemingly different, the approaches just describe the same geometric concept using different languages and at different levels of abstraction.

Wonder if there is a mapping between the components of the different approaches?

Suggestions of other tensor resources appreciated!

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