"tensor libraries" entries

The tensor renaissance in data science

The O'Reilly Data Show Podcast: Anima Anandkumar on tensor decomposition techniques for machine learning.


After sitting in on UC Irvine Professor Anima Anandkumar’s Strata + Hadoop World 2015 in San Jose presentationI 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.

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Let’s build open source tensor libraries for data science

Tensor methods for machine learning are fast, accurate, and scalable, but we'll need well-developed libraries.


Data scientists frequently find themselves dealing with high-dimensional feature spaces. As an example, text mining usually involves vocabularies comprised of 10,000+ different words. Many analytic problems involve linear algebra, particularly 2D matrix factorization techniques, for which several open source implementations are available. Anyone working on implementing machine learning algorithms ends up needing a good library for matrix analysis and operations.

But why stop at 2D representations? In a recent Strata + Hadoop World San Jose presentation, UC Irvine professor Anima Anandkumar described how techniques developed for higher-dimensional arrays can be applied to machine learning. Tensors are generalizations of matrices that let you look beyond pairwise relationships to higher-dimensional models (a matrix is a second-order tensor). For instance, one can examine patterns between any three (or more) dimensions in data sets. In a text mining application, this leads to models that incorporate the co-occurrence of three or more words, and in social networks, you can use tensors to encode arbitrary degrees of influence (e.g., “friend of friend of friend” of a user).

Being able to capture higher-order relationships proves to be quite useful. In her talk, Anandkumar described applications to latent variable models — including text mining (topic models), information science (social network analysis), recommender systems, and deep neural networks. A natural entry point for applications is to look at generalizations of matrix (2D) techniques to higher-dimensional arrays. Read more…