In a

*very*broad sense, I was discussing the technique of putting a potentially nonlinear problem into a linear vector space. This vague idea matches to many techniques in machine learning. A number of well-developed algorithms take advantage of linearity assumptions, including PCA, logistic regression, and SVM.

^{1}A common approach to machine learning is to find a number of features, which are just functions of your data, and use one of these techniques on the features (hoping they are close enough to linear). Another common technique, the kernel trick, projects features into a higher-dimensional space where the linearity assumption is more likely to get good results. Either way, a large part of the work to get good results is "feature engineering": choosing how to represent the data as a set of features to feed into the machine learning algorithm.

We could even argue that probability theory itself is an example: probabilities are always linear, no matter how nonlinear the underlying problem being described. (The probability of an event is the sum of the ways it could happen.) This gives us nice results; for example, there is always a Nash equilibrium for a game if we allow probabilistic strategies. This is not the case if we only consider "pure" strategies.

This theme is interesting to me, but, I was trying to be much more narrow in talking about recent developments in distributed representations. Like feature-based machine learning, a distributed representation will put data into a vector space to make it easier to work with. Unlike approaches relying on feature engineering, there is an emphasis on figuring out how to get the representations to "build themselves", often starting with randomly assigned vector representations.

The beginning of this kind of approach is probably latent semantic analysis (LSA), which is from 1988. LSA assigns 'semantic vectors' to words based on statistical analysis of the contexts those words occur in, based on the idea that words with similar meaning will have very similar statistics.

Given how old this technique is, the excitement around Google's release of the word2vec tool is striking. Reports spun it as deep learning for the masses. Deep learning is a much more recent wave of development. I think the term has lost much of its meaning in becoming a buzzword.

^{2}Calling word2vec "deep" takes this to farcical levels: the techniques of word2vec improve previous models by

*removing*the hidden layer from the network. Using very shallow learning makes the technique faster, allowing it to be trained on (much!) larger amounts of data. This gives a higher-quality result.

One of the exciting things about word2vec is the good results with solving word analogies by vector math. The result of vector computations like France - Paris and Russia - Moscow are very similar, meaning we can approximately find the vector for a capital given the vector for the corresponding nation. The same trick works for a range of word relationships.

However, I've talked with people who had the incorrect impression that this is a new idea. I'm not sure exactly how old it is, but I've heard the idea mentioned before, and I did find a reference from 2004 which appears to use LSA to do the same basic thing. (I can't see the whole article on google books...)

One thing which I thought was really new was the emerging theme of combining vectors to form representations of compound entities. This, too, is quite old. I found a paper from 1994, which cites harder-to-find papers from 1993, 1990, and 1989 that also developed techniques to combine vectors to create representations of compound objects. Recent developments seem much more

*useful*, but, the basic idea is present.

So, all told, it's a fairly long-standing area which has seen large improvements in the actual techniques employed, but, whose central ideas were laid out (in one form or another) over 20 years ago.

**1**: By the way, don't get too hung up about what makes one machine learning technique "linear" and another "nonlinear". This is a false dichotomy. What I really mean is that a technique works in a vector space (which more or less means a space where + is defined and behaves very much like we expect), and relies "largely" on linear operations in this space.*What does "linear" actually mean? A function F is linear if and only if*F(x+y) = F(x) + F(y)

*and for scalar*a

*,*F(ax) = aF(x).

*PCA, for example, is justified by minimizing a squared error (a common theme), where the error is based on euclidean distance, a linear operation. Notice that taking the square isn't linear, but PCA is still thought of as a linear approach.*

**2**: Deep learning has come to mean almost any multi-layer neural network. The term caught on with the success related to Deep Belief Networks, which proposed specific new techniques. Things currently being called "deep learning" often have little in common with this. I feel the term has been watered down by people looking to associate their work with the success of others. This isn't all bad. The work on multi-layered networks seems to have produced real progress in reducing or eliminating the need for feature engineering.
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