## Messing around with word2vec

Word2vec is a way of representing words and phrases as vectors in medium-dimensional space developed by Tomas Mikolov and his team at Google; you can train it on any corpus you like (see Ben Schmidt’s blog for some great examples) but the version of the embedding you can download was trained on about 100 billion words of Google News, and encodes words as unit vectors in 300-dimensional space.

What really got people’s attention, when this came out, was word2vec’s ability to linearize analogies.  For example:  if x is the vector representing “king,” and y the vector representing “woman,” and z the vector representing “man,” then consider

x + y – z

which you might think of, in semantic space, as being the concept “king” to which “woman” has been added and “man” subtracted — in other words, “king made more female.”  What word lies closest in direction to x+y-z?  Just as you might hope, the answer is “queen.”

I found this really startling.  Does it mean that there’s some hidden linear structure in the space of words?

It turns out it’s not quite that simple.  I played around with word2vec a bunch, using Radim Řehůřek’s gensim package that nicely pulls everything into python; here’s what I learned about what the embedding is and isn’t telling you.

Word2Vec distance isn’t semantic distance

The Word2Vec metric tends to place two words close to each other if they occur in similar contexts — that is, w and w’ are close to each other if the words that tend to show up near w also tend to show up near w’  (This is probably an oversimplification, but see this paper of Levy and Goldberg for a more precise formulation.)  If two words are very close to synonymous, you’d expect them to show up in similar contexts, and indeed synonymous words tend to be close:

>>> model.similarity(‘tremendous’,’enormous’)

0.74432902555062841

The notion of similarity used here is just cosine distance (which is to say, dot product of vectors.)  It’s positive when the words are close to each other, negative when the words are far.  For two completely random words, the similarity is pretty close to 0.

On the other hand:

>>> model.similarity(‘tremendous’,’negligible’)

0.37869063705009987

Tremendous and negligible are very far apart semantically; but both words are likely to occur in contexts where we’re talking about size, and using long, Latinate words.  ‘Negligible’ is actually one of the 500 words closest to ’tremendous’ in the whole 3m-word database.

You might ask:  well, what words in Word2Vec are farthest from “tremendous?”  You just get trash:

>>> model.most_similar(negative=[‘tremendous’])

[(u’By_DENISE_DICK’, 0.2792186141014099), (u’NAVARRE_CORPORATION’, 0.26894450187683105), (u’By_SEAN_BARRON’, 0.26745346188545227), (u’LEGAL_NOTICES’, 0.25829464197158813), (u’Ky.Busch_##-###’, 0.2564955949783325), (u’desultorily’, 0.2563159763813019), (u’M.Kenseth_###-###’, 0.2562236189842224), (u’J.McMurray_###-###’, 0.25608277320861816), (u’D.Earnhardt_Jr._###-###’, 0.2547803819179535), (u’david.brett_@_thomsonreuters.com’, 0.2520599961280823)]

If 3 million words were distributed randomly in the unit ball in R^300, you’d expect the farthest one from “tremendous” to have dot product about -0.3 from it.  So when you see a list whose largest score is around that size, you should think “there’s no structure there, this is just noise.”

Antonyms

Challenge problem:  Is there a way to accurately generate antonyms using the word2vec embedding?  That seems to me the sort of thing the embedding is not capturing.  Kyle McDonald had a nice go at this, but I think the lesson of his experiment is that asking word2vec to find analogies of the form “word is to antonym as happy is to?” is just going to generate a list of neighbors of “happy.”  McDonald’s results also cast some light on the structure of word2vec analogies:  for instance, he finds that

waste is to economise as happy is to chuffed

First of all, “chuffed” is a synonym of happy, not an antonym.  But more importantly:  The reason “chuffed” is there is because it’s a way that British people say “happy,” just as “economise” is a way British people spell “economize.”  Change the spelling and you get

waste is to economize as happy is to glad

Non-semantic properties of words matter to word2vec.  They matter a lot.  Which brings us to diction.

Word2Vec distance keeps track of diction

Lots of non-semantic stuff is going on in natural language.  Like diction, which can be high or low, formal or informal, flowery or concrete.    Look at the nearest neighbors of “pugnacity”:

>>> model.most_similar(‘pugnacity’)

[(u’pugnaciousness’, 0.6015268564224243), (u’wonkishness’, 0.6014434099197388), (u’pugnacious’, 0.5877301692962646), (u’eloquence’, 0.5875781774520874), (u’sang_froid’, 0.5873805284500122), (u’truculence’, 0.5838015079498291), (u’pithiness’, 0.5773230195045471), (u’irascibility’, 0.5772287845611572), (u’hotheadedness’, 0.5741063356399536), (u’sangfroid’, 0.5715578198432922)]

Some of these are close semantically to pugnacity, but others, like “wonkishness,” “eloquence”, and “sangfroid,” are really just the kind of elevated-diction words the kind of person who says “pugnacity” would also say.

In the other direction:

>>> model.most_similar(‘psyched’)

[(u’geeked’, 0.7244787216186523), (u’excited’, 0.6678282022476196), (u’jazzed’, 0.666187584400177), (u’bummed’, 0.662735104560852), (u’amped’, 0.6473385691642761), (u’pysched’, 0.6245862245559692), (u’exicted’, 0.6116108894348145), (u’awesome’, 0.5838013887405396), (u’enthused’, 0.581687331199646), (u’kinda_bummed’, 0.5701783299446106)]

“geeked” is a pretty good synonym, but “bummed” is an antonym.  You may also note that contexts where “psyched” is common are also fertile ground for “pysched.”  This leads me to one of my favorite classes of examples:

Misspelling analogies

Which words are closest to “teh”?

>>> model.most_similar(‘teh’)

[(u’ther’, 0.6910992860794067), (u’hte’, 0.6501408815383911), (u’fo’, 0.6458913683891296), (u’tha’, 0.6098173260688782), (u’te’, 0.6042138934135437), (u’ot’, 0.595798909664154), (u’thats’, 0.595078706741333), (u’od’, 0.5908242464065552), (u’tho’, 0.58894944190979), (u’oa’, 0.5846965312957764)]

Makes sense:  the contexts where “teh” is common are those contexts where a lot of words are misspelled.

Using the “analogy” gadget, we can ask; which word is to “because” as “teh” is to “the”?

>>> model.most_similar(positive=[‘because’,’teh’],negative=[‘the’])

[(u’becuase’, 0.6815075278282166), (u’becasue’, 0.6744950413703918), (u’cuz’, 0.6165347099304199), (u’becuz’, 0.6027254462242126), (u’coz’, 0.580410361289978), (u’b_c’, 0.5737690925598145), (u’tho’, 0.5647958517074585), (u’beacause’, 0.5630674362182617), (u’thats’, 0.5605655908584595), (u’lol’, 0.5597798228263855)]

Or “like”?

>>> model.most_similar(positive=[‘like’,’teh’],negative=[‘the’])

[(u’liek’, 0.678846001625061), (u’ok’, 0.6136218309402466), (u’hahah’, 0.5887773633003235), (u’lke’, 0.5840467214584351), (u’probly’, 0.5819578170776367), (u’lol’, 0.5802655816078186), (u’becuz’, 0.5771245956420898), (u’wierd’, 0.5759704113006592), (u’dunno’, 0.5709049701690674), (u’tho’, 0.565370500087738)]

Note that this doesn’t always work:

>>> model.most_similar(positive=[‘should’,’teh’],negative=[‘the’])

[(u’wil’, 0.63351970911026), (u’cant’, 0.6080706715583801), (u’wont’, 0.5967696309089661), (u’dont’, 0.5911301970481873), (u’shold’, 0.5908039212226868), (u’shoud’, 0.5776053667068481), (u’shoudl’, 0.5491836071014404), (u”would’nt”, 0.5474458932876587), (u’shld’, 0.5443994402885437), (u’wouldnt’, 0.5413904190063477)]

What are word2vec analogies?

Now let’s come back to the more philosophical question.  Should the effectiveness of word2vec at solving analogy problems make us think that the space of words really has linear structure?

I don’t think so.  Again, I learned something important from the work of Levy and Goldberg.  When word2vec wants to find the word w which is to x as y is to z, it is trying to find w maximizing the dot product

w . (x + y – z)

But this is the same thing as maximizing

w.x + w.y – w.z.

In other words, what word2vec is really doing is saying

“Show me words which are similar to x and y but are dissimilar to z.”

This notion makes sense applied any notion of similarity, whether or not it has anything to do with embedding in a vector space.  For example, Levy and Goldberg experiment with minimizing

log(w.x) + log(w.y) – log(w.z)

instead, and get somewhat superior results on the analogy task.  Optimizing this objective has nothing to do with the linear combination x+y-z.

None of which is to deny that the analogy engine in word2vec works well in many interesting cases!  It has no trouble, for instance, figuring out that Baltimore is to Maryland as Milwaukee is to Wisconsin.  More often than not, the Milwaukee of state X correctly returns the largest city in state X.  And sometimes, when it doesn’t, it gives the right answer anyway:  for instance, the Milwaukee of Ohio is Cleveland, a much better answer than Ohio’s largest city (Columbus — you knew that, right?)  The Milwaukee of Virginia, according to word2vec, is Charlottesville, which seems clearly wrong.  But maybe that’s OK — maybe there really isn’t a Milwaukee of Virginia.  One feels Richmond is a better guess than Charlottesville, but it scores notably lower.  (Note:  Word2Vec’s database doesn’t have Virginia_Beach, the largest city in Virginia.  That one I didn’t know.)

Another interesting case:  what is to state X as Gainesville is to Florida?  The answer should be “the location of the, or at least a, flagship state university, which isn’t the capital or even a major city of the state,” when such a city exists.  But this doesn’t seem to be something word2vec is good at finding.  The Gainesville of Virginia is Charlottesville, as it should be.  But the Gainesville of Georgia is Newnan.  Newnan?  Well, it turns out there’s a Newnan, Georgia, and there’s also a Newnan’s Lake in Gainesville, FL; I think that’s what’s driving the response.  That, and the fact that “Athens”, the right answer, is contextually separated from Georgia by the existence of Athens, Greece.

The Gainesville of Tennessee is Cookeville, though Knoxville, the right answer, comes a close second.

Why?  You can check that Knoxville, according to word2vec, is much closer to Gainesville than Cookeville is.

>>> model.similarity(‘Cookeville’,’Gainesville’)

0.5457580604439547

>>> model.similarity(‘Knoxville’,’Gainesville’)

0.64010456774402158

But Knoxville is placed much closer to Florida!

>>> model.similarity(‘Cookeville’,’Florida’)

0.2044376252927515

>>> model.similarity(‘Knoxville’,’Florida’)

0.36523836770416895

Remember:  what word2vec is really optimizing for here is “words which are close to Gainesville and close to Tennessee, and which are not close to Florida.”  And here you see that phenomenon very clearly.  I don’t think the semantic relationship between ‘Gainesville’ and ‘Florida’ is something word2vec is really capturing.  Similarly:  the Gainesville of Illinois is Edwardsville (though Champaign, Champaign_Urbana, and Urbana are all top 5) and the Gainesville of Indiana is Connersville.  (The top 5 for Indiana are all cities ending in “ville” — is the phonetic similarity playing some role?)

Just for fun, here’s a scatterplot of the 1000 nearest neighbors of ‘Gainesville’, with their similarity to ‘Gainesville’ (x-axis) plotted against their similarity to ‘Tennessee’ (y-axis):

The Pareto frontier consists of “Tennessee” (that’s the one whose similarity to “Tennessee” is 1, obviously..) Knoxville, Jacksonville, and Tallahassee.

Bag of contexts

One popular simple linear model of word space is given by representing a word as a “bag of contexts” — perhaps there are several thousand contexts, and each word is given by a sparse vector in the space spanned by contexts:  coefficient 0 if the word is not in that context, 1 if it is.  In that setting, certain kinds of analogies would be linearized and certain kinds would not.  If “major city” is a context, then “Houston” and “Dallas” might have vectors that looked like “Texas” with the coodinate of “major city” flipped from 0 to 1.  Ditto, “Milwaukee” would be “Wisconsin” with the same basis vector added.  So

“Texas” + “Milwaukee” – “Wisconsin”

would be pretty close, in that space, to “Houston” and “Dallas.”

On the other hand, it’s not so easy to see what relations antonyms would have in that space. That’s the kind of relationship the bag of contexts may not make linear.

The word2vec space is only 300-dimensional, and the vectors aren’t sparse at all.  But maybe we should think of it as a random low-dimensional projection of a bag-of-contexts embedding!  By the Johnson-Lindenstrauss lemma, a 300-dimensional projection is plenty big enough to preserve the distances between 3 million points with a small distortion factor; and of course it preserves all linear relationships on the nose.

Perhaps this point of view gives some insight into which kind of word relationships manifest as linear relationships in word2vec.  “flock:birds” is an interesting example.  If you imagine “group of things” as a context, you can maybe imagine word2vec picking this up.  But actually, it doesn’t do well:

>> model.most_similar(positive=[‘fish’,’flock’],negative=[‘birds’])
[(u’crays’, 0.4601619839668274), (u’threadfin_salmon’, 0.4553075134754181), (u’spear_fishers’, 0.44864755868911743), (u’slab_crappies’, 0.4483765661716461), (u’flocked’, 0.44473177194595337), (u’Siltcoos_Lake’, 0.4429660737514496), (u’flounder’, 0.4414420425891876), (u’catfish’, 0.4413948059082031), (u’yellowtail_snappers’, 0.4410281181335449), (u’sockeyes’, 0.4395104944705963)]

>> model.most_similar(positive=[‘dogs’,’flock’],negative=[‘birds’])
[(u’dog’, 0.5390862226486206), (u’pooches’, 0.5000904202461243), (u’Eminem_Darth_Vader’, 0.48777419328689575), (u’Labrador_Retrievers’, 0.4792211949825287), (u’canines’, 0.4766522943973541), (u’barked_incessantly’, 0.4709487557411194), (u’Rottweilers_pit_bulls’, 0.4708423614501953), (u’labradoodles’, 0.47032350301742554), (u’rottweilers’, 0.46935927867889404), (u’forbidding_trespassers’, 0.4649636149406433)]

The answers “school” and “pack” don’t appear here.  Part of this, of course, is that “flock,” “school”, and “pack” all have interfering alternate meanings.  But part of it is that the analogy really rests on information about contexts in which the words “flock” and “birds” both appear.  In particular, in a short text window featuring both words, you are going to see a huge spike of “of” appearing right after flock and right before birds.  A statement of the form “flock is to birds as X is to Y” can’t be true unless “X of Y” actually shows up in the corpus a lot.

Challenge problem:  Can you make word2vec do a good job with relations like “flock:birds”?  As I said above, I wouldn’t have been shocked if this had actually worked out of the box, so maybe there’s some minor tweak that makes it work.

Boys’ names, girls’ names

Back to gender-flipping.  What’s the “male version” of the name “Jennifer”?

There are various ways one can do this.  If you use the analogy engine from word2vec, finding the closest word to “Jennifer” + “he” – “she”, you get as your top 5:

David, Jason, Brian, Kevin, Chris

>>> model.most_similar(positive=[‘Jennifer’,’he’],negative=[‘she’])
[(u’David’, 0.6693146228790283), (u’Jason’, 0.6635637283325195), (u’Brian’, 0.6586753129959106), (u’Kevin’, 0.6520106792449951), (u’Chris’, 0.6505492925643921), (u’Mark’, 0.6491551995277405), (u’Matt’, 0.6386727094650269), (u’Daniel’, 0.6294828057289124), (u’Greg’, 0.6267883777618408), (u’Jeff’, 0.6265031099319458)]

But there’s another way:  you can look at the words closest to “Jennifer” (which are essentially all first names) and pick out the ones which are closer to “he” than to “she”.  This gives

Matthew, Jeffrey, Jason, Jesse, Joshua.

>>> [x[0] for x in model.most_similar(‘Jennifer’,topn=2000) if model.similarity(x[0],’he’) > model.similarity(x[0],’she’)]
[u’Matthew’, u’Jeffrey’, u’Jason’, u’Jesse’, u’Joshua’, u’Evan’, u’Brian’, u’Cory’, u’Justin’, u’Shawn’, u’Darrin’, u’David’, u’Chris’, u’Kevin’, u’3/dh’, u’Christopher’, u’Corey’, u’Derek’, u’Alex’, u’Matt’, u’Jeremy’, u’Jeff’, u’Greg’, u’Timothy’, u’Eric’, u’Daniel’, u’Wyvonne’, u’Joel’, u’Chirstopher’, u’Mark’, u’Jonathon’]

Which is a better list of “male analogues of Jennifer?”  Matthew is certainly closer to Jennifer in word2vec distance:

>>> model.similarity(‘Jennifer’,’Matthew’)

0.61308109388608356

>>> model.similarity(‘Jennifer’,’David’)

0.56257556538528708

But, for whatever, reason, “David” is coded as much more strongly male than “Matthew” is; that is, it is closer to “he” – “she”.  (The same is true for “man” – “woman”.)  So “Matthew” doesn’t score high in the first list, which rates names by a combination of how male-context they are and how Jennifery they are.  A quick visit to NameVoyager shows that Matthew and Jennifer both peaked sharply in the 1970s; David, on the other hand, has a much longer range of popularity and was biggest in the 1950s.

Let’s do it again, for Susan.  The two methods give

David, Robert, Mark, Richard, John

Robert, Jeffrey, Richard, David, Kenneth

And for Edith:

Ernest, Edwin, Alfred, Arthur, Bert

Ernest, Harold, Alfred, Bert, Arthur

Pretty good agreement!  And you can see that, in each case, the selected names are “cultural matches” to the starting name.

Sidenote:  In a way it would be more natural to project wordspace down to the orthocomplement of “he” – “she” and find the nearest neighbor to “Susan” after that projection; that’s like, which word is closest to “Susan” if you ignore the contribution of the “he” – “she” direction.  This is the operation Ben Schmidt calls “vector rejection” in his excellent post about his word2vec model trained on student evaluations.

If you do that, you get “Deborah.”  In other words, those two names are similar in so many contextual ways that they remain nearest neighbors even after we “remove the contribution of gender.”  A better way to say it is that the orthogonal projection doesn’t really remove the contribution of gender in toto.  It would be interesting to understand what kind of linear projections actually make it hard to distinguish male surnames from female ones.

Google News is a big enough database that this works on non-English names, too.  The male “Sylvie”, depending on which protocol you pick, is

Alain, Philippe, Serge, Andre, Jean-Francois

or

Jean-Francois, Francois, Stephane, Alain, Andre

The male “Kyoko” is

Kenji, Tomohiko, Nobuhiro, Kazuo, Hiroshi

or

Satoshi, Takayuki, Yosuke, Michio, Noboru

French and Japanese speakers are encouraged to weigh in about which list is better!

Update:  Even a little more messing around with “changing the gender of words” in a followup post.

## Every Noise At Once

Glenn McDonald is the guy who wrote the amazing, obsessive, beautiful music blog The War Against Silence, now mostly dormant.  I admire him for writing tens of thousands of words about Alanis Morissette, whom he, and I, and maybe nobody else, still consider an important cultural figure.  He’s also a pretty hardcore data analyst.  I’ve often fallen down the rabbit hole of his analysis of the Pazz and Jop ballots.

Now he works for Echo Nest, the Greater Boston music startup that sponsored the Music Hack Day I participated in a couple of years ago.  And his latest project, Every Noise At Once, is a map of all music.  Seriously!  A map of all music!  By which I mean: an embedding of the set of genres tracked by EN into the Euclidean plane, and, for each genre, an embedding of bands tagged in that genre.

## Compressed sensing, compressed MDS, compressed clustering, and my talk tomorrow

I’m talking tomorrow at the spanking-new SILO seminar (“Systems, Learning, Information, and Optimization”) in the spanking-new Wisconsin Institute for Discovery.  As you may know, I don’t do any research in any of the four subjects appearing in the acronym!  But I’ve been following some work in the area ever since last year’s very successful Topology and Data seminar.  Here’s one semi-baked idea that will appear in the talk.

World’s most hurried introduction to compressed sensing (see Terry’s blog for a more thorough account.)

Think of an image on an N x N grid of pixels as a vector in R^(N^2).  We know that images can be compressed — that is, we can specify an image with reasonable accuracy using much less information.  That’s because most real-world images lie on or near a manifold in R^(N^2) of dimension much lower than N^2.

It’s already great that we can get away with fewer than N^2 coordinates to specify our image.  But it gets better still.  Maybe it’s expensive to record all N^2 pixels.  If we’re want to end up with (say) only N log N numbers, maybe we should try making only N log N measurements of our image in the first place?  If you just record a subset of N log N pixels, you’re probably in trouble; there may be localized features in the image that happen to be far from any of the pixels in your sparse set.  But if you measure N log N random linear functions on R^(N^2), it turns out that you’re very likely to have pinned down a single point on the manifold of interesting images!

Finite metric spaces

Suppose you’ve got a set S of objects, and a function d: S x S -> R, which you think of as a distance or dissimilarity function.  Could be texts, could be genomes, whatever.  An arbitrary finite metric space is a hard thing to get one’s head around, so it’s desirable to construct an embedding f: S -> R^k such that the Euclidean distance between f(x) and f(y) agrees as closely as possible with d(x,y).  (Think of k as fixed and small.)  The standard algorithm for this is called multidimensional scaling. It works well.  End of story.

Except — note that once again, we’ve compressed our data set.  It takes N^2 coordinates to specify a distance function, but only kN to specify an embedding.  So you can think of MDS as a compression algorithm for metric spaces!  And why does it work?  Because the set of N-point metrics that embed in R^k is a very low-dimensional submanifold of the N^2-dimensional space of general metrics.

So the example of compressed sensing leads us to ask — if you are willing to grant that your metric is approximately embeddable in R^k, can you reconstruct it from a sparse set of distances?  The answer is yes — you can do this with “Landmark MDS,” developed by Vin de Silva and Josh Tenenbaum, and, just as one might want, one can reconstruct the metric from a number of distances that’s linear in N.  I want to call this “Compressed MDS” — is that appropriate?  Will a randomly chosen set of cN distances specify the embedding with high probability?  Seems like it to me.

Multidimensional scaling is just one example of a metric embedding problem — you have a finite set S and a fixed metric space M (or, sometimes, a class of metrics from which M is to be drawn) and you want to find an embedding f: S -> M which respects distance as well as possible.  (This survey by Indyk and Matousek is a good resource for the general problem; and here’s a Newton Institute conference on metric embeddings from two weeks ago, with all the talks online for streaming.)  Of special interest is the case where M is required to be an ultrametric; to embed S in an ultrametric space is to identify S with the leaves of a weighted tree, which is more or less to impose a heirarchical clustering on S.  The space of ultrametrics on S is again of dimension much smaller than N^2, so, the heirarchical construction can be thought of as a compressed representation of the distance function d.  And again, one asks — can we reconstruct the ultrametric from a sparse set of distances?

Yes — a forthcoming paper of Eriksson, Dasarathy, Singh, and Nowak explains how.   N log N distances is enough.  (Update, Feb 2011:  now on the arXiv.)

Can one call this “compressed clustering”?