Joerns complains that in 1911, the average American spend $81.22 on food, $26.02 on clothes, $19.23 on intoxicants, $9.08 on tobacco, and only $6.19 on furniture. “Do you think furniture should be on the bottom of this list?” he asks, implicitly shaking his head. “Wouldn’t you — dealer or manufacturer — rather see it nearer the top, — say at least ahead of tobacco and intoxicants?”

Good news for furniture lovers: by 2012, US spending on “household furnishings and equipment” was at $1,506 per household, almost a quarter as much as we spent on food. (To be fair, it looks like this includes computers, lawnmowers, and many other non-furniture items.) Meanwhile, spending on alcohol is only $438. That’s pretty interesting: in 1911, liquor expenditures were a quarter of food expenditures; now it’s less than a tenth. Looks like a 1911 dollar is roughly 2012$25, so the real dollars spent on alcohol aren’t that different, but we spend a lot more now on food and on furniture.

Anyway, this piece takes a spendidly nuts turn at the end, as Joerns works up a head of steam about the moral peril of discount furniture:

I do not doubt but that fewer domestic troubles would exist if people were educated to a greater understanding of the furniture sentiment. Our young people would find more pleasure in an evening at home — if we made that home more worth while and a source of personal pride; then, perhaps, they would cease joy-riding, card-playing, or drinking and smoking in environments unhealthful to their minds and bodies.

It would even seem reasonable to assume, that if the public mind were educated to appreciate more the sentiment in furniture and its relation to the Ideal Home, we would have fewer divorces. Home would mean more to the boys and girls of today and the men and women of tomorrow. Obviously, if the public is permitted to lose more and more its appreciation of home sentiment, the divorce evil will grow, year by year.

Joerns proposes that the higher sort of furniture manufacturers boost their brand by advertising it, not as furniture, but as “meuble.” This seems never to have caught on.

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Now we know that you can get pretty decent translation and Go without anything like AI. But AI still seems really hard.

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However, Apple CEO Tim Cook pointed out that 60% of people who owned an iPhone before the launch of the iPhone 6 haven’t upgraded to the most recent models, which implies that there is still room to grow, Reuters notes.

Doesn’t it imply that a) people are no longer on contracts incentivizing biannual upgrade; and b) Apple hasn’t figured out a way to make a new phone that’s different enough from the iPhone 5 to make people want to switch?

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How to put all this information together into a preliminary ranking?

The traditional way is to assign to each applicant their mean score. But there’s a problem: different raters have different scales. My 7 might be your 5.

You could just normalize the scores by subtracting that rater’s overall mean. But that’s problematic too. What if one rater actually happens to have looked at stronger files? Or even if not: what if the relation between rater A’s scale and rater B’s scale isn’t linear? Maybe, for instance, rater A gives everyone she doesn’t think should get in a 0, while rater A uses a range of low scores to express the same opinion, depending on just how unsuitable the candidate seems.

Here’s what I did last year. If (r,a,a’) is a triple with r is a rater and a and a’ are two applicants, such that r rated a higher than a’, you can think of that as a judgment that a is more admittable than a’. And you can put all those judgments from all the raters in a big bag, and then see if you can find a ranking of the applicants (or, if you like, a real-valued function f on the applicants) such that, for every judgment a > a’, we have f(a) > f(a’).

Of course, this might not be possible — two raters might disagree! Or there might be more complicated incompatibilities generated by multiple raters. Still, you can ask: what if I tried to minimize the number of “mistakes”, i.e. the number of judgments in your bag that your choice of ranking contradicts?

Well, you can ask that, but you may not get an answer, because that’s a highly non-convex minimization problem, and is as far as we know completely intractable.

But here’s a way out, or at least a way part of the way out — we can use a *convex relaxation*. Set it up this way. Let V be the space of real-valued functions on applicants. For each judgment j, let mistake_j(f) be the step function

mistake_j(f) = 1 if f(a) < f(a’) + 1

mistake_j(f) = 0 if f(a) >= f(a’) + 1

Then “minimize total number of mistakes” is the problem of minimizing

M = sum_j mistake_j(f)

over V. And M is terribly nonconvex. If you try to gradient-descend (e.g. start with a random ranking and then switch two adjacent applicants whenever doing so reduces the total number of mistakes) you are likely to get caught in a local minimum that’s far from optimal. (Or at least that *can* happen; whether this typically actually happens in practice, I haven’t checked!)

So here’s the move: replace mistake_j(f) with a function that’s “close enough,” but is convex. It acts as a sort of tractable proxy for the optimization you’re actually after. The customary choice here is the *hinge loss*:

hinge_j(f) = min(0, f(a)-f(a’) -1).

Then H := sum_j hinge_j(f) is a convex function on f, which you can easily minimize in Matlab or python. If you can actually find an f with H(f) = 0, you’ve found a ranking which agrees with every judgment in your bag. Usually you can’t, but that’s OK! You’ve very quickly found a function H which does a decent job aggregating the committee scores. and which you can use as your starting point.

Now here’s a paper by Nihal Shah and Martin Wainwright commenter Dustin Mixon linked in my last ranking post. It suggests doing something much simpler: using a *linear* function as a proxy for mistake_j. What this amounts to is: score each applicant by the number of times they were placed above another applicant. Should I be doing this instead? My first instinct is no. It looks like Shah and Wainwright assume that each pair of applicants is equally likely to be compared; I think I don’t want to assume that, and I think (but correct me if I’m wrong!) the optimality they get may not be robust to that?

Anyway, all thoughts on this question — or suggestions as to something *totally different* I could be doing — welcome, of course.

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def mascify(y): return [x[0] for x in model.most_similar(y,topn=200) if model.similarity(x[0],’she’) < model.similarity(x[0],’he’)][0]

“femify” is defined similarly. We could put a threshold away from 0 in there, if we wanted to restrict to more strongly gender-coded words.

Anyway, if you start with a word and run mascify and femify alternately, you can ask whether you eventually wind up in a 2-cycle: a pair of words which are each others gender counterparts in this loose sense.

e.g.

gentle -> easygoing -> chatty -> talkative -> chatty -> …..

So “chatty” and “talkative” are a pair, with “chatty” female-coded and “talkative” male-coded.

beautiful -> magnificent -> wonderful -> marvelous -> wonderful -> …

So far, I keep hitting 2-cycles, and pretty quickly, though I don’t see why a longer cycle wouldn’t be possible or likely. **Update:** Kevin in comments explains very nicely why it has to terminate in a 2-cycle!

Some other pairs, female-coded word first:

overjoyed / elated

strident / vehement

fearful / worried

furious / livid

distraught / despondent

hilarious / funny

exquisite / sumptuous

thought_provoking / insightful

kick_ass / badass

Sometimes it’s basically giving the same word, in two different forms or with one word misspelled:

intuitive / intuitively

buoyant / bouyant

sad / Sad

You can do this for names, too, though you have to make the “topn” a little longer to find matches. I found:

Jamie / Chris

Deborah / Jeffrey

Fran / Pat

Mary / Joseph

Pretty good pairs! Note that you hit a lot of gender-mixed names (Jamie, Chris, Pat), just as you might expect: the male-biased name word2vec-closest to a female name is likely to be a name at least some women have! You can deal with this by putting in a threshold:

>> def mascify(y): return [x[0] for x in model.most_similar(y,topn=1000) if model.similarity(x[0],’she’) < model.similarity(x[0],’he’) – 0.1][0]

This eliminates “Jamie” and “Pat” (though “Chris” still reads as male.)

Now we get some new pairs:

Jody / Steve (this one seems to have a big basis of attraction, it shows up from a lot of initial conditions)

Kasey / Zach

Peter / Catherine (is this a Russia thing?)

Nicola / Dominic

Alison / Ian

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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.

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Please interpret and derive relevant policy prescriptions for math pedagogy in comments.

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Why do we do it that way? It seems totally backwards. For my own part, I want everyone to appreciate me and tell me I’m great right now, while I can enjoy it.* *And if you think my work is overrated and anyway I’m kind of a jerk? After I’m dead would be an *awesome *time to bring that up. You have my permission, go for it.

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Here’s an example familiar to tropical geometers: let T be the hyperfield whose elements are , whose multiplication law is real addition, and whose addition law is

a + b = max(a,b) if a <> b

a + b = {c: c < a} if a=b

In other words, each element of T can be thought of as the valuation of an otherwise unspecified element of a field with a non-archimedean valuation, and then the addition law answers the question “what is ord(x+y) if ord(x) = a and ord(y) = b”?

This may sounds at first like an almost aggressively useless generalization, but no! The main point of Matt’s paper is that it makes sense to talk about a matroid with coefficients in a hyperfield, and that lots of well-studied flavors of matroids can be written as “matroids over F” for a suitable hyperfield F; in this way, a lot of different stories about different matroid theories get unified and cleaned up.

In fact, a matroid itself turns out to be the same thing as a matroid over K, where K is the *Krasner hyperfield*: just two elements 0 and 1, with the multiplication law you expect, and addition given by

0 + 0 = 0

0 + 1 = 1

1 + 1 = {0,1}

One thing I like about K is that it repairs the problem (if you see it as a problem) that the category of fields has no terminal object. K is terminal in the category of hyperfields; any hyperfield (and in particular any field) has a unique map to K which sends 0 to 0 and everything else to 1.

More generally, as Matt observes, if R is a commutative ring, a homomorphism f from R to K is nothing other than a prime ideal of R — namely, f^{-1}(0). So once you relax a little and accept the category of hyperfield, the functor Spec: Rings -> Sets is representable! I enjoy that.

**Update:** David Goss points out that this observation about Spec and the Krasner hyperfield is due to Connes and Consani in “The hyperring of adèle classes” JNT 131, (2011) 159-194, p.161. In fact, for any scheme X of finite type over Z, the underlying Zariski set of X is naturally identified with Hom(Spec(K),X); so Spec(K) functions as a kind of generic point that’s agnostic to characteristic.

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