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Wikipedia defines the Jaccard distance between sets A and B as $$J_\delta(A,B)=1-\frac{|A\cap B|}{|A\cup B|}.$$ There's also a book claiming that this is a metric. However, I couldn't find any explanation of why $J_\delta$ obeys the triangle inequality. The naive approach of writing the inequality with seven variables (e.g., $x_{001}$ thru $x_{111}$, where $x_{101}$ is the number of elements in $(A\cap C) \backslash B$) and trying to reduce it seems hopeless for pen and paper. In fact it also seems hopeless for Mathematica, which is trying to find a counterexample for 20 minutes and is still running. (It's supposed to say if there isn't any.)

Is there a simple argument showing that this is a distance? Somehow, it feels like the problem shouldn't be difficult and I'm missing something.

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    $\begingroup$ Would it be possible (and/or appropriate) to put a link on the Wikipedia page to this MO question and answer? I'm hoping someone here knows more about Wikipedia guidelines than I do regarding such links. $\endgroup$ – Joe Silverman Jul 3 '15 at 15:16
  • $\begingroup$ What does that final period communicate in the formula? $\endgroup$ – Gabriel Fair Oct 25 '17 at 0:14
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The trick is to use a transform called the Steinhaus Transform. Given a metric $(X, d)$ and a fixed point $a \in X$, you can define a new distance $D'$ as

$$D'(x,y) = \frac{2D(x,y)}{D(x,a) + D(y,a) + D(x,y)}$$

It's known that this transformation produces a metric from a metric. Now if you take as the base metric $D$ the symmetric difference between two sets, what you end up with is the Jaccard distance (which actually is known by many other names as well).

For more information and references, check out Ken Clarkson's survey (Section 2.3)

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    $\begingroup$ You forgot to specify what $a$ should be, but the empty set seems to do the job. $\endgroup$ – Harald Hanche-Olsen Mar 13 '10 at 19:31
  • $\begingroup$ Steinhaus Transform, is it a standard term? $\endgroup$ – Anton Petrunin Mar 14 '10 at 0:24
  • $\begingroup$ I believe so, at least among the right folks :). It's possible it shows up under other names as well $\endgroup$ – Suresh Venkat Mar 14 '10 at 2:10
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    $\begingroup$ It seems that proving that Steinhaus Transform gives a metric is just as hard as the original problem... $\endgroup$ – Anton Petrunin Mar 18 '10 at 2:26
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    $\begingroup$ The "Steinhaus transform" proof may also be found in The minisum location problem for the Jaccard metric by H. Späth. $\endgroup$ – Chester Aug 30 '17 at 21:39
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Here is an elementary proof of the Steinhaus transform (from which said metricity follows as a special case, as noted in Suresh's answer).

Lemma. Let $p,q,r > 0$ such that $p \le q$. Then, $\frac{p}{q} \le \frac{p+r}{q+r}.$

Corollary. Let $d(x,y)$ be a metric. Then, for arbitrary (but fixed) $a$, \begin{equation*} \delta(x,y) := \frac{2d(x,y)}{d(x,a)+d(y,a)+d(x,y)}, \end{equation*} is a metric.

Proof. Only the triangle inequality for $\delta$ is nontrivial. Let $p=d(x,y)$, $q=d(x,y)+d(x,a)+d(y,a)$, and $r=d(x,z)+d(y,z)-d(x,y)$. Applying the lemma, we obtain \begin{eqnarray*} \delta(x,y) &=& \frac{2d(x,y)}{d(x,a)+d(y,a)+d(x,y)} \le \frac{2d(x,z)+2d(y,z)}{d(x,a)+d(y,a)+d(x,z)+d(y,z)}\\ &=& \frac{2d(x,z)}{d(x,a)+d(z,a)+d(x,z)+d(y,z)+d(y,a)-d(z,a)} + \frac{2d(y,z)}{d(y,a)+d(z,a)+d(y,z)+d(x,z)+d(x,a)-d(z,a)}\\ &\le& \delta(x,z)+\delta(y,z), \end{eqnarray*} where the last inequality again uses triangle inequality for $d$.

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  • $\begingroup$ What does that prime (comma) character mean in the denominator of your corollary equation? $\endgroup$ – Gabriel Fair Oct 25 '17 at 0:16
  • $\begingroup$ I see, it looks visually a bit unfortunately placed; it's just a punctuation after the equation; I don't want to edit the answer just to add a tiny bit more space between the equation and the comma though. $\endgroup$ – Suvrit Oct 25 '17 at 0:46
  • $\begingroup$ oh, thanks. I'm a beginner and I saw the same thing on the wikipedia page and I was very confused. $\endgroup$ – Gabriel Fair Oct 26 '17 at 1:05
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    $\begingroup$ @Gabriel Fair 2017: English sentences, whether or not containing mathematical formulae, are still English sentences, with punctuation working as usual. $\endgroup$ – Qfwfq Nov 21 '18 at 23:51
  • $\begingroup$ @Qfwfq, I think that explanation doesn't work here, since the comma doesn't belong in a plain English sentence either; one wouldn't say "then delta, is a metric". $\endgroup$ – LSpice Nov 22 '18 at 18:04
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I know this is an old question, but I'm surprised that all other answers have overlooked a trivial proof technique. Let's reconsider the naïve approach, suggested by the OP, of splitting the Venn diagram for $A$, $B$, and $C$ into seven pieces $x_{001}, \dots, x_{111}$. The triangle inequality for the Jaccard metric is the following rational inequality:

\begin{multline} 1-\frac{x_{011}+x_{111}}{x_{001}+x_{010}+x_{011}+x_{101}+x_{110}+x_{111}} \\ +1-\frac{x_{110}+x_{111}}{x_{010}+x_{011}+x_{100}+x_{101}+x_{110}+x_{111}} \\ -1+\frac{x_{101}+x_{111}}{x_{001}+x_{011}+x_{100}+x_{101}+x_{110}+x_{111}} \ge 0. \end{multline}

Clear fractions by multiplying through by the product of the denominators, then fully expand the resulting polynomial products. What one obtains is the following monstrosity:

$$ x_{001}^2 x_{010}+x_{001}^2 x_{011}+x_{001}^2 x_{100}+x_{001}^2 x_{101}+x_{001} x_{010}^2+2 x_{001} x_{010} x_{011}+2 x_{001} x_{010} x_{100}+4 x_{001} x_{010} x_{101}+2 x_{001} x_{010} x_{110}+2 x_{001} x_{010} x_{111}+x_{001} x_{011}^2+2 x_{001} x_{011} x_{100}+4 x_{001} x_{011} x_{101}+x_{001} x_{011} x_{110}+x_{001} x_{011} x_{111}+x_{001} x_{100}^2+4 x_{001} x_{100} x_{101}+2 x_{001} x_{100} x_{110}+2 x_{001} x_{100} x_{111}+3 x_{001} x_{101}^2+3 x_{001} x_{101} x_{110}+3 x_{001} x_{101} x_{111}+x_{010}^2 x_{011}+x_{010}^2 x_{100}+2 x_{010}^2 x_{101}+x_{010}^2 x_{110}+2 x_{010}^2 x_{111}+x_{010} x_{011}^2+2 x_{010} x_{011} x_{100}+5 x_{010} x_{011} x_{101}+2 x_{010} x_{011} x_{110}+3 x_{010} x_{011} x_{111}+x_{010} x_{100}^2+4 x_{010} x_{100} x_{101}+2 x_{010} x_{100} x_{110}+2 x_{010} x_{100} x_{111}+4 x_{010} x_{101}^2+5 x_{010} x_{101} x_{110}+6 x_{010} x_{101} x_{111}+x_{010} x_{110}^2+3 x_{010} x_{110} x_{111}+2 x_{010} x_{111}^2+2 x_{011}^2 x_{101}+3 x_{011} x_{100} x_{101}+x_{011} x_{100} x_{110}+4 x_{011} x_{101}^2+4 x_{011} x_{101} x_{110}+4 x_{011} x_{101} x_{111}+x_{100}^2 x_{101}+x_{100}^2 x_{110}+3 x_{100} x_{101}^2+4 x_{100} x_{101} x_{110}+3 x_{100} x_{101} x_{111}+x_{100} x_{110}^2+x_{100} x_{110} x_{111}+2 x_{101}^3+4 x_{101}^2 x_{110}+4 x_{101}^2 x_{111}+2 x_{101} x_{110}^2+4 x_{101} x_{110} x_{111}+2 x_{101} x_{111}^2 \ge 0 $$

Despite its length, this inequality is completely obvious, because every coefficient is positive!

Of course, this technique is beyond the reach of a pen-and-paper calculation, but expanding and simplifying polynomial products should be instantaneous in any modern computer algebra system.

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    $\begingroup$ Actually, going via the Steinhaus transform gives you vastly more than just triangle inequality for the Jaccard distance (because it applies to arbitrary metric spaces!). That said, I think so far the cleanest "venn" diagram proof is in Ryan Moulton's answer. The rational inequality you note above also looks very nice! $\endgroup$ – Suvrit Nov 21 '18 at 13:25
  • $\begingroup$ @Suvrit I definitely agree it's not the most enlightening proof! I was mostly spurred by the OP's remark that the problem "seems hopeless for Mathematica," to which I thought "nonsense --- moderate-sized rational inequalities should be well within reach." And I do find some elegance, admittedly of a different sort, when a simple-minded proof idea can be made to work modulo computational effort. $\endgroup$ – David Zhang Nov 21 '18 at 23:48
  • $\begingroup$ IIRC, the first inequality is exactly what I plugged in some Mathematica function. But now I can't recall which one ... $\endgroup$ – rgrig Jan 10 at 19:11
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Possibly the simplest proof of the triangle inequality for the jaccard distance comes from the fact that it is the collision probability of the MinHash algorithm, and that's all we need. Let $H(X) = \text{argmin}_{i\in X} \pi(i)$ where $\pi(i)$ is a uniformly random permutation.

\begin{align*} J(X,Y) &= \Pr\left[H(X) = H(Y)\right] \\ 1 - J(X,Y) &= \Pr\left[H(X) \neq H(Y)\right].\\ \end{align*} So for any $Z$, \begin{align*} \Pr\left[H(X) = H(Y)\right] &\ge \Pr\left[H(X) = H(Z) \land H(Y) = H(Z)\right] \\ \Pr\left[H(X) \neq H(Y)\right] &\le \Pr\left[H(X) \neq H(Z) \lor H(Y) \neq H(Z)\right] \end{align*} But by the union bound, \begin{align*} \begin{split} \Pr\big[H(X) \neq H(Z) \lor H(Y) \neq H(Z)\big] &\le \Pr\big[H(X) \neq H(Z)\big] + \Pr\big[H(Y) \neq H(Z)\big] \end{split} \end{align*}

My co-author used this to prove that a particular jaccard generalization is a metric after I'd been struggling to prove it for a month, and I couldn't believe it.

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    $\begingroup$ Note that this seems to be the same as the fixed version of Emolga's answer above. $\endgroup$ – Harry Altman Nov 21 '18 at 6:48
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We permute all the elements of $A \cup B \cup C$ and denote by $p_{A,B}$ the probability that the first element of the permutation that is in $A$ or $B$ is not in both. This probability is equal to $1-\frac{A \cap B}{A \cup B}$, which is the Jaccard distance, because we look at the first element which is in $A \cup B$ and the probability that it is in both sets is $\frac{A \cap B}{A \cup B}$.

Now we are only left to prove that $p_{A,B}+p_{B,C} \geq p_{A,C}$. That's true because if the first element of the permutation that is in $A$ is in index $i(A) \neq i(C)$, then it means that $i(A) \neq i(B)$ or $i(B) \neq i(C)$.

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  • $\begingroup$ The definition of $p_{A,B}$ is asymmetric in $A$ and $B$, unlike Jaccard distance. The error in the proof occurs in "we look at the first element which in in $A\cup B$" since the first element which is in $A\cup B$ might be in $B$, but not in $A$; even though the first element in $A$ is also in $B$. $\endgroup$ – Boris Bukh Jul 3 '15 at 7:33
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    $\begingroup$ Howver, the proof can be salvaged. Just define $i(A)$ to be the first element of $A$, $i(B)$ to be the first element of $B$, etc. Define $p_{A,B}$ as "$i(A)\neq i(B)$", and so on. $\endgroup$ – Boris Bukh Jul 3 '15 at 7:38
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It is possible to prove this directly too, without invoking the Steinhaus Transform. But that would probably make the proof longer. However, I did once prove it directly, and I think it went a bit like this:

Assume there exist A, B ,C such that d(A,B) + d(B,C) < d(A,C). For such a counterexample, note that A, C and $A\cap C$ have to be nonempty. Now since the right hand side remains unchanged on changing B, we can remove all elements in B which are not in A or C, since that would only further decrease the left hand side. Thus B is contained in $A\cup C$. The final step involves arguing that we can also remove all those elements in B which are only in A or C, as this operation will also only decrease the left hand side. Finally, we will have a B that is supposedly a counterexample to the metric distance claim, but it lies completely in $A \cap C$. This can also be shown to be not possible.

I hope I remember it right, I haven't worked this out recently.

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