Abstract
The “relative earth mover distance” is a technical term introduced by Valiant and Valiant (43rd STOC, 2011), and extensively used in their work. They claimed that, for every two distributions, the relative earth mover distance upper-bounds the variation distance up to relabeling, but this claim was not used in their work. The claim appears as a special case of a result proved by Valiant and Valiant in a later work (48th STOC, 2016), but we found their proof too terse. The proof presented here is merely an elaboration of (this special case of) their proof.
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Notes
- 1.
- 2.
Specifically, for two distributions presented by their probability functions \(p,q:D\!\rightarrow \![0,1]\), their variation distance equals \(0.5\cdot \sum _{i\in D}|p(i)-q(i)|\), which in turn equals \(\min _{S\subseteq D}\{p(S)-q(S)\}\), where \(p(S)=\sum _{i\in S}p(i)\). The set S may be viewed as the set of samples on which an observer (as discussed next) outputs the verdict 1.
- 3.
Here and in the sequel, the logarithm is to base 2. The proof of Theorem 3.1 as presented in Sect. 3 remains valid for any base \(b\in (1,e]\); our only reference to this base is that it (i.e., b) should satisfy \(\log _bz>1-(1/z)\) for every \(z>1\). It seems that Valiant and Valiant do mean to take \(b=2\) (although other parts of their text suggest \(b=e\)). Indeed, both \(b=2\) and \(b=e\) seems natural choices.
- 4.
As stated in Footnote 3, we assume that the logarithm is to base \(b\in (1,e]\). Indeed, here we use \(\log _b z+(1/z)>1\) for all \(z>1\), and this is the only place in the proof in which the choice of b matters.
- 5.
That is, letting \(\pi _p\) and \(\pi _q\) be permutations over [n] such that \(p(\pi _p(j))\le p(\pi _p(j+1))\) and \(q(\pi _q(j))\le q(\pi _q(j+1))\) for every \(j\in [n-1]\), in the \(i^\mathrm{th}\) iteration we transport one unit from location \(p(\pi _p(i))\) of \(h_p\) to location \(q(\pi _q(i))\) of \(h_q\).
- 6.
To see that (1) holds, note that the cost of \(\ell '\) equals the cost of \(\ell \) plus \(c\cdot |x^*-y^*|-c\cdot |x'-y^*|-c\cdot |x^*-y'|+c\cdot |x'-y'|\). Hence, we need to verify that the added value is not positive; equivalently, that \(|x^*-y^*|-|x^*-y'|\le |x'-y^*|-|x'-y'|\). Consider the following cases:
-
1.
The diagonal line \(y=x\) does not cross the rectangle spanned by \((x^*,y^*)\) (i.e., either \(y'\le x^*\) or \(y^*\ge x'\)). If \(y'\le x^*\), then \(|y^*-x^*|-|y'-x^*|=y'-y^*=|y^*-x'|-|y'-x'|\), and otherwise \(|y^*-x^*|-|y'-x^*|=-(y'-y^*)=|y^*-x'|-|y'-x'|\).
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2.
The diagonal line \(y=x\) separates one corner-point of the rectangle from the other three corner-points (e.g., \(y'>x^*\) but \(y<x\) for \((x,y)\in \{(x^*,y^*),(x',y^*),(x',y')\}\)). If \(y'>x^*\), then \(|y^*-x^*|-|y'-x^*| < y'-y^* = |y^*-x'|-|y'-x'|\), and similarly for the case that \((x',y^*)\) is separated.
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3.
The diagonal line \(y=x\) crosses both horizontal lines of the rectangle (i.e., \(y^*,y'\in [x^*,x']\)). In this case, \(|y^*-x^*|-|y'-x^*|=-(y'-y^*)\) and \(|y^*-x'|-|y'-x'|=y'-y^*\).
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4.
The diagonal line \(y=x\) crosses both vertical lines of the rectangle (i.e., \(x^*,x'\in [y^*,y']\)). In this case \(|y^*-x^*|-|y'-x^*|<|y^*-x'|-|y'-x'|\), since \(|y^*-x^*|<|y^*-x'|\) and \(|y'-x^*|>|y'-x'|\).
To see that (2) holds, recall that \(\ell '(x,y)=\ell (x,y)=m(x,y)\) for every \((x,y)<(x^*,y^*)\).
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1.
References
Goldreich, O.: Introduction to Property Testing. Cambridge University Press, Cambridge (2017)
Goldreich, O., Ron, D.: On sample-based testers. In: 6th Innovations in Theoretical Computer Science, pp. 337–345 (2015)
Valiant, G., Valiant, P.: Estimating the unseen: an \(n/\log (n)\)-sample estimator for entropy and support size, shown optimal via new CLTs. In: 43rd ACM Symposium on the Theory of Computing, pp. 685–694 (2011). See ECCC TR10-180 for the algorithm, and TR10-179 for the lower bound
Valiant, G., Valiant, P.: Instance optimal learning. CoRR abs/1504.05321 (2015)
Valiant, G., Valiant, P.: Instance optimal learning of discrete distributions. In: 48th ACM Symposium on the Theory of Computing, pp. 142–155 (2016)
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Goldreich, O., Ron, D. (2020). On the Relation Between the Relative Earth Mover Distance and the Variation Distance (an Exposition). In: Goldreich, O. (eds) Computational Complexity and Property Testing. Lecture Notes in Computer Science(), vol 12050. Springer, Cham. https://doi.org/10.1007/978-3-030-43662-9_9
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