Negative type diversities, a multi-dimensional analogue of negative type metrics

09/18/2018
by   Pei Wu, et al.
University of Otago
Simon Fraser University
0

Diversities are a generalization of metric spaces in which a non-negative value is assigned to all finite subsets of a set, rather than just to pairs of points. Here we provide an analogue of the theory of negative type metrics for diversities. We introduce negative type diversities, and show that, as in the metric space case, they are a generalization of L_1-embeddable diversities. We provide a number of characterizations of negative type diversities, including a geometric characterisation. Much of the recent interest in negative type metrics stems from the connections between metric embeddings and approximation algorithms. We extend some of this work into the diversity setting, showing that lower bounds for embeddings of negative type metrics into L_1 can be extended to diversities by using recently established extremal results on hypergraphs.

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1 Introduction

A finite metric has negative type

if for all vectors

indexed by the elements of with zero sum we have

(1)

Negative type metrics were introduced by Schoenberg [26, 27], who showed that a finite metric has negative type exactly when the square of that metric is embeddable in Euclidean space (see, e.g. [12]). The concept can be extended to metrics with -negative type, which satisfy

(2)

for all with zero sum (e.g. [25]).

Negative type metrics have received recent attention as a tool for combinatorial optimization based on metric embedding. Arora et al. 

[2] showed that any metric of negative type on an -point set can be embedded in with at most distortion, an upper bound that closely matches the provable lower bound of . Their results provide near-optimal approximation algorithms for Sparsest Cut and other key graph problems.

This application of negative type metrics to combinatorial optimization continues a large body of work tracing back to the influential paper of Linial et al. [21]. A finite metric can be embedded in with distortion if there exists an isometrically -embeddable such that

for all . Bourgain [4] showed that every finite metric on points can be embedded in with distortion . Linial et al. [21] showed how this provides an approximation algorithm for Sparsest Cut.

In recent work, results on metric embedding and -embeddable metrics have been generalized to diversities [8]. A diversity is like a metric that assigns values to finite subsets of points, rather than just pairs. More formally, given a set and a map from the finite subsets of to , a diversity satisfies

(D1) , and if and only if ;
(D2) If then

The first property corresponds to the metric axiom that for all , and . The second property corresponds to the triangle inequality. Indeed if is a diversity and for all then (D1) and (D2) imply that is a metric. Every diversity induces a metric in this fashion. The mathematics of diversities has been explored in [7, 8, 9, 10, 6, 13, 15, 23, 24]. Much of this work parallels developments in metric theory.

Many of the important results on embeddings and their applications have analogues for diversities. There is a natural diversity analogue of -embeddable metrics: A finite diversity is -embeddable if there is a map for some such that for all ,

Note that the induced metric of an -embeddable diversity is an -embeddable metric. A diversity is said to be -embeddable with distortion if there is an -embeddable diversity such that

The main result of [8] is that links between embedding of metrics and Sparsest Cut generalise to a link between embeddings of diversities and Hypergraph Sparsest Cut. A bound of on the distortion required to embed an -point diversity gives an approximation for Hypergraph Sparsest Cut. Unfortunately, we have been unable to prove a distortion bound better that , although much tighter bounds hold for specific instances [10].

Given that the tightest approximation bounds for Sparsest Cut are achieved by methods based on negative type metrics, an obvious question is whether equivalent results might be obtained for diversities. The first step is to determine what the appropriate definition of a negative type diversity might be. There are two main characterisations for negative type metrics: the negativity condition in (1) and the fact that the squares of negative type metrics are Euclidean. We do not, yet, have an sufficiently convincing analogue of Euclidean diversities, though (1) appears to generalize quite naturally.

Definition 1.

A finite diversity is of negative type if for all zero sum vectors with we have

In Section 2 we present multiple characterisations of negative type diversities. Our first main theorem (Theorem 2) gives an explicit characterisation of negative type diversities in terms of a finite set of linear inequalities, thereby demonstrating that the collection of negative type diversities on a set forms a polyhedral cone. This property does not hold for negative type metrics. We use the result to show that the space of negative type diversities spans the space of all diversities on that set. In contrast, the space of -embeddable diversities, which is contained in the space of negative type diversities, has dimension roughly half of that of diversities.

The induced metric of any -embeddable diversity is an -embeddable metric and, conversely, every -embeddable metric is the induced metric of some -embeddable diversity [8]. Turning to negative type diversities, we see that the induced metric of any negative type diversity is not just a negative type metric, it is also -embeddable (Theorem 5). This means that negative type metrics which are not -embeddable are not the induced metrics of any negative type diversity. We prove Theorem 5 by first establishing a characterisation of negative type diversities based on a metric on the power set.

The next two sections return to the problem of embedding negative type diversities. In Section 3 we derive a geometric representation of a negative type diversity. Define by

The expression for bears substantial resemblance to that for an -embeddable diversity. We prove in Theorem 7 that a finite diversity is negative type if and only if it can be embedded in for some .

With this result in hand, we look at the problem of embedding (finite) negative type diversities into . There is a lower bound of for the metric case, however this does not directly imply a bound for negative type diversities as the induced metric of any negative type diversity is already -embeddable. We follow a different strategy and use results on Cheegar constants for hypergraphs to prove a bound in the diversity case.

2 Characterising negative type diversities

In this section, we establish some basic characterisations and properties of negative type diversities on finite sets.

Lemma 1.

For a finite set and real-valued function defined on , let be the matrix with rows and columns indexed by and for all . Define the vector by

(3)

Then for all we have

(4)
Proof.

Applying Moebius inversion to (3) we obtain

for all (see [1]). Hence for we have

We now prove the first characterisation theorem for negative type diversities.

Theorem 2.

Let be a finite set and let be a real-valued function on such that whenever . For all define

(5)

Then is a negative type diversity if and only if for all . Furthermore,

(6)

for all .

Proof.

Suppose that is a negative type diversity. Fix and define by

Then, by Moebius inversion,

while

since . Also, because . Hence by the definition of negative type diversities and Lemma 1 we have

For the converse, suppose that for all . Suppose is any vector in with and . By Lemma 1 we have

Suppose . Define the vector by , and for all . Then

It follows that is monotonic.

Now consider arbitrary and . Then

This, together with monotonicity, implies the triangle inequality for diversities. Hence is a diversity with negative type. ∎

A direct consequence of Theorem 2 is that the space of negative type diversities on a finite set forms a polyhedral cone. This cone has dimension , the same as the dimension of the cone of diversities on .

We present several examples of negative type diversities.

Proposition 3.
  1. Every diversity on three points is negative type.

  2. There is a diversity on four points which is not negative type.

  3. Every finite -embeddable diversity is negative type, though there are negative type diversities which are not -embeddable.

  4. For finite , if is the diversity with whenever then is negative.

Proof.
  1. Given a diversity we have from (5) that

    By symmetry, and Theorem 2, is negative.

  2. Let be the diversity with for with . Then so is not negative.

  3. For any cut diversity on we have , if or , and otherwise. By Theorem 2, is negative, and since every -embeddable diversity is a non-negative combination of cut diversities, so is every -embeddable diversity. Any diversity on three points which does not satisfy is negative but not -embeddable.

  4. For any with we have

    so by Theorem 2, is negative.

Schoenberg’s theorem states that every negative type metric is isometric to the square of a Euclidean metric. We do not have a direct analogue of this result for diversities, however the metric result leads to an appealing property of negative type diversities.

In what follows let be the set of all non-empty subsets of .

Proposition 4.

Let be a finite diversity and let be the symmetric real-valued function defined on given by

The following are equivalent

  1. is a negative type diversity;

  2. is an -embeddable metric;

  3. is a metric of negative type;

  4. is isometric to the square of a Euclidean metric.

Proof.

(1) (2). From (6) in Theorem 2 we have

Here is the cut metric for the cut . So is a non-negative linear combination of split metrics, and is therefore an -embeddable metric.
(2) (3). Every -embeddable metric is a negative type metric.
(3) (1). For all such that and we have

where for all . Hence is of negative type.
(3) (4) Schoenberg’s theorem [26, 27]. ∎

If we look at restricted to singletons we obtain a surprising result.

Theorem 5.

The induced metric of a negative type diversity is -embeddable.

Proof.

Suppose is negative. When we restrict to singleton sets, we see

Hence the induced metric of is isometric to restricted to singletons and, by Proposition 4, is -embeddable. ∎

The relationship between -embeddable metrics and -embeddable diversities is straight-forward: the induced metric of any -embeddable diversity is an -embeddable metric and, conversely, every -embeddable metric is the induced metric of some -embeddable diversity.

The situation for negative type diversities is a bit more nuanced. The induced metric of any negative type diversity is an -embeddable metric, and hence a negative type metric. But metrics which are negative but not -embeddable are not the induced metrics of any negative type diversity.

Furthermore, as there are metrics which require distortion to embed into , there are diversities which will require distortion to embed into a negative type diversity. In contrast, any metric can be embedded in a negative type metric with only distortion.

3 A universal embedding for negative type diversities

We provide an example of a diversity is that is universal for negative type diversities, in the sense that every (finite) negative type diversity can be embedded into this diversity, and every finite subset of this diversity induces a negative type diversity. Define by

First we show that we can restrict our attention to embeddings where all vectors have zero sum.

Lemma 6.

There is an embedding of into such that for all .

Proof.

The map

satisfies the condition that for all . For all finite we have

Theorem 7.

A finite diversity is of negative type if and only if can be embedded in for some .

Proof.

First we show that if is any finite subset of then is of negative type. By Lemma 6 we can assume, without loss of generality, that for all .

For each , we have

By Theorem 2 we need to show that for . Fix , and suppose that that is ordered as such that

For define . Then

If there is such that , or if then
otherwise

Hence for all such that , and is of negative type.

For the converse, suppose and is of negative type. Let be given by (5), let and suppose that the dimensions of are indexed by nonempty subsets of . Let be the map with

for all and nonempty . Then
For all ,
and for all we have

where we have used expression (6) in Theorem 2. ∎

4 embedding

Much of the recent interest in negative type metrics relates to embedding into [22, 18, 16, 17, 3, 14, 19]. Every -point negative type metric can be embedded into with distortion at most , with a lower bound of [2]. These results lead to the current best approximation bound for sparsest cut and other problems.

Here we investigate the problem of embedding negative type diversities into . The aim is to investigate whether the approximation algorithms based on embeddings of negative type metrics can be extended to algorithms based on embedding negative type diversities. We have already shown that algorithms of [21] and others based on embedding metrics in have direct analogies for diversities [8].

Every -embeddable diversity is negative; we start by characterising which negative type diversities are -embeddable.

Proposition 8.

Let be a negative type diversity and let be given by (5). Then is -embeddable if and only if for all .

Proof.

Suppose . First, suppose . For any with we have too. Then

which is unless , in which case it is . Now suppose that . We write using as

which is unless , in which case it is . Similarly for , if and is otherwise. Summarizing, for cut diversities , , and all other . So, if is a cut diversity then for all . This will hold also for any diversity, since diversities are non-negative linear combinations of cut diversities.

For the converse, suppose that the diversity of negative type is such that for all . Let be the -embeddable diversity

and with given by

Then, when ,

So and have the same vector. As the map from to is invertible, and is -embeddable. ∎

There are negative type metrics on points which cannot be embedded in with distortion less than . However, as the induced metric of any negative type diversity is -embeddable the general bound for metrics does not imply a bound for diversities. We instead follow a different strategy to show that there are negative type diversities which still require at least distortion to embed the diversity into . Our bound is based on connections in [8] between embedding of diversities and sparsest cut problems in hypergraphs.

Given let and let be the set of all subsets of of cardinality . Let be the diversity with

for all non-empty .

Proposition 9.

is of negative type.

Proof.

A diversity isomorphism from to is given by mapping each set in to the corresponding vector of ones and zeros. The diversity is then of negative type by Theorem 7. ∎

Our lower bound for embedding into is based on a lower bound for hypergraph cuts. Suppose that is a non-empty subset of . Define

We derive a lower bound for . Without loss of generality we assume , swapping for if this is not the case. Hence

(7)
Lemma 10.
Proof.

Let be the graph with vertex set and edge set

The graph is a Johnson graph [5]. For any subset we define the vertex boundary by

Theorem 2 of [11] provides a lower bound on the size of . Let , then

(8)

Let . Then implies that but that there is such that . Let . As we have . Since we have and so . The only remaining possibility is that and hence

(9)

Since , we have that . Then from (7) we have and from (8) and (9) we have

Reversing the inequality we now obtain

Thus

(10)

Inequality (10) holds since there are elements of contained in each , and each of these could be contained in at most other sets . A bound for follows immediately:

The link between sparsest cuts of hypergraph and diversity embeddings was established by [8]. We make use of the same ideas.

Theorem 11.

Let be the minimum distortion required to embed into . Then

Proof.

Define vectors and , indexed by , by

Then

since for all .

Also

Let be any -embeddable diversity such that for all ,

We then have

which implies: