We study in this paper an asymptotic parameter of -partite -uniform hypergraphs: the asymptotic induced matching number. For , a -partite -uniform hypergraph, or -graph for short, is a tuple of finite sets together with a subset of their cartesian product:
Whenever possible we will leave the vertex sets implicit and refer to the -graph by its edge set . For any we use the notation . Let be a -graph. We say a subset of is induced if where for each we define the marginal set . We call a matching if any two distinct elements are distinct in all coordinates, that is, . The subrank111The term subrank originates from an analogous parameter in the theory of tensors, see Section 1.4.1. or induced matching number is defined as the size of the largest subset of that is an induced matching, that is,
For example, consider the 3-graph . Here is itself an induced matching, and so . Next, let . Now the subset is an induced matching and there is no larger induced matching in , and so .
We define the Kronecker product of two -graphs and as the -graph
and we naturally define the power . The asymptotic subrank or the asymptotic induced matching number of the -graph is defined as
This limit exists and equals the supremum by Fekete’s lemma (see, e.g., [PS98, No. 98]).
We study the following basic question:
Given what is the value of ?
A priori, for we have the upper bound and therefore holds that , since .
creftype 1.1 has been studied for several families of -graphs, in several different contexts: the cap set problem [EG17, Tao16, KSS16, Nor16, Peb16], approaches to fast matrix multiplication [Str91, BCC17a, BCC17b, Saw17], arithmetic removal lemmas [LS18, FLS18], property testing [FK14, HX17], quantum information theory [VC15, VC17], and the general study of asymptotic properties of tensors [TS16, CVZ18a, CVZ18c]. We finally mention the related result of Ruzsa and Szemerédi which says that the largest subset such that is a matching, has size when goes to infinity [RS78], see also [AS06, Equation 2].
We solve creftype 1.1 for a family of -graphs that are structured but nontrivial. For let be an integer partition of with nonzero parts, that is, and . We define the -graph
where the expression means that is a permutation of the -tuple
For example, the partition corresponds to the 2-graph
and the partition corresponds to the 4-graph
It was shown in [CVZ18a] that for every where is the Shannon entropy in base 2. As a natural continuation of that work we study for even . Since we have . Clearly, the 2-graph is itself a matching, and so . It was shown in [CVZ18a] that also . Our new result is the following extension:
Let be even and large enough. Then .
In other words, we prove that for every large enough even there is an induced matching of size when goes to infinity.
equals the Shannon entropy of the probability distribution obtained by normalising the partition. We will discuss further motivation and background in Section 1.4.
We prove Theorem 1.2 by applying the higher-order Coppersmith–Winograd (CW) method from [CVZ18a] to the -graph . This method is an extension of the work of Coppersmith and Winograd [CW87] and Strassen [Str91] from the case to the case . It provides a construction of large induced matchings in -graphs via the probabilistic method, and we prove Theorem 1.2 by analysing the size of these induced matchings.
Theorem 1.3 (Higher-order CW method [CVZ18a]).
Let be a nonempty -graph for which there exist injective maps such that for all the equality
holds. For any let be the rank over of the matrix with rows
where . Then
where the parameters , and are taken over the following domains:
is the set of probability distributions on
is the set of subsets of that are not a subset of and moreover satisfy
is the set of probability distributions on with marginal distributions equal to respectively.
Here for we denote by the marginal probability distributions of on the components respectively, and denotes Shannon entropy.
Let be any integer partition of with nonzero parts. We can apply Theorem 1.3 to the -graph as follows. For every the equality
holds, since the element occurs times in . Let be identity maps and let . Then, because of (2), . (Note that with this choice of maps we have that equals for every .) Therefore Theorem 1.3 can be applied to obtain a lower bound on for any partition . The difficulty now lies in evaluating the right-hand side of (1).
Let us return to the case . To prove Theorem 1.2 via Theorem 1.3 we will show for every large enough even and that the right-hand side of (1) is at least 2, using the aforementioned choice of injective maps . In Section 2 we prove that this follows from the following statement, which may be of interest on its own.
For any large enough even and subspace the inequality
holds. Here denotes the Hamming weight of .
In Section 3 we prove Theorem 1.4 for low-dimensional by carefully splitting the left-hand side of (3) into two parts and upper bounding these parts. In Section 4 we prove Theorem 1.4 for high-dimensional using Fourier analysis, Krawchouk polynomials and the Kahn–Kalai–Linial (KKL) inequality [KKL88]. We thus prove Theorem 1.4 and hence Theorem 1.2. While in our current proof the tools for the low- and high-dimensional cases are used complementarily, it may be possible that the full Theorem 1.2 can be proven by cleverly using only the low-dimensional tools or only the high-dimensional tools.
1.4. Motivation and background
Our original motivation to study the asymptotic induced matching number of -graphs comes from a connection to the study of asymptotic properties of tensors. In fact, the interplay in this connection goes both directions. The purpose of this section is to discuss the asymptotic study of tensors and the connection with the asymptotic induced matching number. Reading this section is not required to understand the rest of the paper.
1.4.1. Asymptotic rank and asymptotic subrank of tensors
The asymptotic study of tensors is a field of its own that started with the work of Strassen [Str87, Str88, Str91] in the context of fast matrix multiplication. We begin by introducing two fundamental asymptotic tensor parameters: asymptotic rank and asymptotic subrank.
Let be a field. Let and be -tensors. We write if there are linear maps for such that . For let be the standard basis of . For define the -tensor
The rank of the -tensor is defined as . The subrank of the -tensor is defined as
One can think of tensor rank as a measure of the complexity of a tensor, namely the “cost” of the tensor in terms of the diagonal tensors . It has been studied in several contexts, see, e.g., [BCS97, Lan12]. In this language, the subrank is the “value” of the tensor in terms of and as such is the natural companion to tensor rank. It has its own applications, which we will elaborate on after having discussed the asymptotic viewpoint.
Writing and in the standard basis as , , the tensor Kronecker product is the -tensor defined by
In other words, the -tensor is the image of the -tensor under the natural regrouping map . The asymptotic rank of is defined as and the asymptotic subrank of is defined as . These limits exist and equal the infimum and the supremum , respectively. This follows from Fekete’s lemma and the fact that and .
Tensor rank is known to be hard to compute [Hås90] (the natural tensor rank decision problem is NP-hard). Not much is known about the complexity of computing subrank, asymptotic subrank and asymptotic rank. It is a long-standing open problem in algebraic complexity theory to compute the asymptotic rank of the matrix multiplication tensor. The asymptotic rank of the matrix multiplication tensor corresponds directly to the asymptotic algebraic complexity of matrix multiplication. The asymptotic subrank of 3-tensors also plays a central role in the context of matrix multiplication, for example in recent work on barriers for upper bound methods on the asymptotic rank of the matrix multiplication tensor [CVZ18b, Alm18]. As another example, in combinatorics, the resolution of the cap set problem [EG17, Tao16] can be phrased in terms of the asymptotic subrank of a well-chosen 3-tensor, cf. [CVZ18a], via the general connection to the asymptotic induced matching number that we will review now.
It is readily verified that the subrank of the -graph is at most the subrank of the -tensor , that is, . The reader may also verify directly that . Therefore, the asymptotic subrank of the support of is at most the asymptotic subrank of the -tensor , that is,
We can read (5) in two ways. On the one hand, given any -tensor we may find lower bounds on by finding lower bounds on . On the other hand, given any -graph the asymptotic subrank is upper bounded by for any tensor (over any field ) with support equal to , that is,
We do not know whether the inequality in (6) can be strict. We will discuss these two directions in the following two sections.
1.4.2. Upper bounds on asymptotic subrank of -tensors
Let us focus on the task of finding upper bounds on the asymptotic subrank of -tensors. One natural strategy is to construct maps that are sub-multiplicative under the tensor Kronecker product , normalised on to , and monotone under , that is, for any -tensors and and for any :
The reader verifies directly that for any such map the inequality holds.
Strassen in [Str91], motivated by the study of the algebraic complexity of matrix multiplication, introduced an infinite family of maps
parametrised by probability vectors, . The maps are called the upper support functionals. We will not define them here. Strassen proved that each map satisfies conditions (7), (8) and (9). Thus
Tao, motivated by the study of the cap set problem, proved in [Tao16] that subrank is upper bounded by a parameter called slice rank, that is, . We do not define slice rank here. While slice rank is easily seen to be normalised on and monotone under , slice rank is not sub-multiplicative (see, e.g., [CVZ18c]). However, it still holds that
No examples are known for which this inequality is strict. It is known that for so-called oblique tensors holds [CVZ18c].
1.4.3. Lower bounds on asymptotic subrank of -graphs
We now consider the task of finding lower bounds on the asymptotic subrank of -graphs. For the CW method introduced by Coppersmith and Winograd [CW87] and extended by Strassen [Str91] gives the following. Let be a 3-graph for which there exist injective maps such that . Then
where is the set of probability distributions on . The inequality
follows from using (5) and using the support functionals as upper bound on the asymptotic subrank of tensors. Thus, the CW method is optimal whenever it can be applied.
1.4.4. Type tensors
As an investigation of the power of the higher-order CW method (Theorem 1.3) and of the power of the support functionals (Section 1.4.2) we study the asymptotic subrank of the following family of tensors and their support. While we do not have any immediate “application” for these tensors, we feel that they provide enough structure to make progress while still showing interesting behaviour.
Let be an integer partition of with nonzero parts. Recall the definition of the -graph from Section 1.1. We define the tensor as the -tensor with support and all nonzero coefficients equal to 1, that is,
It was shown in [CVZ18a] that
We conjecture that (12) holds for all even . We numerically verified this up to . More generally we conjecture that holds for all partitions , where denotes the Shannon entropy and denotes the probability vector .
In quantum information theory, the tensors , when normalized, correspond to so-called Dicke states (see [Dic54, SGDM03, VC15], and, e.g., [BE19]). Namely, in quantum information language, Dicke states are -partite pure quantum states given by
where the sum is over all permutations of the parties. Roughly speaking, our result, Theorem 1.2, amounts to an asymptotically optimal -party stochastic local operations and classical communication (slocc) protocol for the problem of distilling GHZ-type entanglement from a subfamily of the Dicke states. More precisely, letting be the -party GHZ state, Theorem 1.2 says that for large enough the maximal rate such that copies of can be transformed via slocc to copies of equals 1 when goes to infinity, that is,
and this rate is optimal.
2. Reduction to counting
We will use the higher-order CW method Theorem 1.3 to show that Theorem 1.4 implies Theorem 1.2. Let . Let be the identity map and let . With this definition of we have for all satisfied the condition from Theorem 1.3. As in the statement of Theorem 1.3, for let be the dimension of the -vector space
be the uniform distribution on. Then Theorem 1.3 gives
For any we have that is at most the Shannon entropy of the uniform distribution on . We thus obtain
It remains to upper bound the maximisation over in (13). We define the set
For let be the dimension of the -vector space
By assumption Theorem 1.4 is true. This means
For any there is a subset with and . Namely, one constructs as follows. Without loss of generality . For every , if , then add to , and if , then add the negated tuple to . Therefore, (14) implies
3. Case: low dimension
For any even and subspace such that , the inequality