arXiv subjects

Jeroen Zuiddam

Publications and source records attributed to Jeroen Zuiddam.

At least 19 recordsLinked to original sources

arXiv2026

Concise tensors with maximal symmetries

Conner, Gesmundo, Landsberg and Ventura (2019) determined the largest stabilizer dimension of concise $n\times n \times n$ tensors that are binding, and they determined the corresponding maximizing tensors to be the null algebra tensors. They left as an open problem to extend this to all concise $n \times n \times n$ tensors (i.e. dropping binding). We solve this problem: We prove that the largest stabilizer dimension of concise $n\times n\times n$ tensors is $n^2 + 1$ and the maximizers are the null algebra tensors (as in the binding case) and the skew symmetric tensor $e_1 \wedge e_2 \wedge e_3$. As part of our approach we obtain upper bounds on the stabilizer dimension of matrix tuples under left-right action (generalized Kronecker quiver representations), which we think are of independent interest.

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arXiv2026

Lean-verified lower bounds for the Shannon capacity of odd cycles

We give new lower bounds for the Shannon capacities of small odd cycles: $Θ(C_7)\geq3.258805369885\ldots$, $Θ(C_{11})\geq5.294502522149\ldots$, $Θ(C_{13})\geq6.302455083464\ldots$, $Θ(C_{15})\geq7.301600534487\ldots$, $Θ(C_{19})\geq9.357192705918\ldots$, $Θ(C_{21})\geq10.342455853338\ldots$, and $Θ(C_{23})\geq11.328224257774\ldots$. The bounds are obtained by an iterative procedure due to Gao (2026) which is based on a method by Itty, Rosin, Carstensen and Reichman (2026). The bounds are fully formalised in Lean.

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arXiv2026

The asymptotic spectrum distance, graph limits, and the Shannon capacity

Determining the Shannon capacity of graphs is a long-standing open problem in information theory, graph theory and combinatorial optimization. Over decades, a wide range of upper and lower bound methods have been developed to analyze this problem. However, despite tremendous effort, even small instances of the problem have remained open. In recent years, a new dual characterization of the Shannon capacity of graphs, asymptotic spectrum duality, has unified and extended known upper bound methods and structural theorems. In this paper, building on asymptotic spectrum duality, we develop a new theory of graph distance, that we call asymptotic spectrum distance, and corresponding limits (reminiscent of, but different from, the celebrated theory of cut-norm, graphons and flag algebras). We propose a graph limit approach to the Shannon capacity problem: to determine the Shannon capacity of a graph, construct a sequence of easier to analyse graphs converging to it. (1) We give a very general construction of non-trivial converging sequences of graphs (in a family of circulant graphs). (2) We construct Cauchy sequences of finite graphs that do not converge to any finite graph, but do converge to an infinite graph. We establish strong connections between convergence questions of finite graphs and the asymptotic properties of Borsuk-like infinite graphs on the circle. (3) We observe that all best-known lower bound constructions for Shannon capacity of small odd cycles can be obtained from a "finite" version of the graph limit approach. We develop computational and theoretical aspects of this approach and use these to obtain a new Shannon capacity lower bound for the fifteen-cycle. The theory of asymptotic spectrum distance applies not only to Shannon capacity of graphs; indeed, we will develop it for a general class of mathematical objects and their asymptotic properties.

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arXiv2026

Asymptotic tensor rank is characterized by polynomials

Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the $2\times 2$ matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other hand, Strassen's asymptotic rank conjecture makes the bold claim that asymptotic tensor rank equals the largest dimension of the tensor and is thus as easy to compute as matrix rank. Despite tremendous interest, much is still unknown about the structural and computational properties of asymptotic rank; for instance whether it is computable. We prove that asymptotic tensor rank is "computable from above", that is, for any real number $r$ there is an (efficient) algorithm that determines, given a tensor $T$, if the asymptotic tensor rank of $T$ is at most $r$. The algorithm has a simple structure; it consists of evaluating a finite list of polynomials on the tensor. Indeed, we prove that the sublevel sets of asymptotic rank are Zariski-closed (just like matrix rank). While we do not exhibit these polynomials explicitly, their mere existence has strong implications on the structure of asymptotic rank. As one such implication, we find that the values that asymptotic tensor rank takes, on all tensors, is a well-ordered set. In other words, any non-increasing sequence of asymptotic ranks stabilizes ("discreteness from above"). In particular, for the matrix multiplication exponent (which is an asymptotic rank) there is no sequence of exponents of bilinear maps that approximates it arbitrarily closely from above without being eventually constant. In other words, any such upper bound on the matrix multiplication exponent that is close enough, will "snap" to it. Previously such discreteness results were only known for finite fields or for other tensor parameters (e.g., asymptotic slice rank). We obtain them for infinite fields like the complex numbers.

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arXiv2026

Border subrank of higher order tensors and algebras

We determine the border subrank of higher order structure tensors of several families of algebras, and in particular obtain the following results. (1) We determine tight bounds on the border subrank of $k$-fold matrix multiplication and $k$-fold upper triangular matrix multiplication for all $k$. (2) We determine the border subrank of the higher order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric. (3) We determine the border subrank of the higher order structure tensors of the Lie algebra $\mathfrak{sl}_2$ for all orders. (4) We prove that degeneration of structure tensors of algebras propagates from higher to lower order. Along the way, we investigate which upper bound methods (geometric rank, $G$-stable rank, socle degree) are effective in which settings, and how they relate. Our work extends the results of Strassen (J.~Reine Angew.~Math., 1987, 1991), who determined the asymptotic subrank of these algebras for tensors of order three, in two directions: we determine the border subrank itself rather than its asymptotic version, and we consider higher order structure tensors.

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arXiv2025

Universality of asymptotic graph homomorphism

The Shannon capacity of graphs, introduced by Shannon in 1956 to model zero-error communication, asks for determining the rate of growth of independent sets in strong powers of graphs. Much is still unknown about this parameter, for instance whether it is computable. Recent work has established a dual characterization of the Shannon capacity in terms of the asymptotic spectrum of graphs. A core step in this duality theory is to shift focus from Shannon capacity itself to studying the asymptotic relations between graphs, that is, the asymptotic cohomomorphisms. Towards understanding the structure of Shannon capacity, we study the "combinatorial complexity" of asymptotic cohomomorphism. As our main result, we prove that the asymptotic cohomomorphism order is universal for all countable preorders. That is, we prove that any countable preorder can be order-embedded into the asymptotic cohomomorphism order (i.e. appears as a suborder). Previously this was only known for (non-asymptotic) cohomomorphism. Our proof is based on techniques from asymptotic spectrum duality and convex structure of the asymptotic spectrum of graphs. Our approach in fact leads to a new proof of the universality of (non-asymptotic) cohomomorphism.

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arXiv2025

Barriers for rectangular matrix multiplication

We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give algorithms with complexity $n^{p + 1}$ for $n \times n$ by $n \times n^p$ matrix multiplication. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously known barriers, both in the numerical sense, as well as in its generality. In particular, we prove that any lower bound on the dual exponent of matrix multiplication $α$ via the big Coppersmith-Winograd tensors cannot exceed 0.6218.

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arXiv2025

Computing moment polytopes -- with a focus on tensors, entanglement and matrix multiplication

Tensors are fundamental in mathematics, computer science, and physics. Their study through algebraic geometry and representation theory has proved very fruitful in the context of algebraic complexity theory and quantum information. In particular, moment polytopes have been understood to play a key role. In quantum information, moment polytopes (also known as entanglement polytopes) provide a framework for the single-particle quantum marginal problem and offer a geometric characterization of entanglement. In algebraic complexity, they underpin quantum functionals that capture asymptotic tensor relations. More recently, moment polytopes have also become foundational to the emerging field of scaling algorithms in computer science and optimization. Despite their fundamental role and interest from many angles, much is still unknown about these polytopes, and in particular for tensors beyond $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ and $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ only sporadically have they been computed. We give a new algorithm for computing moment polytopes of tensors (and in fact moment polytopes for the general class of reductive algebraic groups) based on a mathematical description by Franz (J. Lie Theory 2002). This algorithm enables us to compute moment polytopes of tensors of dimension an order of magnitude larger than previous methods, allowing us to compute with certainty, for the first time, all moment polytopes of tensors in $\mathbb{C}^3\otimes\mathbb{C}^3\otimes\mathbb{C}^3$, and with high probability those in $\mathbb{C}^4\otimes\mathbb{C}^4\otimes\mathbb{C}^4$ (which includes the $2\times 2$ matrix multiplication tensor). We discuss how these explicit moment polytopes have led to several new theoretical directions and results.

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arXiv2025

Discreteness of asymptotic tensor ranks

Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters. We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field (and in fact any finite set of coefficients in any field), the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points. Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any concise three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank. Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor, the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank.

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arXiv2025

A group-theoretic approach to Shannon capacity of graphs and a limit theorem from lattice packings

We develop a group-theoretic approach to the Shannon capacity problem. Using this approach we extend and recover, in a structured and unified manner, various families of previously known lower bounds on the Shannon capacity. Bohman (2003) proved that, in the limit $p\to\infty$, the Shannon capacity of cycle graphs $Θ(C_p)$ converges to the fractional clique covering number, that is, $\lim_{p \to \infty} p/2 - Θ(C_p) = 0$. We strengthen this result by proving that the same is true for all fraction graphs: $\lim_{p/q \to \infty} p/q - Θ(E_{p/q}) = 0$. Here the fraction graph $E_{p/q}$ is the graph with vertex set $\mathbb{Z}/p\mathbb{Z}$ in which two distinct vertices are adjacent if and only if their distance mod $p$ is strictly less than $q$. We obtain the limit via the group-theoretic approach. In particular, the independent sets we construct in powers of fraction graphs are subgroups (and, in fact, lattices). Our approach circumvents known barriers for structured ("linear") constructions of independent sets of Calderbank-Frankl-Graham-Li-Shepp (1993) and Guruswami-Riazanov (2021).

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arXiv2025

The next gap in the subrank of 3-tensors

Recent works of Costa-Dalai, Christandl-Gesmundo-Zuiddam, Blatter-Draisma-Rupniewski, and Briët-Christandl-Leigh-Shpilka-Zuiddam have investigated notions of discreteness and gaps in the possible values that asymptotic tensor ranks can take. In particular, it was shown that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, or at least 2 (over any field), and that the set of possible values of these parameters is discrete (in several regimes). We determine exactly the next gap, showing that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, equal to 2, or at least 2.68.

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arXiv2025

The moment polytope of matrix multiplication is not maximal

Moment polytopes of tensors, the study of which is deeply rooted in invariant theory, representation theory and symplectic geometry, have found relevance in numerous places, from quantum information (entanglement polytopes) and algebraic complexity theory (GCT program and the complexity of matrix multiplication) to optimization (scaling algorithms). Towards an open problem in algebraic complexity theory, we prove separations between the moment polytopes of matrix multiplication tensors and unit tensors. As a consequence, we find that matrix multiplication moment polytopes are not maximal, i.e. are strictly contained in the corresponding Kronecker polytope. As another consequence, we obtain a no-go result for a natural operational characterization of moment polytope inclusion in terms of asymptotic restriction. We generalize the separation and non-maximality to moment polytopes of iterated matrix multiplication tensors. Our result implies that tensor networks where multipartite entanglement structures beyond two-party entanglement are allowed can go beyond projected entangled-pair states (PEPS) in terms of expressivity. Our proof characterizes membership of uniform points in moment polytopes of tensors, and establishes a connection to polynomial multiplication tensors via the minrank of matrix subspaces. As a result of independent interest, we extend these techniques to obtain a new proof of the optimal border subrank bound for matrix multiplication.

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arXiv2025

Explicit non-free tensors

Free tensors are tensors which, after a change of bases, have free support: any two distinct elements of its support differ in at least two coordinates. They play a distinguished role in the theory of bilinear complexity, in particular in Strassen's duality theory for asymptotic rank. Within the context of quantum information theory, where tensors are interpreted as multiparticle quantum states, freeness corresponds to a type of multiparticle Schmidt decomposition. In particular, if a state is free in a given basis, the reduced density matrices are diagonal. Although generic tensors in $\mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^n$ are non-free for $n \geq 4$ by parameter counting, no explicit non-free tensors were known until now. We solve this hay in a haystack problem by constructing explicit tensors that are non-free for every $n \geq 3$. In particular, this establishes that non-free tensors exist in $\mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^n$, where they are not generic. To establish non-freeness, we use results from geometric invariant theory and the theory of moment polytopes. In particular, we show that if a tensor $T$ is free, then there is a tensor $S$ in the GL-orbit closure of $T$, whose support is free and whose moment map image is the minimum-norm point of the moment polytope of $T$. This implies a reduction for checking non-freeness from arbitrary basis changes of $T$ to unitary basis changes of $S$. The unitary equivariance of the moment map can then be combined with the fact that tensors with free support have diagonal moment map image, in order to further restrict the set of relevant basis changes.

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arXiv2023

A Gap in the Subrank of Tensors

The subrank of tensors is a measure of how much a tensor can be ''diagonalized''. This parameter was introduced by Strassen to study fast matrix multiplication algorithms in algebraic complexity theory and is closely related to many central tensor parameters (e.g. slice rank, partition rank, analytic rank, geometric rank, G-stable rank) and problems in combinatorics, computer science and quantum information theory. Strassen (J. Reine Angew. Math., 1988) proved that there is a gap in the subrank when taking large powers under the tensor product: either the subrank of all powers is at most one, or it grows as a power of a constant strictly larger than one. In this paper, we precisely determine this constant for tensors of any order. Additionally, for tensors of order three, we prove that there is a second gap in the possible rates of growth. Our results strengthen the recent work of Costa and Dalai (J. Comb. Theory, Ser. A, 2021), who proved a similar gap for the slice rank. Our theorem on the subrank has wider applications by implying such gaps not only for the slice rank, but for any ``normalized monotone''. In order to prove the main result, we characterize when a tensor has a very structured tensor (the W-tensor) in its orbit closure. Our methods include degenerations in Grassmanians, which may be of independent interest.

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arXiv2023

Geometric rank of tensors and subrank of matrix multiplication

Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs. We prove that the geometric rank is smaller than the slice rank of Tao, and relate geometric rank to the analytic rank of Gowers and Wolf in an asymptotic fashion. As a first application, we use geometric rank to prove a tight upper bound on the (border) subrank of the matrix multiplication tensors, matching Strassen's well-known lower bound from 1987.

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arXiv2023

Weighted slice rank and a minimax correspondence to Strassen's spectra

Structural and computational understanding of tensors is the driving force behind faster matrix multiplication algorithms, the unraveling of quantum entanglement, and the breakthrough on the cap set problem. Strassen's asymptotic spectra program (FOCS 1986) characterizes optimal matrix multiplication algorithms through monotone functionals. Our work advances and makes novel connections among two recent developments in the study of tensors, namely - the slice rank of tensors, a notion of rank for tensors that emerged from the resolution of the cap set problem (Ann. of Math. 2017), - and the quantum functionals of tensors (STOC 2018), monotone functionals defined as optimizations over moment polytopes. More precisely, we introduce an extension of slice rank that we call weighted slice rank and we develop a minimax correspondence between the asymptotic weighted slice rank and the quantum functionals. Weighted slice rank encapsulates different notions of bipartiteness of quantum entanglement. The correspondence allows us to give a rank-type characterization of the quantum functionals. Moreover, whereas the original definition of the quantum functionals only works over the complex numbers, this new characterization can be extended to all fields. Thereby, in addition to gaining deeper understanding of Strassen's theory for the complex numbers, we obtain a proposal for quantum functionals over other fields. The finite field case is crucial for combinatorial and algorithmic problems where the field can be optimized over.

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arXiv2022

Tensor rank is not multiplicative under the tensor product

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specifically, if a tensor t has border rank strictly smaller than its rank, then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power. The "tensor Kronecker product" from algebraic complexity theory is related to our tensor product but different, namely it multiplies two k-tensors to get a k-tensor. Nonmultiplicativity of the tensor Kronecker product has been known since the work of Strassen. It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, lower bounds on border rank obtained from generalised flattenings (including Young flattenings) multiply under the tensor product.

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arXiv2022

Subrank and Optimal Reduction of Scalar Multiplications to Generic Tensors

Since the seminal works of Strassen and Valiant it has been a central theme in algebraic complexity theory to understand the relative complexity of algebraic problems, that is, to understand which algebraic problems (be it bilinear maps like matrix multiplication in Strassen's work, or the determinant and permanent polynomials in Valiant's) can be reduced to each other (under the appropriate notion of reduction). In this paper we determine precisely how many independent scalar multiplications can be reduced to a given bilinear map (this number is called the subrank, and extends the concept of matrix diagonalization to tensors), for essentially all (i.e. generic) bilinear maps. Namely, we prove for a generic bilinear map $T : V \times V \to V$ where $\dim(V) = n$ that $θ(\sqrt{n})$ independent scalar multiplications can be reduced to $T$. Our result significantly improves on the previous upper bound from the work of Strassen (1991) and Bürgisser (1990) which was $n^{2/3 + o(1)}$. Our full result is much more general and applies not only to bilinear maps and 3-tensors but also to $k$-tensors, for which we find that the generic subrank is $θ(n^{1/(k-1)})$. Moreover, as an application we prove that the subrank is not additive under the direct sum. The subrank plays a central role in several areas of complexity theory (matrix multiplication algorithms, barrier results) and combinatorics (e.g., the cap set problem and sunflower problem). As a consequence of our result we obtain several large separations between the subrank and tensor methods that have received much interest recently, notably the slice rank (Tao, 2016), analytic rank (Gowers--Wolf, 2011; Lovett, 2018; Bhrushundi--Harsha--Hatami--Kopparty--Kumar, 2020), geometric rank (Kopparty--Moshkovitz--Zuiddam, 2020), and G-stable rank (Derksen, 2020).

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