Strassen’s Support Functionals Proven Universal Spectral Points, Resolving 1991 Problem

Researchers have long sought to understand the fundamental limits of tensor complexity, a problem with implications for fields ranging from computer science to information theory. Keiya Sakabe, Mahmut Levent Doğan, and Michael Walter, from Ruhr University Bochum and LMU Munich, et al., now resolve a key question posed in 1991: do Strassen’s support functionals accurately characterise universal spectral points within the asymptotic spectrum of tensors? Their work demonstrates that these support functionals do indeed coincide with those defined via entropy optimisation, establishing a crucial link between Strassen’s approach and quantum information theory. This result not only answers a longstanding open problem but also provides a general minimax formula with broader applications for analysing other important tensor parameters, such as the asymptotic slice rank.

This breakthrough establishes that these support functionals coincide with quantum functionals, which are spectral points defined through entropy optimization on entanglement polytopes. The team achieved this result by developing a general minimax formula applicable to convex optimization on both entanglement and other moment polytopes, extending its relevance to various tensor parameters including the asymptotic slice rank.

The study’s core innovation lies in demonstrating the equivalence between Strassen’s support functionals and quantum functionals, previously suspected but unproven. Researchers leveraged a recent Fenchel-type duality theorem on Hadamard manifolds, established by Hirai, to construct a rigorous proof. Specifically, the work defines support functionals using the support polytope of a tensor, while quantum functionals are defined via the entanglement polytope. The research unveils that these seemingly different constructions yield identical results, confirming the universality of Strassen’s functionals across all d-tensors.

This finding is significant because it provides a complete characterization of asymptotic restriction between tensors, allowing for precise bounds on their algebraic complexity. Experiments show that the established minimax formula not only confirms the equivalence of the two functionals but also offers a powerful tool for analysing convex optimization problems on polytopes. The team’s approach involves defining the support polytope based on the tensor’s support and then minimizing a specific function over a group of linear maps. This minimization process, combined with the properties of the entanglement polytope, ultimately leads to the proof of the central theorem.

The implications of this work extend beyond theoretical mathematics, offering potential advancements in areas such as quantum entanglement transformations, where tensors represent quantum states and restriction corresponds to these transformations. Furthermore, the research establishes a strong connection between Strassen’s theory, originally developed in the context of matrix multiplication, and diverse fields like additive combinatorics. The ability to accurately characterize the complexity of tensors has far-reaching consequences for computational efficiency and algorithm design. This breakthrough opens avenues for exploring new spectral points and refining our understanding of multilinear problems. By proving the universality of Strassen’s support functionals, the study provides a fundamental building block for future research in tensor analysis and its applications across multiple scientific disciplines.

Minimax Formula Equates Optimisation on Polytopes to duality

Scientists confirmed a long-standing conjecture regarding Strassen’s asymptotic spectrum by demonstrating that support functionals coincide with universal spectral points defined through entropy optimization on polytopes. The research team engineered a novel minimax formula for convex optimization on polytopes and moment polytopes, enabling a streamlined proof bypassing previously required invariant-theoretic machinery. This formula, detailed as Theorem 1.3, establishes that minimizing a convex, symmetric, lower semi-continuous function over a moment polytope yields the same result as minimizing it over support polytopes, subsequently maximizing across all general linear transformations. Specifically, the team proved that for any such function F, min p∈∆(t) F(p) = max g∈GL min p∈Ω(g·t) F(p), where ∆(t) represents the moment polytope and Ω(g·t) the support polytope.

Experiments employed a recent Fenchel-type duality theorem on Hadamard manifolds, developed by Hirai, as a foundational element of the proof. The study pioneered a method for relating the asymptotic slice rank of tensors to the asymptotic vertex cover number of hypergraphs, constructing a d-uniform, d-partite hypergraph Ht from any d-tensor t, defining hyperedges based on zero tensor elements. Researchers then defined τ(H) as the vertex cover number of Ht and its asymptotic counterpart as τ(H) := lim n→∞τ(H×n)1/n, where H×n denotes the n-th power of the hypergraph. The team demonstrated that the asymptotic slice rank can be computed as f SR(t) = ming∈GL τ(Hg·t), a result previously known only for free tensors.

The approach enables computation of both the asymptotic slice rank and the asymptotic vertex cover number via minimization of the quantum and support functional over a parameter θ, expressed as f SR(t) = minθ∈Θ Fθ(t) and τ(H) = minθ∈Θ ζθ(t). Theorem 1.3 extends beyond entanglement polytopes, applying to any convex, symmetric, and lower semi-continuous function on moment polytopes associated with arbitrary actions of the group GL. The proof leverages properties of gradient flow and relies on a duality theorem for convex optimization on Hadamard manifolds, where the team considered geodesically convex functions and functions Q defined on the cotangent bundle, ensuring invariance under parallel transport. This innovative methodology establishes a strong connection between tensor parameters and hypergraph properties, offering new insights into algebraic complexity and additive combinatorics.

Strassen Functionals are universal in tensor asymptotic spectrum

Scientists have definitively proven that Strassen’s support functionals are universal spectral points within the asymptotic spectrum of tensors, resolving a problem open since 1991. The research demonstrates that these support functionals precisely coincide with functionals defined via entropy optimization on polytopes. This breakthrough stems from a general minimax formula for convex optimization on polytopes and moment polytopes, offering broader applications to other tensor parameters like the asymptotic slice rank. Experiments revealed the equality of quantum functionals Fθ(t) and Strassen’s upper support functional ζθ(t) for every tensor t and every θ ∈Θ, as confirmed by Theorem 1.1.

The team measured this equality by establishing a direct link between the asymptotic slice rank of tensors and the asymptotic vertex cover number of hypergraphs. Specifically, the asymptotic slice rank, denoted as f SR(t), can be computed as the minimum over all g in GL of the vertex cover number of the hypergraph Hg·t. Data shows this computation is possible because both f SR(t) and τ(H) are minimized by the quantum and support functional, respectively, over the parameter θ. Measurements confirm that f SR(t) = minθ∈Θ Fθ(t) and τ(H) = minθ∈Θ ζθ(t), providing a novel computational approach. Results demonstrate that Theorem 1.1 is a special case of a more general result applicable to any convex, symmetric, and lower semi-continuous function on moment polytopes.

The team proved that for such functions, minimizing over the moment polytope yields the same value as maximizing the minimum over support polytopes under GL transformations. Theorem 1.3, the minimax formula, states that min p∈∆(t) F(p) = max g∈GL min p∈Ω(g·t) F(p), where ∆(t) and Ω(g·t) represent moment and support polytopes. The proof relies on a recent Fenchel-type duality theorem on Hadamard manifolds established by Hirai, leveraging the unique properties of these manifolds, such as the existence of unique geodesics. Scientists found that this theorem allows for a streamlined proof, bypassing the need for complex invariant-theoretic machinery previously used to establish the equality of quantum functionals. This work also establishes new connections between tensor invariants, including the non-commutative rank, G-stable rank, and symmetric quantum functional, opening avenues for further research and applications.

Strassen’s Functionals Confirmed as Universal Spectral Points

Scientists have confirmed a long-standing conjecture regarding Strassen’s asymptotic spectrum, a key framework for analysing tensor complexity. Researchers demonstrated that Strassen’s support functionals are indeed universal spectral points, establishing their equivalence to functionals defined through entropy optimisation on polytopes. This result was achieved as a specific instance of a broader minimax formula applicable to convex optimisation on polytopes and moment polytopes, extending to other tensor parameters like asymptotic slice rank. The findings significantly advance our understanding of tensor decomposition and its applications in fields such as computer science, additive combinatorics, and information theory.

By proving the universality of Strassen’s support functionals, this work clarifies a fundamental aspect of tensor analysis and provides a powerful tool for studying tensor rank and related properties. The authors acknowledge a limitation in that the current proof relies on a recent Fenchel-type duality theorem, highlighting an area where further theoretical development could offer alternative approaches. Future research directions include exploring the broader implications of the derived minimax formula for other tensor parameters and investigating its potential for developing more efficient algorithms for tensor computations.