This paper extends the Jaccard distance to arbitrary lattices and real valuations, proving that the triangle inequality holds when the valuation is strictly positive, monotone, and modular. It further shows that in relatively complemented distributive lattices the inequality remains valid under positive, monotone, supermodular, and log‑submodular valuations, and adapts the symmetric‑difference Jaccard formulation for submodular valuations in sectionally complemented distributive lattices. The authors also establish that supermodularity is a necessary condition for the generalized Jaccard distance to be a metric and discuss the implications of these relaxed structural constraints for fields such as quantum information theory, formal concept analysis, and machine learning.
arXiv:2607. 23500v1 Announce Type: cross Abstract: Razborov's flag algebra method is a powerful tool for proving asymptotic inequalities in extremal graph theory, often reducing the task to finding a finite certificate by semidefinite programming.
arXiv:2607. 05492v1 Announce Type: cross Abstract: Quantum information theory is built on entropic quantities; among them, the sandwiched R\'enyi relative entropy is a fundamental divergence with various applications, and its data processing inequality (DPI) under quantum channels is a cornerstone result.
arXiv:2609.05641v1 Announce Type: cross Abstract: We consider the problem of minimizing error in quantized matrix multiplication $C=AB$. Scalar quantization of the factors introduces rounding errors...
arXiv:2609.23094v1 Announce Type: cross Abstract: We study the number of prototypes needed to represent Boolean functions by nearest-neighbour classification. There are two distinct settings: the pro...
The paper introduces the concept of supermodular rank, defined as the minimal number of supermodular components required to express a function on a lattice. It characterizes the maximum possible rank and the structure of functions with a fixed rank, and analogously defines submodular rank. Leveraging submodular decompositions, the authors propose an algorithm that partitions a set‑function optimization problem into submodular subproblems, thereby improving approximation guarantees for monotone maximization and ratio minimization while incurring a computational cost tied to the submodular rank.
arXiv:2608. 02533v1 Announce Type: cross Abstract: We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $\Omega(n^2)$.
arXiv:2607. 09032v1 Announce Type: cross Abstract: Quantum logic is usually presented as a non-classical departure from ordinary reasoning forced on us by quantum mechanics, with classical logic kept as the secure starting point.
We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $Ω(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity.
arXiv:2602.02940v2 Announce Type: replace-cross Abstract: Formal semantics and distributional semantics are distinct approaches to linguistic meaning: the former models meaning as reference via model...
arXiv:2607. 09632v1 Announce Type: cross Abstract: Quantum information theory (QIT) characterizes the capabilities and fundamental limits of quantum information processing, underpinning quantum communication, computation, and error correction.
arXiv:2606. 18236v1 Announce Type: new Abstract: In learning theory, the sign rank of a binary concept class captures the smallest dimension in which it can be represented by points and halfspaces.
arXiv:2606. 12211v1 Announce Type: cross Abstract: A central principle in quantum machine learning is that an ansatz should be expressive enough to represent the quantum data of interest.