matrix rank arguments

This work addresses the longstanding challenge of deriving strong lower bounds for the weak pigeonhole principle in proof complexity by introducing its algebraic generalization—the Weak Rank principle (WRank)—and constructing multiple encodings, including perfect matching and bamboo-tree CNF formulations. By innovatively designing a scalable lower-bound generator tailored for Polynomial Calculus Resolution over $F_2$ (PCR$_{F_2}$) and a novel pseudo-expectation method adapted to the Sherali–Adams system, the paper establishes the first exponential proof-length lower bounds for WRank in PCR$_{F_2}$, resolving a major open problem. It further demonstrates that circuit lower bound formulas admit no short proofs in this system. These results establish WRank as both necessary and sufficient for proving lower bounds against NC$^2$ and AC$^{0}[p]$, thereby cementing its central role in structured proof complexity analysis.

This paper investigates the inequality theory for the number of linear extensions of finite partially ordered sets (posets), focusing on tight bounds, necessary and sufficient conditions for equality, and computational complexity. Methodologically, it unifies classical inequalities—including the BFS, LYM, and Stanley bounds—systematically characterizing their domains of applicability and tightness thresholds; integrates combinatorial inequality analysis, poset structural theory, and asymptotic enumeration techniques to derive several novel tight bounds; and establishes a comprehensive computational complexity classification framework for linear extension counting, proving #P-completeness for multiple natural variants. The contributions include: (i) the first systematic delineation of tightness boundaries across major linear extension inequalities; (ii) complete characterizations of equality conditions for several foundational inequalities; and (iii) the first complexity-theoretic taxonomy of the problem, resolving long-standing gaps in both extremal analysis and hardness classification. The work also identifies key open problems at the intersection of extremal poset theory and counting complexity.

This work investigates the connection between multiplication complexity in algebraic circuits and tensor-rank–type measures in the high-order multilinear setting. Introducing Naslund’s partition rank as a generalized notion of tensor rank, we establish for the first time a direct link between partition rank and the multiplication complexity of algebraic circuits, thereby extending classical rank-based lower-bound techniques—originally developed for bilinear computation, such as Strassen’s seminal results—to arbitrary constant-order multilinear computations. This framework not only yields a streamlined proof of the NP-hardness of symmetric slice rank but also elucidates intrinsic relationships among partition rank, slice rank, and symmetric slice rank, offering a new rank-based toolkit for fine-grained complexity questions such as the hyperclique conjecture.

This work addresses the long-standing open problem of proving super-polynomial lower bounds for multilinear algebraic branching programs (mABPs), a barrier to separating multilinear complexity classes. Focusing on the widely used min-partition rank method, the paper establishes—unconditionally—the inherent limitation of this technique by showing it cannot yield super-polynomial lower bounds for mABPs. By constructing a 1-balanced chain-set system via the probabilistic method and combining martingale analysis with multi-scale recursion, the authors reduce the system’s size to polynomial in $n$, thereby exhibiting a full-rank multilinear polynomial computable by a polynomial-size mABP. This result not only demonstrates a fundamental barrier for current lower-bound approaches but also resolves an open question concerning the minimum size of 1-balanced chain-set systems, establishing $N(n) = n^{O(1)}$.

This study systematically investigates the mathematical structure and computational complexity of Boolean rank and binary rank, as well as their relationship to real rank. By integrating key techniques from linear algebra, combinatorics, and graph theory—alongside tools such as fooling sets, probabilistic methods, kernelization, communication protocols, and query-to-communication lifting—the work provides a deep analysis of their interdisciplinary applications in communication complexity, parameterized algorithms, and approximate matrix factorization. The paper clarifies existing theoretical boundaries, establishes nontrivial upper and lower bounds for several matrix families, and surveys recent algorithmic advances in Boolean matrix factorization, thereby charting a clear path for future research.

This work investigates the affine rank minimization problem ARM(k)—determining whether there exists a real matrix of rank at most a fixed constant \(k\) satisfying given affine constraints. By constructing a polynomial-time reduction from the Existential Theory of the Reals (ETR), it establishes for the first time that ARM(3) is ETR-complete, and further proves that ARM(k) remains ETR-complete for every fixed integer \(k \geq 1\). The core technical contributions include the development of a non-interfering embedding gadget for rank enforcement and a canonical decomposition encoding scheme. These innovations precisely characterize the computational complexity boundary of low-rank affine feasibility problems, revealing that even systems consisting solely of affine equalities together with a fixed-rank constraint already possess the full expressive power of real algebraic computation.

This work investigates the computational hardness of finding low-rank matrices within a given linear subspace that is guaranteed to contain a rank‑1 matrix. Under the assumption that NP does not admit subexponential-time algorithms, it is shown that no polynomial-time algorithm can find a matrix of rank below $n^{o(1/\log\log n)}$, even when such a rank‑1 matrix exists. The paper introduces a novel combination of hypergraph consistency and moment-matrix techniques to establish strong inapproximability results without relying on the PCP framework, extending these results to arbitrary finite fields. Via two distinct reductions, the authors achieve a 1 vs. $k$ rank gap inapproximability within running times $n^{O(\log k)}$ and $n^{O(k)}$, respectively, substantially advancing the understanding of inapproximability in coding theory and lattice problems.

This work addresses the challenge of effectively characterizing the complexity of matrices that simultaneously exhibit combinatorial and algebraic structure—a key issue in understanding matrix rigidity and circuit lower bounds for neural networks. We introduce a novel parameter, “spiky rank,” which expresses a matrix as a sum of block-diagonal structures whose diagonal blocks are arbitrary rank-one matrices, thereby unifying combinatorial patterns with linear-algebraic flexibility. As a more robust measure of complexity, spiky rank is analyzed through both random matrix theory and explicit constructions, establishing theoretical connections to matrix rigidity, depth-2 ReLU circuit lower bounds, and the γ₂-norm. Our results demonstrate that high spiky rank implies high rigidity and yield tight circuit lower bounds for Hamming distance matrices and spectral expanders.

This work addresses the long-standing open question of whether the circuit diameter of polyhedra admits a strong polynomial upper bound—the circuit analogue of the polynomial Hirsch conjecture. By integrating circuit direction analysis, the geometric structure of linear programming, and combinatorial optimization techniques, the authors construct, for the first time, a monotone circuit walk and establish that the circuit diameter of any polyhedron in standard form is at most \(O(m^2 \log m)\). This result breaks through the previously known weakly polynomial bounds and provides the first strong polynomial upper bound on circuit diameters, offering crucial theoretical support for the existence of strongly polynomial-time algorithms for linear programming.

Computing the nonnegative rank of a nonnegative matrix is an NP-hard problem, necessitating efficient methods to determine its lower bounds. This work addresses this challenge by introducing, within a nonconvex optimization framework, the first practically implementable algorithm for the self-scaled bound (SSB) and unifying the efficient computation of four classical lower bounds—FSB, RCB, HSB, and SSB. Evaluated on multiple standard benchmark matrices, the proposed method improves upon the best-known lower bounds in the literature; in several cases, these bounds match existing upper bounds, thereby establishing the exact nonnegative rank for the first time. The study thus provides a unified and practical numerical tool for advancing research on nonnegative rank lower bounds.