This paper compares two theoretical frameworks for hierarchical neural networks with a finite but large number of hidden units and shows that training input-to-hidden weights reduces generalization er…

This paper investigates the closeness of singularity for $n imes n$ unimodular matrices, specifically for $(2k+1)$-ary and $(k+1)$-ary cases of $4 imes 4$ nonnegative unimodular matrices.

The paper introduces Singularity-aware Adam (S-Adam), a novel optimizer that stabilizes deep learning training in non-smooth loss landscapes by dynamically damping updates based on local geometric ins…

The paper presents a novel and significantly faster algorithm for computing the second discrete homology group of a graph by identifying five basic quotient shapes of the 3-cube.

This paper proves that a majority of filled-in Sudoku grids require a logarithmic fraction of cells to be filled by clues, and constructs grids requiring 18 and 80 clues for 9x9 and 16x16 Sudoku, resp…

The paper introduces a computational framework using Hodge zero-modes to track the geometry of topological features in parameter-dependent data, providing metrics like curvature and holonomy to quanti…

The paper proves the real-rootedness and ultra-log-concavity of the Poincaré polynomials for the moduli space $\overline{\mathcal M}_{0,n}$ and the Fulton--MacPherson space $\mathbb{P}^1[n]$ using a n…

This paper presents an algorithmic framework for exhaustively generating and tabulating knot and link diagrams on the thickened torus.

This paper studies the expressivity of shallow polynomial neural networks with monomial activation functions over finite fields, quantifying it by the cardinality of the neuromanifold and deriving low…

This paper proves conditions for efficiently approximating expectations of certain functions with respect to standard Gaussian or symmetric exponential probability measures.

The paper systematically studies the trade-offs between the number of slopes, bends per edge, and required area for planar drawings of bounded-degree graphs, providing new constructions for high-degre…

This paper proves DP-completeness of Exact-3-DNP and Exact-4-DNP by reducing 3SAT to these problems.

This paper proves that the satisfiability problem of existential Presburger arithmetic extended with divisibility predicates (EPAD) is PP-hard.

The paper introduces NewtPhys, a novel 4D dataset of real-world scenes with dense physical annotations, to systematically evaluate and reveal the limitations of foundation models in low-level Newtonia…

The paper shows that every problem asking if a given graph has an edge-coloring without an edge-colored clique from a fixed family is poly-time equivalent to a Constraint Satisfaction Problem, resulti…

This paper establishes mathematical limits of AGI safety, proving structural unverifiability as the core barrier.