arXiv AI
Aug 10
A Finite E-Group of Nilpotency Class Three
arXiv:2608. 07275v1 Announce Type: cross Abstract: A group is an E-group if every element commutes with each of its endomorphic images.
Computations and ML for surjective rational maps
arXiv:2510. 08093v2 Announce Type: replace-cross Abstract: The present note studies \emph{surjective rational endomorphisms} $f: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ with \emph{cubic} terms and the indeterminacy locus $I_f \ne \emptyset$.
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arXiv Statistics ML
Sep 4
Algebraic Invariants of Lightning Self-Attention
The paper investigates the polynomial coefficients of lightning self‑attention, treating them as coordinates of an algebraic variety. In the single‑token case it identifies the coefficient variety as a rank‑constrained Chow‑type variety and derives algebraic equations; for multiple tokens it shows that linear relations reduce the geometry to coefficients involving interactions between distinct tokens, characterized by a common linear factor and a low‑rank condition. The authors provide explicit families of determinantal, Veronese‑type, and Sylvester resultant‑based invariants, and in the rank‑one case give pencil and flattening equations that define the variety set‑theoretically, with small‑dimension computations confirming the theoretical generators.
arXiv Machine Learning
Sep 25
Algebraic Expressivity Certificates for Shallow Polynomial Neural Networks
arXiv:2609. 28500v1 Announce Type: cross Abstract: We study exact representability by bias-free shallow polynomial neural networks using algebraic geometry.
arXiv Computation and Language
Sep 11
Formalizing building-up constructions of self-dual codes through isotropic lines in Lean
arXiv:2604. 08485v2 Announce Type: replace-cross Abstract: The purpose of this paper is two-fold.
arXiv Machine Learning
Sep 4
Grassmann--Pl\"ucker Parametrization of Convolutional Filter Subspaces: Regularity and Closed Embeddings
arXiv:2609. 03361v1 Announce Type: cross Abstract: We propose a geometric parametrization of the filters in a single convolutional layer: the parameter is no longer an ordered family of filter vectors, but a fixed-dimensional subspace of the filter space.
arXiv Statistics ML
Sep 17
Generalized DCCQ: From Binary Quotients to Multinomial Simplex Geometry and Critical-Strip Coordinates
arXiv:2609. 17899v1 Announce Type: new Abstract: We extend the discrete complex complement quotient (DCCQ) framework from binary Bernoulli counts to multinomial count compositions.
arXiv AI
Jun 12
Real-rootedness of the Poincar\'e polynomials of $\overline{\mathcal M}_{0,n}$: an AI-assisted proof
arXiv:2605. 29151v2 Announce Type: replace-cross Abstract: We prove real-rootedness for the Poincar\'e polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli.
arXiv Machine Learning
Aug 4
Polyatomic Complexes: A topologically-informed learning representation for atomistic systems
arXiv:2409. 15600v3 Announce Type: replace Abstract: A representation of a molecule or material should be invariant to the symmetries of physics, unique, continuous, efficient and general.
arXiv Machine Learning
Jul 28
Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
arXiv:2607. 24251v1 Announce Type: cross Abstract: We investigate the extent to which the reduced minimal Weierstrass coefficients of an elliptic curve over $\mathbb{Q}$ may be computed from it's Frobenius traces.
arXiv Machine Learning
Aug 18
High-Dimensional Nonparametric Change-Point Detection via Low-Rank Degree-Three Density Projection
arXiv:2608. 15466v1 Announce Type: new Abstract: Distributional changes can be invisible to means and covariances yet appear in skewness, asymmetric interactions, or other third-order structure.
arXiv Machine Learning
Jun 11
Minimal surfaces, Knots, and Neural Networks
arXiv:2605. 26234v2 Announce Type: replace-cross Abstract: A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot $K$ in the 3-sphere $S^3$, and the signed count of minimal surfaces in hyperbolic 4-space $\mathrm{H}^4$ meeting the sphere at infinity at $K$, with prescribed genus and self-intersection number.
arXiv AI
Jul 1
Improved Upper Bounds for Slicing the Hypercube
arXiv:2602. 16807v2 Announce Type: replace Abstract: A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior.