Related papers: Expressivity of Transformers: A Tropical Geometry …
This paper investigates the learning theory of Transformer networks for regression tasks on the compact Euclidean domain $[0,1]^d$ and $d$-dimensional compact Riemannian manifolds. We propose a novel constructive approximation framework for…
This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…
Continuing our recent work we study polynomial masks of multivariate tight wavelet frames from two additional and complementary points of view: convexity and system theory. We consider such polynomial masks that are derived by means of the…
We solve Talagrand's entropy problem: the L_2-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley's theorem on classes of {0,1}-valued functions, for which the…
We quantify the rigidity of branching microstructures in shape memory alloys undergoing cubic-to-tetragonal transformations in the geometrically linearized theory by making use of Tartar's H-measures. The main result is a…
An analytical MHD model of coronal loops with compressible flows and including heating is compared to observational data. The model is constructed via a systematic nonlinear separation of the variables method used to calculate several…
Vision Transformers are very popular nowadays due to their state-of-the-art performance in several computer vision tasks, such as image classification and action recognition. Although their performance has been greatly enhanced through…
Transformers have become the go-to architecture for language and vision tasks, yet their theoretical properties, especially memorization capacity, remain elusive. This paper investigates the memorization abilities of multi-head attention…
We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing…
We focus on computing certified upper bounds for the positive maximal singular value (PMSV) of a given matrix. The PMSV problem boils down to maximizing a quadratic polynomial on the intersection of the unit sphere and the nonnegative…
We show how the theory of the critical behaviour of $d$-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain…
In recent years, transformer architectures have revolutionized the field of language processing, opening the door to previously unforeseen possibilities. However, from a theoretical point of view, the mathematical models proposed in the…
A few models have tried to tackle the link prediction problem, also known as knowledge graph completion, by embedding knowledge graphs in comparably lower dimensions. However, the state-of-the-art results are attained at the cost of…
This paper proposes a tropical geometry-based edge detection framework that reformulates convolution and gradient computations using min-plus and max-plus algebra. The tropical formulation emphasizes dominant intensity variations,…
The tropical row span and column span of a real matrix are, from the polyhedral point of view, different objects living in different ambient spaces. These polytopes are known to be combinatorially isomorphic as polyhedral complexes; we…
We prove that estimating the ground state energy of a translationally-invariant, nearest-neighbour Hamiltonian on a 1D spin chain is QMAEXP-complete, even for systems of low local dimension (roughly 40). This is an improvement over the best…
A comprehensive study of a reduced version of Lust's equations, the extended magnetohydrodynamic (XMHD) model obtained from the two-fluid theory for electrons and ions with the enforcement of quasineutrality, is given. Starting from the…
We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral…
This research introduces a novel dual-attention transformer architecture for predicting soil electrical resistivity, a critical parameter for high-voltage substation construction. Our model employs attention mechanisms operating across both…
Do the rich representations of multi-modal diffusion transformers (DiTs) exhibit unique properties that enhance their interpretability? We introduce ConceptAttention, a novel method that leverages the expressive power of DiT attention…