arXiv research

Computed linking number of modular knots and Lorenz links.

Modular networks learn to decompose neural computation efficiently.

We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type $X_6$ and $X_7$. We verify that the cluster modular groups of finite mutation type $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E…

PICLE uses probabilistic models to efficiently evaluate and compose modules for continual learning.

A general question behind this paper is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80s. It has only recently begun (2014) to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this paper, we exte…

A new modularity density measure improves community detection in heterogeneous networks.

Defines map from skein module to Habiro ring for quantum modularity.

Empirical risk minimization frequently employs convex surrogates to underlying discrete loss functions in order to achieve computational tractability during optimization. However, classical convex surrogates can only tightly bound modular loss functions, sub-modular functions or supermodular functions separately while …

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the co…

Revealing a community structure in a network or dataset is a central problem arising in many scientific areas. The modularity function $Q$ is an established measure quantifying the quality of a community, being identified as a set of nodes having high modularity. In our terminology, a set of nodes with positive modular…

Method constructs fundamental domains for Picard modular groups.

New method for clustering hypergraphs using modularity maximization.

New quantum modular forms for specific 3-manifolds discovered.

We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth ro…

Quantum theory uses modular group representations to assign invariants to 3-manifolds.

Modular method simplifies curvature computation in neural nets.

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Con…

Noise-driven neural networks emerge modular structures, improving robustness and generalization.

New proof of rearrangement lemma for noncommutative tori using hypergeometric functions.

Study explores how neural networks and Transformers learn modular arithmetic with multiple inputs.

BNNs enhance reservoir computing by acting as generalization filters.

Mathematical study supports connection between 3D manifolds and modular tensor categories.

Modular deep learning framework using pairwise labels without backpropagation.

Survey of methods to train deep architectures without E2EBP.

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones polynomial and arise in Witten-Chern-Simons theory. The braiding and fusion of anyonic excitations in quantum Hall electron liqu…

Routing networks tackle challenges in modular and compositional computation.

Revisits SYM theory to compute Donaldson invariants using mock modular forms.

RLgraph separates RL tasks into modular components for stability and efficiency.

We compute the Euler characteristics of the recently discovered series of Gothic Teichmüller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichmüller curves. Contrary to all previously known examples, the Euler characteristic is not proportional to the Eul…

E-string theory reveals modular properties of 4-manifold invariants.

In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modul…

Method calculates systolic length of modular curves.

Characterizes slopes for Markov ordering on prime pairs.

Quantum invariants of three-manifolds linked to mock theta functions.

Modularity-aware GAE and VGAE improve community detection and link prediction.

Estimating graphical model structure from high-dimensional and undersampled data is a fundamental problem in many scientific fields. Existing approaches, such as GLASSO, latent variable GLASSO, and latent tree models, suffer from high computational complexity and may impose unrealistic sparsity priors in some cases. We…

Modular curves $X_{1}(N)$ parametrize elliptic curves with a point of order $N$. They can be identified with connected components of projectivized strata $\mathbb{P}\mathcal{H}(a,-a)$ of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the …

Researchers create projective representations of Hecke groups using TQFT.

A modular GP framework for efficient transfer learning.

The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact lo…

Discoveries new symmetries in 3d topological and physical systems.

New modular data from torus bundles via particle-hole equivariantization.

Researchers found the global topology of the Eisenstein-Picard modular surface.

Study the Mexican stock market's interdependency structure from 2000-2019.

Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.

Modular neural networks generalize better with less data.

New mapping class group actions on Hochschild complexes for modular categories.

This review establishes a taxonomy for modular neural networks.