Related papers: Combinatorial generation via permutation languages…
Mathematical models play an increasingly important role in the interpretation of biological experiments. Studies often present a model that generates the observations, connecting hypothesized process to an observed pattern. Such generative…
Previous work on symmetric group equivariant neural networks generally only considered the case where the group acts by permuting the elements of a single vector. In this paper we derive formulae for general permutation equivariant layers,…
Counting distinct permutations with replacement, especially when involving multiple subwords, is a longstanding challenge in combinatorial analysis, with critical applications in cryptography, bioinformatics, and statistical modeling. This…
We present a novel and effective approach for generating new clothing on a wearer through generative adversarial learning. Given an input image of a person and a sentence describing a different outfit, our model "redresses" the person as…
The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data…
Like the letters in the alphabet forming words, reusing components of a heterogeneous mixture is an efficient strategy for assembling a large number of target structures. Examples range from synthetic DNA origami to proteins self-assembling…
Symmetry is a guiding principle in physics that allows to generalize conclusions between many physical systems. In the ongoing search for new topological phases of matter, symmetry plays a crucial role because it protects topological…
A generic construction of linear codes over finite fields has recently received a lot of attention, and many one-weight, two-weight and three-weight codes with good error correcting capability have been produced with this generic approach.…
Extraction of structure, in particular of group symmetries, is increasingly crucial to understanding and building intelligent models. In particular, some information-theoretic models of parsimonious learning have been argued to induce…
For a given permutation or set partition there is a natural way to assign a genus. Counting all permutations or partitions of a fixed genus according to cycle lengths or block sizes, respectively, is the main content of this article. After…
In [4] we describe a variation of the classical permutation decoding algorithm that can be applied to any binary affine-invariant code; in particular, it can be applied to first-order Reed-Muller codes successfully. In this paper we study…
The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple…
We present a deep generative scene modeling technique for indoor environments. Our goal is to train a generative model using a feed-forward neural network that maps a prior distribution (e.g., a normal distribution) to the distribution of…
A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs,…
We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results we develop a…
We introduce the use of high order automatic differentiation, implemented via the algebra of truncated Taylor polynomials, in genetic programming. Using the Cartesian Genetic Programming encoding we obtain a high-order Taylor representation…
Generative Pretrained Transformers (GPTs) are foundational Large Language Models (LLMs) for text generation. However, individual LLMs often produce inconsistent outputs and exhibit biases, limiting their representation of diverse language…
Humans have the ability to rapidly understand rich combinatorial concepts from limited data. Here we investigate this ability in the context of auditory signals, which have been evolved in a cultural transmission experiment to study the…
Polynomial reconstruction on Cartesian grids is fundamental in many scientific and engineering applications, yet it is still an open problem how to construct for a finite subset $K$ of $\mathbb{Z}^{\textsf{D}}$ a lattice $\mathcal{T}\subset…
Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces.…