Related papers: Geometry, combinatorics, and algebra of inductivel…
We study the rational Kontsevich integral of torus knots. We construct explicitely a series of diagrams made of circles joined together in a tree-like fashion and colored by some special rational functions. We show that this series codes…
We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of…
Convolutional codes are constructed, designed and analysed using row and/or block structures of unit algebraic schemes. Infinite series of such codes and of codes with specific properties are derived. Properties are shown algebraically and…
The cut polytope ${\rm CUT}(n)$ is the convex hull of the cut vectors in a complete graph with vertex set $\{1,\ldots,n\}$. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation…
Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…
Relying on the combinatorial classification of toric ideals using their bouquet structure, we focus on toric ideals of hypergraphs and study how they relate to general toric ideals. We show that hypergraphs exhibit a surprisingly general…
Convex codes were recently introduced as models for neural codes in the brain. Any convex code $\C$ has an associated minimal embedding dimension $d(\C)$, which is the minimal Euclidean space dimension such that the code can be realized by…
A convex partition of a point set P in the plane is a planar partition of the convex hull of P with empty convex polygons or internal faces whose extreme points belong to P. In a convex partition, the union of the internal faces give the…
We propose a new family of neural networks to predict the behaviors of physical systems by learning their underpinning constraints. A neural projection operator lies at the heart of our approach, composed of a lightweight network with an…
Despite their success, modern convolutional neural networks (CNNs) exhibit fundamental limitations, including data inefficiency, poor out-of-distribution generalization, and vulnerability to adversarial perturbations. These shortcomings can…
Current deep-learning models for object recognition are known to be heavily biased toward texture. In contrast, human visual systems are known to be biased toward shape and structure. What could be the design principles in human visual…
In pure integer linear programming it is often desirable to work with polyhedra that are full-dimensional, and it is well known that it is possible to reduce any polyhedron to a full-dimensional one in polynomial time. More precisely, using…
This paper introduces a new Convolutional Neural Network (ConvNet) architecture inspired by a class of partial differential equations (PDEs) called quasi-linear hyperbolic systems. With comparable performance on the image classification…
We solve a long-standing open problem about the optimal codebook structure of codes in $n$-dimensional Euclidean space that consist of $n+1$ codewords subject to a codeword energy constraint, in terms of minimizing the average decoding…
Accurate constitutive models of soft materials are crucial for understanding their mechanical behavior and ensuring reliable predictions in the design process. To this end, scientific machine learning research has produced flexible and…
Associated to any hypergraph is a toric ideal encoding the algebraic relations among its edges. We study these ideals and the combinatorics of their minimal generators, and derive general degree bounds for both uniform and non-uniform…
Discrete structures are currently second-class in differentiable programming. Since functions over discrete structures lack overt derivatives, differentiable programs do not differentiate through them and limit where they can be used. For…
We describe the combinatorics of equilibria and steady states of neurons in threshold-linear networks that satisfy Dale's law. The combinatorial code of a Dale network is characterized in terms of two conditions: (i) a condition on the…
Convex geometries are closure systems satisfying the anti-exchange axiom. Every finite convex geometry can be embedded into a convex geometry of finitely many points in an n-dimensional space equipped with a convex hull operator, by the…
A major area in neuroscience research is the study of how the brain processes spatial information. Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps,…