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Circuit algebras are a symmetric version of Jones's planar algebras. They originated in quantum topology as a framework for encoding virtual crossings. This paper extends existing results for modular operads to construct a graphical…
Neural network width and depth are fundamental aspects of network topology. Universal approximation theorems provide that with increasing width or depth, there exists a neural network that approximates a function arbitrarily well. These…
Connected networks are a fundamental structure of neurobiology. Understanding these networks will help us elucidate the neural mechanisms of computation. Mathematically speaking these networks are `graphs' - structures containing objects…
In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with geometric structures. We take particular care to establish a notion of smooth variation of such…
For any category ${\mathcal E}$ and monad $T$ thereon, we introduce the notion of $T$-simplicial object in ${\mathcal E}$. Any $T$-category in the sense of Burroni induces a $T$-simplicial object as its nerve. This nerve construction…
Many physical and biological systems can be studied using complex network theory, a new statistical physics understanding of graph theory. The recent application of complex network theory to the study of functional brain networks generated…
Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to…
Although neural networks traditionally are typically used to approximate functions defined over $\mathbb{R}^n$, the successes of graph neural networks, point-cloud neural networks, and manifold deep learning among other methods have…
Cover's function counting theorem is a milestone in the theory of artificial neural networks. It provides an answer to the fundamental question of determining how many binary assignments (dichotomies) of $p$ points in $n$ dimensions can be…
We characterize the geometry and topology of the set of all weight vectors for which a linear neural network computes the same linear transformation $W$. This set of weight vectors is called the fiber of $W$ (under the matrix multiplication…
In this paper a parameterized generalization of a good cover filtration is introduced called an {\epsilon}-good cover, defined as a cover filtration in which the reduced homology groups of the image of the inclusions between the…
A fully tensorial theoretical framework for hypercomplex-valued neural networks is presented. The proposed approach enables neural network architectures to operate on data defined over arbitrary finite-dimensional algebras. The central…
We deal with the symmetries of a (2-term) graded vector space or bundle. Our first theorem shows that they define a (strict) Lie 2-groupoid in a natural way. Our second theorem explores the construction of nerves for Lie 2-categories,…
We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in…
We introduce the notion of cylinder of a relation in the context of posets, extending the construction of the mapping cylinder. We establish a local-to-global result for relations, generalizing Quillen's Theorem A for order preserving maps,…
In recent years, it has become common practice in neuroscience to use networks to summarize relational information in a set of measurements, typically assumed to be reflective of either functional or structural relationships between regions…
For many fundamental problems in computational topology, such as unknot recognition and $3$-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms…
Morphological neurons, that is morphological operators such as dilation and erosion with learnable structuring elements, have intrigued researchers for quite some time because of the power these operators bring to the table despite their…
We define a notion of Morse function and establish Morse theory-like theorems over offsets of any compact set in a Euclidean space at regular values of their distance function. Using non-smooth analysis and tools from geometric measure…
Many important theorems in differential topology relate properties of manifolds to properties of their underlying homotopy types -- defined e.g. using the total singular complex or the \v{C}ech nerve of a good open cover. Upon embedding the…