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This article presents a continuous model for hierarchical networks based on a combination of mathematically derived models of receptive fields and biologically inspired computations. Based on a functional model of complex cells in terms of…
Predicting stable and metastable structures is central to molecular and materials discovery, but remains limited by the cost of searching high-dimensional energy landscapes. Deep generative models offer efficient structure sampling, yet…
In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports h-adaptivity on computational domains represented as…
In this paper we offer a new perspective on the well established agglomerative clustering algorithm, focusing on recovery of hierarchical structure. We recommend a simple variant of the standard algorithm, in which clusters are merged by…
Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular…
Representations of small quantum groups $u_q({\mathfrak{g}})$ at a root of unity and their extensions provide interesting tensor categories, that appear in different areas of algebra and mathematical physics. There is an ansatz by Lusztig…
Treemaps have been widely applied to the visualization of hierarchical data. A treemap takes a weighted tree and visualizes its leaves in a nested planar geometric shape, with sub-regions partitioned such that each sub-region has an area…
We say that a finitely generated group $G$ has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. A quasi-tree is a connected graph with path metric quasi-isometric…
In Bayesian learning of Gaussian graphical model structure, it is common to restrict attention to certain classes of graphs and approximate the posterior distribution by repeatedly moving from one graph to another, using MCMC or methods…
Higher order data is modeled using matrices whose entries are numerical arrays of a fixed size. These arrays, called t-scalars, form a commutative ring under the convolution product. Matrices with elements in the ring of t-scalars are…
We establish an identification between the spaces of $\alpha$-fusion trees in non-semisimple topological quantum computation (NSS TQC) and a family of homological representations of the braid group known as the Lawrence representations…
Hessian captures important properties of the deep neural network loss landscape. Previous works have observed low rank structure in the Hessians of neural networks. In this paper, we propose a decoupling conjecture that decomposes the…
Scalar field comparison is a fundamental task in scientific visualization. In topological data analysis, we compare topological descriptors of scalar fields -- such as persistence diagrams and merge trees -- because they provide succinct…
The transition matrix of a Markov chain $(X_k,k\geq 0)$ on a finite or infinite rooted tree is said to be almost upper-directed if, given $X_k$, the node $X_{k+1}$ is either a descendant of $X_k$ or the parent of $X_k$. It is said to be…
In this paper, we establish the quasi-compactness of the transfer operator associated with skew product systems that are semi-conjugate to piecewise convex maps with a countably infinite number of branches. These non-invertible skew…
In this semi-expository paper, we first explain key notions from current quantum information theory and criteria for them in a coherent way. These include separability/entanglement, Schmidt numbers of bi-partite states and block-positivity,…
We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give…
We propose a hierarchical learning strategy aimed at generating sparse representations and associated models for large noisy datasets. The hierarchy follows from approximation spaces identified at successively finer scales. For promoting…
For an $n$-tuple $s$ of non-negative integers, the $s$-weak order is a lattice structure on $s$-trees, generalizing the weak order on permutations. We first describe the join irreducible elements, the canonical join representations, and the…
The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated…