arXiv research

GMT improves interpretability of XGNNs by approximating SubMT.

We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building …

Optimizing quantum graphs yields geodesic nets on surfaces.

Proposes MLDP for modeling multilinear data.

New method forecasts multilinear data using tensor autoregression.

Geometrically, tensors of fixed rank form a minimal submanifold.

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of …

Unified multilinear model for causal factor disentanglement.

Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.

Develops method for learning signed graphs from smooth signals.

New algorithm solves $\ell_0$-norm constrained multilinear logistic regression for tensor data.

Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging problem is related to determination of model complexity (i.e., multilinear rank), e…

In this paper we present a new model and an algorithm for unsupervised clustering of 2-D data such as images. We assume that the data comes from a union of multilinear subspaces (UOMS) model, which is a specific structured case of the much studied union of subspaces (UOS) model. For segmentation under this model, we de…

Wide neural networks become linear, but adding bottlenecks makes them bilinear or multilinear.

The main results of our paper deal with the lifting problem for multilinear differential operators between complexes of horizontal de Rham forms on the infinite jet bundle. We answer the question when does an n-multilinear differential operator from the space of (N,0)-forms (where N is the dimension of the base) to the…

Generative Adversarial Network (GAN) and its variants exhibit state-of-the-art performance in the class of generative models. To capture higher-dimensional distributions, the common learning procedure requires high computational complexity and a large number of parameters. The problem of employing such massive framewor…

Study uses random matrix theory to improve tensor approximation accuracy.

Theorem shows generic metrics yield non-degenerate geodesic nets.

Principal component analysis (PCA) is an unsupervised method for learning low-dimensional features with orthogonal projections. Multilinear PCA methods extend PCA to deal with multidimensional data (tensors) directly via tensor-to-tensor projection or tensor-to-vector projection (TVP). However, under the TVP setting, i…

Extends string-net theory to 3D TQFT via surface graphs and surgery.

Knot theory is the study of isotopy classes of embeddings of the circle $S^1$ into a 3-manifold, specifically $R^3$. The Fáry-Milnor Theorem says that any curve in $R^3$ of total curvature less than $4π$ is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topologica…

Matrix factorizations and their extensions to tensor factorizations and decompositions have become prominent techniques for linear and multilinear blind source separation (BSS), especially multiway Independent Component Analysis (ICA), NonnegativeMatrix and Tensor Factorization (NMF/NTF), Smooth Component Analysis (Smo…

Proposes FMPCA for federated tensor data dimensionality reduction.

Graph Neural Nets (GNNs) have received increasing attentions, partially due to their superior performance in many node and graph classification tasks. However, there is a lack of understanding on what they are learning and how sophisticated the learned graph functions are. In this work, we propose a dissection of GNNs …

We consider the problem of representation learning for graph data. Convolutional neural networks can naturally operate on images, but have significant challenges in dealing with graph data. Given images are special cases of graphs with nodes lie on 2D lattices, graph embedding tasks have a natural correspondence with i…

Numerous pattern recognition applications can be formed as learning from graph-structured data, including social network, protein-interaction network, the world wide web data, knowledge graph, etc. While convolutional neural network (CNN) facilitates great advances in gridded image/video understanding tasks, very limit…

Paper proposes JDR to denoise graph features and rewire graphs for better node classification.

Graph data widely exist in many high-impact applications. Inspired by the success of deep learning in grid-structured data, graph neural network models have been proposed to learn powerful node-level or graph-level representation. However, most of the existing graph neural networks suffer from the following limitations…

BCD Nets use variational inference to estimate DAGs with uncertainty.

Given a multifunction from $X$ to the $k-$fold symmetric product $Sym_k(X)$, we use the Dold-Thom Theorem to establish a homological selection Theorem. This is used to establish existence of Nash equilibria. Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixe…

Derives a primal-dual MLSVD formulation for multilinear data.

New model generates unseen attribute combinations from limited data.

New algorithms bound graph structure sampling and learning high-dimensional graphical models.

We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.

An embedded cubic graph consisting of segments of geodesics such that the angles at any vertex are equal to $2π/3$ is a closed local minimal net. This net is regular if all segments of geodesics are equal. The problem of classification of closed local minimal nets on surfaces of constant negative curvature has been for…

Dimensionality reduction is a main step in the learning process which plays an essential role in many applications. The most popular methods in this field like SVD, PCA, and LDA, only can be applied to data with vector format. This means that for higher order data like matrices or more generally tensors, data should be…

Critical nets in $\mathbb{R}^k$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices (leaves) have a fixed position. In contrast to what happens on generic manifolds, w…

HGNet improves GNNs' ability to handle long-range interactions in graphs.

Knowledge graph reasoning is a critical task in natural language processing. The task becomes more challenging on temporal knowledge graphs, where each fact is associated with a timestamp. Most existing methods focus on reasoning at past timestamps and they are not able to predict facts happening in the future. This pa…

MCCA extracts shared structure from multiple tensor datasets.

The existence of a balanced vertex is proven for geodesic nets with three boundary vertices.

New method for probabilistic modeling of integer submodular functions.

New neural nets respect triangle inequality, improving graph and reinforcement learning performance.

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph …

Efficiently optimizes boolean functions using multilinear polynomials and exponential weight updates.

We study algebraic varieties of ReLU networks to understand their representable functions.

The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H. Gloeckner and K.-H. Neeb), without any restriction on the dimension or on the cha…

SLAM-net learns to navigate visually in challenging indoor environments.