Related papers: On the Gap between Scalar and Vector Solutions of …
We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these…
This work provides a theoretical framework for assessing the generalization error of graph neural networks in the over-parameterized regime, where the number of parameters surpasses the quantity of data points. We explore two widely…
We define a special case of tree decompositions for planar graphs that respect a given embedding of the graph. We study the analogous width of the resulting decomposition we call the embedded-width of a plane graph. We show both upper…
Many of the classic problems of coding theory are highly symmetric, which makes it easy to derive sphere-packing upper bounds and sphere-covering lower bounds on the size of codes. We discuss the generalizations of sphere-packing and…
For each integer $m \geq 2$, a network is constructed which is solvable over an alphabet of size $m$ but is not solvable over any smaller alphabets. If $m$ is composite, then the network has no vector linear solution over any $R$-module…
Graph neural networks are widely used tools for graph prediction tasks. Motivated by their empirical performance, prior works have developed generalization bounds for graph neural networks, which scale with graph structures in terms of the…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
As an alternative to classical numerical solvers for partial differential equations (PDEs) subject to boundary value constraints, there has been a surge of interest in investigating neural networks that can solve such problems efficiently.…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet…
This paper aims at studying the sample complexity of graph convolutional networks (GCNs), by providing tight upper bounds of Rademacher complexity for GCN models with a single hidden layer. Under regularity conditions, theses derived…
The problem of finding network codes for general connections is inherently difficult in capacity constrained networks. Resource minimization for general connections with network coding is further complicated. Existing methods for…
Networks are a powerful tool to model complex systems, and the definition of many Graph Neural Networks (GNN), Deep Learning algorithms that can handle networks, has opened a new way to approach many real-world problems that would be hardly…
Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat such issues as constraint systems, optimality and equilibrium conditions, variational inequalities, differential…
A fundamental problem in network science is the normalization of the topological or physical distance between vertices, that requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the…
Motivated by the learned iterative soft thresholding algorithm (LISTA), we introduce a general class of neural networks suitable for sparse reconstruction from few linear measurements. By allowing a wide range of degrees of weight-sharing…
We study in this paper lower bounds for the generalization error of models derived from multi-layer neural networks, in the regime where the size of the layers is commensurate with the number of samples in the training data. We show that…
We consider the problem of layering general directed graphs under height and possibly also width constraints. Given a directed graph G = (V,A) and a maximal height, we propose a layering approach that minimizes a weighted sum of the number…
While previous optimization results have suggested that deep neural networks tend to favour low-rank weight matrices, the implications of this inductive bias on generalization bounds remain underexplored. In this paper, we apply Maurer's…
The clear understanding of the non-convex landscape of neural network is a complex incomplete problem. This paper studies the landscape of linear (residual) network, the simplified version of the nonlinear network. By treating the gradient…