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In this work, we propose an efficient algorithm for the calculation of the Betti matching, which can be used as a loss function to train topology aware segmentation networks. Betti matching loss builds on techniques from topological data…

Thanks to the availability of large scale digital datasets and massive amounts of computational power, deep learning algorithms can learn representations of data by exploiting multiple levels of abstraction. These machine learning methods…

Our theoretical understanding of deep learning has not kept pace with its empirical success. While network architecture is known to be critical, we do not yet understand its effect on learned representations and network behavior, or how…

Complex systems, ranging from soft materials to wireless communication, are often organised as random geometric networks in which nodes and edges evenly fill up the volume of some space. Studying such networks is difficult because they…

We survey results on neural network expressivity described in "On the Expressive Power of Deep Neural Networks". The paper motivates and develops three natural measures of expressiveness, which all display an exponential dependence on the…

Network embedding, which aims to learn low-dimensional representations of nodes, has been used for various graph related tasks including visualization, link prediction and node classification. Most existing embedding methods rely solely on…

The topology of many real complex networks has been conjectured to be embedded in hidden metric spaces, where distances between nodes encode their likelihood of being connected. Besides of providing a natural geometrical interpretation of…

The information conveyed by a hierarchical attractor neural network is examined. The network learns sets of correlated patterns (the examples) in the lowest level of the hierarchical tree and can categorize them at the upper levels. A way…

Cellular neural circuit and networks consisting of interconnected neurons and glia are ulti- mately responsible for the information processing associated with information processing in the brain. While there are major efforts aimed at…

The paper aims to investigate relevant computational issues of deep neural network architectures with an eye to the interaction between the optimization algorithm and the classification performance. In particular, we aim to analyze the…

We give a covering number bound for deep learning networks that is independent of the size of the network. The key for the simple analysis is that for linear classifiers, rotating the data doesn't affect the covering number. Thus, we can…

Complex molecules and mesoscopic structures are naturally described by general networks of elementary building blocks and tight-binding is one of the simplest quantum model suitable for studying the physical properties arising from the…

To explore the relation between network structure and function, we studied the computational performance of Hopfield-type attractor neural nets with regular lattice, random, small-world and scale-free topologies. The random net is the most…

Deep neural networks have usually to be compressed and accelerated for their usage in low-power, e.g. mobile, devices. Recently, massively-parallel hardware accelerators were developed that offer high throughput and low latency at low power…

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of…

Artificial neural networks have gone through a recent rise in popularity, achieving state-of-the-art results in various fields, including image classification, speech recognition, and automated control. Both the performance and…

In recent years, deep neural networks have had great success in machine learning and pattern recognition. Architecture size for a neural network contributes significantly to the success of any neural network. In this study, we optimize the…

Modularity has been widely studied as a mechanism to improve the capabilities of neural networks through various techniques such as hand-crafted modular architectures and automatic approaches. While these methods have sometimes shown…

Deep Neural Networks (DNNs) can be represented as graphs whose links and vertices iteratively process data and solve tasks sub-optimally. Complex Network Theory (CNT), merging statistical physics with graph theory, provides a method for…

We study the excess capacity of deep networks in the context of supervised classification. That is, given a capacity measure of the underlying hypothesis class - in our case, empirical Rademacher complexity - to what extent can we (a…