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In this paper we study dual bases functions in subspaces. These are bases which are dual to functionals on larger linear space. Our goal is construct and derive properties of certain bases obtained from the construction, with primary focus…
In this paper, we focus on approximating a natural class of functions that are compositions of smooth functions. Unlike the low-dimensional support assumption on the covariate, we demonstrate that composition functions have an intrinsic…
Although the neural network (NN) technique plays an important role in machine learning, understanding the mechanism of NN models and the transparency of deep learning still require more basic research. In this study, we propose a novel…
Density-functional theory is a formally exact description of a many-body quantum system in terms of its density; in practice, however, approximations to the universal density functional are required. In this work, a model based on deep…
Bent Boolean functions are important objects in cryptography and coding theory, and there are several general approaches for constructing such functions. Metaheuristics proved to be a strong choice as they can provide many bent functions,…
The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…
We introduce a novel embedding method diverging from conventional approaches by operating within function spaces of finite dimension rather than finite vector space, thus departing significantly from standard knowledge graph embedding…
Deep Neural Networks (DNNs) are generated by sequentially performing linear and non-linear processes. Using a combination of linear and non-linear procedures is critical for generating a sufficiently deep feature space. The majority of…
Understanding how convolutional neural networks (CNNs) can efficiently learn high-dimensional functions remains a fundamental challenge. A popular belief is that these models harness the local and hierarchical structure of natural data such…
Neural networks posses the crucial ability to generate meaningful representations of task-dependent features. Indeed, with appropriate scaling, supervised learning in neural networks can result in strong, task-dependent feature learning.…
Task-based modeling with recurrent neural networks (RNNs) has emerged as a popular way to infer the computational function of different brain regions. These models are quantitatively assessed by comparing the low-dimensional neural…
We derive a spectral interpretation of the pivot operation on a graph and generalise this operation to hypergraphs. We establish lower bounds on the number of flat spectra of a Boolean function, depending on internal structures, with…
Traditional estimators of the galaxy power spectrum and bispectrum are sensitive to the survey geometry. They yield spectra that differ from the true underlying signal since they are convolved with the window function of the survey. For the…
The approximation properties of infinitely wide shallow neural networks heavily depend on the choice of the activation function. To understand this influence, we study embeddings between Barron spaces with different activation functions.…
Models of neural networks have proven their utility in the development of learning algorithms in computer science and in the theoretical study of brain dynamics in computational neuroscience. We propose in this paper a spatial neural…
Deep neural networks have achieved remarkable success, yet our understanding of how they learn remains limited. These models can learn high-dimensional tasks, which is generally statistically intractable due to the curse of dimensionality.…
We study entropy-bounded computational geometry, that is, geometric algorithms whose running times depend on a given measure of the input entropy. Specifically, we introduce a measure that we call range-partition entropy, which unifies and…
A fundamental challenge in neuroscience is to understand what structure in the world is represented in spatially distributed patterns of neural activity from multiple single-trial measurements. This is often accomplished by learning a…
In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there…
This paper introduces a hypothesis space for deep learning based on deep neural networks (DNNs). By treating a DNN as a function of two variables - the input variable and the parameter variable - we consider the set of DNNs where the…