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Neural networks are known to give better performance with increased depth due to their ability to learn more abstract features. Although the deepening of networks has been well established, there is still room for efficient feature…
We present new algorithms to compute fundamental properties of a Boolean function given in truth-table form. Specifically, we give an O(N^2.322 log N) algorithm for block sensitivity, an O(N^1.585 log N) algorithm for `tree decomposition,'…
Although neural networks can solve very complex machine-learning problems, the theoretical reason for their generalizability is still not fully understood. Here we use Wang-Landau Mote Carlo algorithm to calculate the entropy (logarithm of…
Understanding how deep neural networks learn useful internal representations from data remains a central open problem in the theory of deep learning. We introduce Neural Low-Degree Filtering (Neural LoFi), a stylized limit of gradient-based…
Recently, it has been shown that many functions on sets can be represented by sum decompositions. These decompositons easily lend themselves to neural approximations, extending the applicability of neural nets to set-valued inputs---Deep…
In this paper we propose a function space approach to Representation Learning and the analysis of the representation layers in deep learning architectures. We show how to compute a weak-type Besov smoothness index that quantifies the…
We propose a representation of boolean bent functions by bent rectangles, that is, by special matrices with restrictions on rows and columns. Using this representation, we exhibit new classes of bent functions, give an algorithm to…
Ability of deep networks to extract high level features and of recurrent networks to perform time-series inference have been studied. In view of universality of one hidden layer network at approximating functions under weak constraints, the…
Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate…
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…
Deep learning techniques are increasingly being adopted for classification tasks over the past decade, yet explaining how deep learning architectures can achieve state-of-the-art performance is still an elusive goal. While all the training…
Deep neural networks are renowned for their ability to generalise well across diverse tasks, even when heavily overparameterized. Existing works offer only partial explanations (for example, the NTK-based task-model alignment explanation…
This paper studies the mathematical properties of collectively canalizing Boolean functions, a class of functions that has arisen from applications in systems biology. Boolean networks are an increasingly popular modeling framework for…
Deep convolutional networks provide state of the art classifications and regressions results over many high-dimensional problems. We review their architecture, which scatters data with a cascade of linear filter weights and non-linearities.…
In this paper, we hypothesize that internal function sharing is one of the reasons to weaken o.o.d. or systematic generalization in deep learning for classification tasks. Under equivalent prediction, a model partitions an input space into…
The problem of computing functions of values at the nodes in a network in a totally distributed manner, where nodes do not have unique identities and make decisions based only on local information, has applications in sensor, peer-to-peer,…
In order to estimate the specific intrinsic volumes of a planar Boolean model from a binary image, we consider local digital algorithms based on weighted sums of $2\times 2$ configuration counts. For Boolean models with balls as grains,…
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…
The problem of extending a function $f$ defined on a training data $\mathcal{C}$ on an unknown manifold $\mathbb{X}$ to the entire manifold and a tubular neighborhood of this manifold is considered in this paper. For $\mathbb{X}$ embedded…
We propose a deep neural network architecture and a training algorithm for computing approximate Lyapunov functions of systems of nonlinear ordinary differential equations. Under the assumption that the system admits a compositional…