Related papers: Copula Representations and Error Surface Projectio…

Graph Neural Networks (GNNs) are increasingly adopted across domains such as molecular biology and social network analysis, yet their black-box nature hinders interpretability and trust. This is especially problematic in high-stakes…

To estimate cosmological parameters from a given dataset, we need to construct a likelihood function, which sometimes has a complicated functional form. We introduce the copula, a mathematical tool to construct an arbitrary multivariate…

Regression analysis is one of the most popularly used statistical technique which only measures the direct effect of independent variables on dependent variable. Path analysis looks for both direct and indirect effects of independent…

A very popular model in machine learning is the feedforward neural network (FFN). The FFN can approximate general functions and mitigate the curse of dimensionality. Here we introduce FFNs which represent sections of holomorphic line…

Generative models that produce point clouds have emerged as a powerful tool to represent 3D surfaces, and the best current ones rely on learning an ensemble of parametric representations. Unfortunately, they offer no control over the…

We study how neural networks trained by gradient descent extrapolate, i.e., what they learn outside the support of the training distribution. Previous works report mixed empirical results when extrapolating with neural networks: while…

Continuous signal representations are naturally suited for inverse problems, such as magnetic resonance imaging (MRI) and computed tomography, because the measurements depend on an underlying physically continuous signal. While classical…

We construct a cellular sheaf from any feedforward ReLU neural network by placing one vertex for each intermediate quantity in the forward pass and encoding each computational step - affine transformation, activation, output - as a…

Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus non-random weights. To address this issue, we study the mapping between…

For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute…

Rectified linear units, or ReLUs, have become the preferred activation function for artificial neural networks. In this paper we consider two basic learning problems assuming that the underlying data follow a generative model based on a…

Many areas of machine learning and science involve large linear algebra problems, such as eigendecompositions, solving linear systems, computing matrix exponentials, and trace estimation. The matrices involved often have Kronecker,…

A frequent task in exploratory data analysis consists in examining pairwise dependencies between data variables. Popular approaches include visualizing correlation or scatter plot matrices. However, both methods can be misleading. The…

There is a longstanding debate whether the Kolmogorov-Arnold representation theorem can explain the use of more than one hidden layer in neural networks. The Kolmogorov-Arnold representation decomposes a multivariate function into an…

We exploit Gaussian copulas to specify a class of multivariate circular distributions and obtain parametric models for the analysis of correlated circular data. This approach provides a straightforward extension of traditional multivariate…

A common approach in neuroscience is to study neural representations as a means to understand a system -- increasingly, by relating the neural representations to the internal representations learned by computational models. However, a…

We propose a new copula model for replicated multivariate spatial data. Unlike classical models that assume multivariate normality of the data, the proposed copula is based on the assumption that some factors exist that affect the joint…

In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded. However, for constant depths, existing…

Online continual learning suffers from an underfitted solution due to insufficient training for prompt model update (e.g., single-epoch training). To address the challenge, we propose an efficient online continual learning method using the…

In this article we present new results on neural networks with linear threshold activation functions. We precisely characterize the class of functions that are representable by such neural networks and show that 2 hidden layers are…