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Graph-theoretical analyses of complex brain networks is a rapidly evolving field with a strong impact for neuroscientific and related clinical research. Due to a number of confounding variables, however, a reliable and meaningful…
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.…
In this paper we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix-variate location mixture of normal…
The learning speed of feed-forward neural networks is notoriously slow and has presented a bottleneck in deep learning applications for several decades. For instance, gradient-based learning algorithms, which are used extensively to train…
Random matrices acting on structured sets play a fundamental role in high-dimensional geometry, compressed sensing, and randomized algorithms. Existing results primarily focus on subgaussian models, when random matrices act as…
We investigate the dynamics of large-scale interacting neural populations, composed of conductance based, spiking model neurons with modifiable synaptic connection strengths, which are possibly also subjected to external noisy currents. The…
There remain theoretical gaps in deep neural network estimators for the nonparametric Cox proportional hazards model. In particular, it is unclear how gradient-based optimization error propagates to population risk under partial likelihood,…
Consider the problem of learning a large number of response functions simultaneously based on the same input variables. The training data consist of a single independent random sample of the input variables drawn from a common distribution…
Recent works have characterized the function-space inductive bias of infinite-width bounded-norm single-hidden-layer neural networks as a kind of bounded-variation-type space. This novel neural network Banach space encompasses many…
We propose a theoretical understanding of neural networks in terms of Wilsonian effective field theory. The correspondence relies on the fact that many asymptotic neural networks are drawn from Gaussian processes, the analog of…
We study sampling from a target distribution ${\nu_* = e^{-f}}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm. For any potential function $f$ whose tails behave like ${\|x\|^\alpha}$ for ${\alpha \in [1,2]}$, and has…
This paper extends the universal approximation property of single-hidden-layer feedforward neural networks beyond compact domains, which is of particular interest for the approximation within weighted $C^k$-spaces and weighted Sobolev…
Semiclassical asymptotics for linear Schr\"odinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment…
Networks of disparate phenomena-- be it the global ecology, human social institutions, within the human brain, or in micro-scale protein interactions-- exhibit broadly consistent architectural features. To explain this, we propose a new…
Recent work by Baratin et al. (2021) sheds light on an intriguing pattern that occurs during the training of deep neural networks: some layers align much more with data compared to other layers (where the alignment is defined as the…
In this paper, we study the asymptotic posterior distribution of linear functionals of the density. In particular, we give general conditions to obtain a semiparametric version of the Bernstein-Von Mises theorem. We then apply this general…
An asymptotic theory is established for linear functionals of the predictive function given by kernel ridge regression, when the reproducing kernel Hilbert space is equivalent to a Sobolev space. The theory covers a wide variety of linear…
We study the influence of boundary conditions on self-affine random functions u(t) in the interval t/L \in [0,1], with independent Gaussian Fourier modes of variance ~ 1/q^{\alpha}. We consider the probability distribution of the mean…
Flexible models for probability distributions are an essential ingredient in many machine learning tasks. We develop and investigate a new class of probability distributions, which we call a Squared Neural Family (SNEFY), formed by squaring…
We study Markov processes on weighted directed hypergraphs where the state of at most one vertex can change at a time. Our setting is general enough to include simplicial epidemic processes, processes on multilayered networks or even the…