Related papers: The Generalized Persistent Nerve Theorem

Given a simplicial complex and a collection of subcomplexes covering it, the nerve theorem, a fundamental tool in topological combinatorics, guarantees a certain connectivity of the simplicial complex when connectivity conditions on the…

Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to…

We analyze the Nystr\"om approximation of a positive definite kernel associated with a probability measure. We first prove an improved error bound for the conventional Nystr\"om approximation with i.i.d. sampling and singular-value…

The goal of this thesis is to improve our understanding of the internal mechanisms by which deep artificial neural networks create meaningful representations and are able to generalize. We focus on the challenge of characterizing the…

In this paper, we extend original Neural Collapse Phenomenon by proving Generalized Neural Collapse hypothesis. We obtain Grassmannian Frame structure from the optimization and generalization of classification. This structure maximally…

We investigate the parameterization of deep neural networks that by design satisfy the continuity equation, a fundamental conservation law. This is enabled by the observation that any solution of the continuity equation can be represented…

We present a method for segmenting neuron membranes in 2D electron microscopy imagery. This segmentation task has been a bottleneck to reconstruction efforts of the brain's synaptic circuits. One common problem is the misclassification of…

We show that at the level of linear response the low frequency limit of a strongly coupled field theory at finite temperature is determined by the horizon geometry of its gravity dual, i.e. by the "membrane paradigm" fluid of classical…

In this work, we improve upon the guarantees for sparse random embeddings, as they were recently provided and analyzed by Freksen at al. (NIPS'18) and Jagadeesan (NIPS'19). Specifically, we show that (a) our bounds are explicit as opposed…

Common measures of neural representational (dis)similarity are designed to be insensitive to rotations and reflections of the neural activation space. Motivated by the premise that the tuning of individual units may be important, there has…

The notion of metric plays a key role in machine learning problems such as classification, clustering or ranking. However, it is worth noting that there is a severe lack of theoretical guarantees that can be expected on the generalization…

Deep neural networks exhibit a simplicity bias, a well-documented tendency to favor simple functions over complex ones. In this work, we cast new light on this phenomenon through the lens of the Minimum Description Length principle,…

Classical results on the statistical complexity of linear models have commonly identified the norm of the weights $\|w\|$ as a fundamental capacity measure. Generalizations of this measure to the setting of deep networks have been varied,…

This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…

Why do deep networks generalize well? In contrast to classical generalization theory, we approach this fundamental question by examining not only inputs and outputs, but the evolution of internal features. Our study suggests a phenomenon of…

Many algorithms have been recently proposed for causal machine learning. Yet, there is little to no theory on their quality, especially considering finite samples. In this work, we propose a theory based on generalization bounds that…

To the coverage problem of sensor networks, V. de Silva and R. Ghrist (2007) developed several approaches based on (persistent) homology theory. Their criteria for the coverage are formulated on the Rips complexes constructed by the…

Thin fluid or elastic films and membranes are found in nature and technology, for instance, as confinements of living cells or in loudspeakers. When applying a net force, resulting flows in an unbounded two-dimensional incompressible…

Neural networks have been very successful in many applications; we often, however, lack a theoretical understanding of what the neural networks are actually learning. This problem emerges when trying to generalise to new data sets. The…

We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss $\ell$ can control the test error under all Moreau…