Abstract We study the k -th nearest neighbor distance function from a finite point-set in $${\mathbb {R}}^d$$ R d . We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order- k Delaunay mosaics, and random k -fold coverage.

We propose a novel approach for comparing the persistent homology representations of two spaces (or filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes, along with suitable metrics (e.g., Wasserstein). These summaries are useful for computational purposes, but they are merely a marginal of the actual topological information that persistent homology can provide. Instead, our approach compares between two topological representations directly in the data space. We do so by defining a correspondence relation between individual persistent cycles of two different spaces, and devising a method for computing this correspondence. Our matching of cycles is based on both the persistence intervals and the spatial placement of each feature. We demonstrate our new framework in the context of topological inference, where we use statistical bootstrap methods in order to differentiate between real features and noise in point cloud data.

We present a novel approach to alarm sound detection using topological data analysis. Our main focus is on proposing a new set of robust features, based on algebraic topology, that are aimed at capturing global structural information about the dynamical system underlying each input signal. In short, we convert each signal into a point cloud and compute its corresponding persistent homology, from which we can extract a variety of useful numerical features. We demonstrate the power of this framework using the UrbanSound8K dataset and show that, by combining topological features with a classical classification method, we achieve state-of-the-art results.

A key requirement facing organisms acting in uncertain dynamic environments is the real-time estimation and prediction of environmental states, based on which effective actions can be selected. While it is becoming evident that organisms employ exact or approximate Bayesian statistical calculations for these purposes, it is far less clear how these putative computations are implemented by neural networks in a strictly dynamic setting. In this work, we make use of rigorous mathematical results from the theory of continuous time point process filtering and show how optimal real-time state estimation and prediction may be implemented in a general setting using simple recurrent neural networks. The framework is applicable to many situations of common interest, including noisy observations, non-Poisson spike trains (incorporating adaptation), multisensory integration, and state prediction. The optimal network properties are shown to relate to the statistical structure of the environment, and the benefits of adaptation are studied and explicitly demonstrated. Finally, we recover several existing results as appropriate limits of our general setting.

It is becoming increasingly evident that organisms acting in uncertain dynamical environments often employ exact or approximate Bayesian statistical calculations in order to continuously estimate the environmental state, integrate information from multiple sensory modalities, form predictions and choose actions. What is less clear is how these putative computations are implemented by cortical neural networks. An additional level of complexity is introduced because these networks observe the world through spike trains received from primary sensory afferents, rather than directly. A recent line of research has described mechanisms by which such computations can be implemented using a network of neurons whose activ- ity directly represents a probability distribution across the possible “world states”. Much of this work, however, uses various approximations, which severely re- strict the domain of applicability of these implementations. Here we make use of rigorous mathematical results from the theory of continuous time point process filtering, and show how optimal real-time state estimation and prediction may be implemented in a general setting using linear neural networks. We demonstrate the applicability of the approach with several examples, and relate the required network properties to the statistical nature of the environment, thereby quantify- ing the compatibility of a given network with its environment.