Omer Bobrowski

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We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the(More)
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(More)
We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either become trivial as the number n of vertices grows, or can contain more and more complex structures. The different behaviours(More)
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(More)
We compute the homology of random Čech complexes over a homogeneous Poisson process on the d-dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erdős -Rényi phase transition, where the Čech complex becomes connected. The second transition is where all the other homology groups are computed(More)
Consider an agent observing the environment through a set of noisy (possibly multimodal) sensory neurons. Based on these observations it needs to estimate the state of the environment (more generally, the state distribution) with the highest accuracy possible. It is well known that if the stochastic dynamics of the environment and the observation process(More)
s of the 16th Annual Meeting of The Israel Society for Neuroscience Eilat, Israel, November 25–27, 2007 Received 9 October 2007; Accepted 9 October 2007 The Israel Society for Neuroscience—ISFN—was founded in 1993 by a group of Israeli leading scientists conducting research in the area of neurobiology. The primary goal of the society was to promote and(More)
This talk fits into the general topic of 'stratification learning,' wherein one tries to make inferences about data based on some assumption that it is sampled from a mixture of manifolds glued together in some nicely-structured way. The theoretical tool of persistent local homology (PLH), now more than five years old, provides a useful way to understand(More)
[1] Robert J. Adler, Omer Bobrowski, and Shmuel Weinberger. Crackle: The Homology of Noise. Discrete & Computational Geometry, 52(4):680–704, December 2014. [2] Noga Alon. On the edge-expansion of graphs. Combinatorics, Probability and Computing, 6(02):145–152, 1997. [3] Noga Alon and Joel H. Spencer. The probabilistic method. John Wiley & Sons, 2004. [4](More)