arXiv subjects

Ido Cohen

Publications and source records attributed to Ido Cohen.

22 records · Page 2Linked to original sources

arXiv2020

Modes of Homogeneous Gradient Flows

Finding latent structures in data is drawing increasing attention in diverse fields such as image and signal processing, fluid dynamics, and machine learning. In this work we examine the problem of finding the main modes of gradient flows. Gradient descent is a fundamental process in optimization where its stochastic version is prominent in training of neural networks. Here our aim is to establish a consistent theory for gradient flows $ψ_t = P(ψ)$, where $P$ is a nonlinear homogeneous operator. Our proposed framework stems from analytic solutions of homogeneous flows, previously formalized by Cohen-Gilboa, where the initial condition $ψ_0$ admits the nonlinear eigenvalue problem $P(ψ_0)=λψ_0 $. We first present an analytic solution for \ac{DMD} in such cases. We show an inherent flaw of \ac{DMD}, which is unable to recover the essential dynamics of the flow. It is evident that \ac{DMD} is best suited for homogeneous flows of degree one. We propose an adaptive time sampling scheme and show its dynamics are analogue to homogeneous flows of degree one with a fixed step size. Moreover, we adapt \ac{DMD} to yield a real spectrum, using symmetric matrices. Our analytic solution of the proposed scheme recovers the dynamics perfectly and yields zero error. We then proceed to show that in the general case the orthogonal modes $\{ ϕ_i \}$ are approximately nonlinear eigenfunctions $P(ϕ_i) \approxλ_i ϕ_i $. We formulate Orthogonal Nonlinear Spectral decomposition (\emph{OrthoNS}), which recovers the essential latent structures of the gradient descent process. Definitions for spectrum and filtering are given, and a Parseval-type identity is shown.

math.DS↗

arXiv2019

Supervised and Unsupervised End-to-End Deep Learning for Gene Ontology Classification of Neural In Situ Hybridization Images

In recent years, large datasets of high-resolution mammalian neural images have become available, which has prompted active research on the analysis of gene expression data. Traditional image processing methods are typically applied for learning functional representations of genes, based on their expressions in these brain images. In this paper, we describe a novel end-to-end deep learning-based method for generating compact representations of in situ hybridization (ISH) images, which are invariant-to-translation. In contrast to traditional image processing methods, our method relies, instead, on deep convolutional denoising autoencoders (CDAE) for processing raw pixel inputs, and generating the desired compact image representations. We provide an in-depth description of our deep learning-based approach, and present extensive experimental results, demonstrating that representations extracted by CDAE can help learn features of functional gene ontology categories for their classification in a highly accurate manner. Our methods improve the previous state-of-the-art classification rate (Liscovitch, et al.) from an average AUC of 0.92 to 0.997, i.e., it achieves 96% reduction in error rate. Furthermore, the representation vectors generated due to our method are more compact in comparison to previous state-of-the-art methods, allowing for a more efficient high-level representation of images. These results are obtained with significantly downsampled images in comparison to the original high-resolution ones, further underscoring the robustness of our proposed method.

cs.CV↗

arXiv2019

Introducing the p-Laplacian Spectra

In this work we develop a nonlinear decomposition, associated with nonlinear eigenfunctions of the p-Laplacian for p \in (1, 2). With this decomposition we can process signals of different degrees of smoothness. We first analyze solutions of scale spaces, generated by γ-homogeneous operators, γ \in R. An analytic solution is formulated when the scale space is initialized with a nonlinear eigenfunction of the respective operator. We show that the flow is extinct in finite time for γ \in [0, 1). A main innovation in this study is concerned with operators of fractional homogeneity, which require the mathematical framework of fractional calculus. The proposed transform rigorously defines the notions of decomposition, reconstruction, filtering and spectrum. The theory is applied to the p-Laplacian operator, where the tools developed in this framework are demonstrated. Keywords: Nonlinear spectra, filtering, shape preserving flows, p-Laplacian, nonlinear eigenfunctions.

math.AP↗

arXiv2017

DeepBrain: Functional Representation of Neural In-Situ Hybridization Images for Gene Ontology Classification Using Deep Convolutional Autoencoders

This paper presents a novel deep learning-based method for learning a functional representation of mammalian neural images. The method uses a deep convolutional denoising autoencoder (CDAE) for generating an invariant, compact representation of in situ hybridization (ISH) images. While most existing methods for bio-imaging analysis were not developed to handle images with highly complex anatomical structures, the results presented in this paper show that functional representation extracted by CDAE can help learn features of functional gene ontology categories for their classification in a highly accurate manner. Using this CDAE representation, our method outperforms the previous state-of-the-art classification rate, by improving the average AUC from 0.92 to 0.98, i.e., achieving 75% reduction in error. The method operates on input images that were downsampled significantly with respect to the original ones to make it computationally feasible.

cs.CV↗