PROJECTS
Manifold Approximation
Enhancing the Moving Least-Squares (MLS) mechanism to approximate low dimensional manifolds embedded in a high dimensional space - the Manifold-MLS.
We created a framework, which enables us to operate on a sampled manifold directly (with linear complexity w.r.t the ambient dimension), instead of performing dimension reduction. In this work, we do not assume any knowledge regarding the manifold other than its dimension (Paper). This approach is extended naturally to the approximation of functions over manifold domains (Paper). Furthermore, we show that using this framework one can reconstruct geodesic distances (Paper).
In all three works, we show that the approximation is smooth and has optimal approximation order.
Furthermore, similar to local polynomial regression (i.e., Moving Least-Squares for function approximation) under some assumptions regarding the noise distribution we can show that the algorithm converges in probability when the sample size reaches infinity (in preparation).
In the process of these investigations, we have encountered a fascinating connection between least-squares linear regression and Principal Component Analysis (arxiv).
Computerized Paleography
Studying the development of ancient Hebrew writing dating to First Temple Period. Here I was a part of an interdisciplinary team of researchers tackling various questions related to the subject. Our most prominent publications here were (a) finding an empirical evidence of high literacy rates in the Kingdom of Judah on the eve of Nebuchadnezzar's destruction of Jerusalem (PNAS) ; (b) showing a negative evidence to the dissemination of writing skills 200 years earlier in the northern kingdom of Israel (in preparation); and (c) the discovery of a new inscription on the back side of an existing one, which was situated for half a century in Israel Museum (PLoS One). For many other publications in this subject, which I took part in, see the project's homepage.
Machine Learning in Art Investigation
X-ray images play a crucial role in the process of understanding the way a painting was created as well as locating areas that need to go through preservation/restoration processes. However, X-ray images of artworks, painted on both sides of their support, contain in one image content from both paintings, making them difficult for experts to “read”. We have constructed a self-supervised Deep Neural Network (i.e., without any training data) that learns the separation of the mixed signal. Our results on two panels of the Ghent Altarpiece show spectacular improvement over previous attempts (Science Advances). In order to assist the conservators in their work process, we had to scale up the algorithmic capabilities and devise a more efficient Neural Network architecture (conference paper, journal paper). We are currently working in collaboration with the Belgian Royal Institute for Cultural Heritage (KIK-IRPA) on applying these methodologies on problematic areas of the Ghent Altarpiece. This project is a part of a large collaboration with researchers from the British National Library, University College London and Imperial College London (ARTICT website).
Expressing Fractals via Deep Neural Networks
Fractals originating from Iterative Function Systems (IFS) can be realized through a sequence of piecewise linear functions Fk with a number of linear regions exponential in k (the number of iterations used). We show that, under some mild assumptions, Fk can be represented by a neural network function comprising of O(k) parameters only. It has been shown that the human visual system is locking on self-similarities in images, to some extent. This fact, combined with the approximation scheme of IFS through neural network functions, may give some intuition to the reason of the undoubted success of DNNs in image processing tasks. (Paper)