David Levin

Anat Amir, David Levin

Approximations of non-smooth multivariate functions return low-order approximations in the vicinities of the singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form $f = g + r_+$ where $g,r \in C^{M+1}(\mathbb{R}^n)$ and the function $r_+$ is defined by \[ r_+(y) = \left\{ \begin{array}{ll} r(y), & r(y) \geq 0 \\ 0, & r(y) < 0 \end{array} \right. \ , \ \forall y \in \mathbb{R}^n \ . \] Given scattered (or uniform) data points $X \subset \mathbb{R}^n$, we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.

N. Dyn, O. Kounchev, D. Levin et al.

We investigate non-stationary orthogonal wavelets based on a non-stationary interpolatory subdivision scheme reproducing a given set of exponentials. The construction is analogous to the construction of Daubechies wavelets using the subdivision scheme of Deslauriers-Dubuc. The main result is the smoothness of these Daubechies type wavelets.

David Levin, Gonen Singer

This paper introduces a novel graph-based filter method for automatic feature selection (abbreviated as GB-AFS) for multi-class classification tasks. The method determines the minimum combination of features required to sustain prediction performance while maintaining complementary discriminating abilities between different classes. It does not require any user-defined parameters such as the number of features to select. The methodology employs the Jeffries-Matusita (JM) distance in conjunction with t-distributed Stochastic Neighbor Embedding (t-SNE) to generate a low-dimensional space reflecting how effectively each feature can differentiate between each pair of classes. The minimum number of features is selected using our newly developed Mean Simplified Silhouette (abbreviated as MSS) index, designed to evaluate the clustering results for the feature selection task. Experimental results on public data sets demonstrate the superior performance of the proposed GB-AFS over other filter-based techniques and automatic feature selection approaches. Moreover, the proposed algorithm maintained the accuracy achieved when utilizing all features, while using only $7\%$ to $30\%$ of the features. Consequently, this resulted in a reduction of the time needed for classifications, from $15\%$ to $70\%$.

David Levin, Gonen Singer

In this paper, we present Automatic Complementary Separation Pruning (ACSP), a novel and fully automated pruning method for convolutional neural networks. ACSP integrates the strengths of both structured pruning and activation-based pruning, enabling the efficient removal of entire components such as neurons and channels while leveraging activations to identify and retain the most relevant components. Our approach is designed specifically for supervised learning tasks, where we construct a graph space that encodes the separation capabilities of each component with respect to all class pairs. By employing complementary selection principles and utilizing a clustering algorithm, ACSP ensures that the selected components maintain diverse and complementary separation capabilities, reducing redundancy and maintaining high network performance. The method automatically determines the optimal subset of components in each layer, utilizing a knee-finding algorithm to select the minimal subset that preserves performance without requiring user-defined pruning volumes. Extensive experiments on multiple architectures, including VGG-16, ResNet-50, and MobileNet-V2, across datasets like CIFAR-10, CIFAR-100, and ImageNet-1K, demonstrate that ACSP achieves competitive accuracy compared to other methods while significantly reducing computational costs. This fully automated approach not only enhances scalability but also makes ACSP especially practical for real-world deployment by eliminating the need for manually defining the pruning volume.

Barak Sober, Yariv Aizenbud, David Levin

We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated directly on that point. We prove that our construction yields a smooth function, and in case of noiseless samples the approximation order is $\mathcal{O}(h^{m+1})$, where $h$ is a local density of sample parameter (i.e., the fill distance) and $m$ is the degree of a local polynomial approximation, used in our algorithm. In addition, the proposed algorithm has linear time complexity with respect to the ambient-space's dimension. Thus, we are able to avoid the computational complexity, commonly encountered in high dimensional approximations, without having to perform non-linear dimension reduction, which inevitably introduces distortions to the geometry of the data. Additionaly, we show numerical experiments that the proposed approach compares favorably to statistical approaches for regression over manifolds and show its potential.

Barak Sober, David Levin

This work suggests a new variational approach to the task of computer aided restoration of incomplete characters, residing in a highly noisy document. We model character strokes as the movement of a pen with a varying radius. Following this model, a cubic spline representation is being utilized to perform gradient descent steps, while maintaining interpolation at some initial (manually sampled) points. The proposed algorithm was utilized in the process of restoring approximately 1000 ancient Hebrew characters (dating to ca. 8th-7th century BCE), some of which are presented herein and show that the algorithm yields plausible results when applied on deteriorated documents.