In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions and, on the other hand, very rough functions such as the Weierstrass function, which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU neural networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.

Handwriting is considered a unique "fingerprint" that characterizes a scribe (it is even used as evidence in modern forensics). In paleography (the study of ancient writing), it is presumed that each writer has a one prototype for each letter in the alphabet. Commonly, for ancient inscriptions, letters are organized into paleographic tables (where the rows are the alphabet letters, and the columns represent the examined inscriptions). These tables play a significant role in dating inscriptions based on their resemblance to columns in the table. In this paper, we argue that each scribe "fingerprint" is not represented by a single character prototype, but in fact by a distribution of characters. We introduce a framework for automatically identifying the writer style and constructing paleographic tables based on character histograms. Subsequently, we propose a method for comparing short documents utilizing letter distribution. We demonstrate the validity of the methods on two handwritten datasets: Modern and Ancient Hebrew pertaining to the First Temple period. Our methodology on the ancient dataset enables us to provide additional evidence concerning the level of literacy in the kingdom of Judah ca. 600 BCE.

Document binary images, created by different algorithms, are commonly evaluated based on a pre-existing ground truth. Previous research found several pitfalls in this methodology and suggested various approaches addressing the issue. This article proposes an alternative binarization quality evaluation solution for binarized glyphs, circumventing the ground truth. Our method relies on intrinsic properties of binarized glyphs. The features used for quality assessment are stroke width consistency, presence of small connected components (stains), edge noise, and the average edge curvature. Linear and tree-based combinations of these features are also considered. The new methodology is tested and shown to be nearly as sound as human experts' judgments.