Arad is a well preserved desert fort on the southern frontier of the biblical kingdom of Judah. Excavation of the site yielded over 100 Hebrew ostraca (ink inscriptions on potsherds) dated to ca. 600 ...BCE, the eve of Nebuchadnezzar's destruction of Jerusalem. Due to the site's isolation, small size and texts that were written in a short time span, the Arad corpus holds important keys to understanding dissemination of literacy in Judah. Here we present the handwriting analysis of 18 Arad inscriptions, including more than 150 pair-wise assessments of writer's identity. The examination was performed by two new algorithmic handwriting analysis methods and independently by a professional forensic document examiner. To the best of our knowledge, no such large-scale pair-wise assessments of ancient documents by a forensic expert has previously been published. Comparison of forensic examination with algorithmic analysis is also unique. Our study demonstrates substantial agreement between the results of these independent methods of investigation. Remarkably, the forensic examination reveals a high probability of at least 12 writers within the analyzed corpus. This is a major increment over the previously published algorithmic estimations, which revealed 4-7 writers for the same assemblage. The high literacy rate detected within the small Arad stronghold, estimated (using broadly-accepted paleo-demographic coefficients) to have accommodated 20-30 soldiers, demonstrates widespread literacy in the late 7.sup.th century BCE Judahite military and administration apparatuses, with the ability to compose biblical texts during this period a possible by-product.
Past excavations in Samaria, capital of biblical Israel, yielded a corpus of Hebrew ink on clay inscriptions (ostraca) that documents wine and oil shipments to the palace from surrounding localities. ...Many questions regarding these early 8th century BCE texts, in particular the location of their composition, have been debated. Authorship in countryside villages or estates would attest to widespread literacy in a relatively early phase of ancient Israel's history. Here we report an algorithmic investigation of 31 of the inscriptions. Our study establishes that they were most likely written by two scribes who recorded the shipments in Samaria. We achieved our results through a method comprised of image processing and newly developed statistical learning techniques. These outcomes contrast with our previous results, which indicated widespread literacy in the kingdom of Judah a century and half to two centuries later, ca. 600 BCE.
The relationship between the expansion of literacy in Judah and composition of biblical texts has attracted scholarly attention for over a century. Information on this issue can be deduced from ...Hebrew inscriptions from the final phase of the first Temple period. We report our investigation of 16 inscriptions from the Judahite desert fortress of Arad, dated ca. 600 BCE—the eve of Nebuchadnezzar’s destruction of Jerusalem. The inquiry is based on new methods for image processing and document analysis, as well as machine learning algorithms. These techniques enable identification of the minimal number of authors in a given group of inscriptions. Our algorithmic analysis, complemented by the textual information, reveals a minimum of six authors within the examined inscriptions. The results indicate that in this remote fort literacy had spread throughout the military hierarchy, down to the quartermaster and probably even below that rank. This implies that an educational infrastructure that could support the composition of literary texts in Judah already existed before the destruction of the first Temple. A similar level of literacy in this area is attested again only 400 y later, ca. 200 BCE.
Most surviving biblical period Hebrew inscriptions are ostraca-ink-on-clay texts. They are poorly preserved and once unearthed, fade rapidly. Therefore, proper and timely documentation of ostraca is ...essential. Here we show a striking example of a hitherto invisible text on the back side of an ostracon revealed via multispectral imaging. This ostracon, found at the desert fortress of Arad and dated to ca. 600 BCE (the eve of Judah's destruction by Nebuchadnezzar), has been on display for half a century. Its front side has been thoroughly studied, while its back side was considered blank. Our research revealed three lines of text on the supposedly blank side and four "new" lines on the front side. Our results demonstrate the need for multispectral image acquisition for both sides of all ancient ink ostraca. Moreover, in certain cases we recommend employing multispectral techniques for screening newly unearthed ceramic potsherds prior to disposal.
In order to avoid the curse of dimensionality, frequently encountered in big data analysis, there has been vast development in the field of linear and nonlinear dimension reduction techniques in ...recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower-dimensional manifold; thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real-life applications, data is often very noisy. In this work, we propose a method to approximate
M
a
d
-dimensional
C
m
+
1
smooth submanifold of
R
n
(
d
≪
n
) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located “near” the lower-dimensional manifold and suggest a nonlinear moving least-squares projection on an approximating
d
-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e.,
O
(
h
m
+
1
)
, where
h
is the fill distance and
m
is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension
n
. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high-dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.
Over of the last century, Principal Component Analysis (PCA) has become one of the pillars of modern scientific methods. Although PCA is typically viewed as a statistical tool aiming at finding ...orthogonal directions on which the variance is maximized, its first introduction by Pearson at 1901 was in the framework of the non-linear least-squares minimization problem of fitting a plane to scattered data points. Since linear least-squares regression also fits a plane to scattered data points, PCA and linear least-squares regression have thus a natural kinship, which we explore in this paper. In particular, we present an iterated linear least-squares approach, yielding a sequence of subspaces that converges to the space spanned by the leading principal components. The key observation, by which we establish our result, is that each iteration of the Power (or Subspace) Iterations, applied to the covariance matrix, can be interpreted as a solution to a linear least-squares problem.
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 knowledge about the local geometry of the manifold or its local parameterizations. We do require, however, knowledge regarding the manifold’s intrinsic dimension d. We use the Manifold Moving Least-Squares approach of Sober and Levin (2019) 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 O(hm+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. Additionally, we show numerically that our approach compares favorably to some well-known approaches for regression over manifolds.
This work suggests a new variational approach to the task of computer aided segmentation and restoration of incomplete characters, residing in a highly noisy document image. We model character ...strokes as the movement of a pen with a varying radius. Following this model, in order to fit the digital image, 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 used 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.
•Computer aided image segmentation for handwritten noisy documents.•Combination of techniques pertaining to CAD and image processing.•Gradient descent for B-Splines, which maintains interpolation at predefined points.
To help understand the underlying mechanisms of neural networks (NNs), several groups have studied the number of linear regions <inline-formula> <tex-math notation="LaTeX">\ell ...</tex-math></inline-formula> of piecewise linear (PwL) functions, generated by deep neural networks (DNN). In particular, they showed that <inline-formula> <tex-math notation="LaTeX">\ell </tex-math></inline-formula> can grow exponentially with the number of network parameters <inline-formula> <tex-math notation="LaTeX">p </tex-math></inline-formula>, a property often used to explain the advantages of deep over shallow NNs. Nonetheless, a dimension argument shows that DNNs cannot generate all PwL functions with <inline-formula> <tex-math notation="LaTeX">\ell </tex-math></inline-formula> linear regions when <inline-formula> <tex-math notation="LaTeX">\ell > p </tex-math></inline-formula>. It is thus natural to seek to characterize specific families of functions with <inline-formula> <tex-math notation="LaTeX">\ell > p </tex-math></inline-formula> linear regions that can be constructed by DNNs. Iterated Function Systems (IFS) recursively construct a sequence of PwL functions <inline-formula> <tex-math notation="LaTeX">F_{k} </tex-math></inline-formula> with a number of linear regions which is exponential in <inline-formula> <tex-math notation="LaTeX">k </tex-math></inline-formula>. We show that <inline-formula> <tex-math notation="LaTeX">F_{k} </tex-math></inline-formula> can be generated by a NN using only <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(k) </tex-math></inline-formula> parameters. IFS are used extensively to generate natural-looking landscape textures in artificial images as well as for compression of natural images. The surprisingly good performance of this compression suggests that human visual system may lock in on self-similarities. The combination of this phenomenon with the capacity of DNNs to efficiently approximate IFS may contribute to the success of DNNs, particularly striking for image processing tasks.
Neural Network Approximation of Refinable Functions Daubechies, Ingrid; DeVore, Ronald; Dym, Nadav ...
IEEE transactions on information theory,
2023-Jan., 2023-1-00, Letnik:
69, Številka:
1
Journal Article
Recenzirano
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.
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