Optimal

optimal

Constructing the Aligning Matrix for Optimal ESPRIT Convergence

Strong Eigenvector Comparison for Optimal ESPRIT Convergence

Proof of Optimal Error Scaling for ESPRIT Algorithm

Table of Links Abstract and 1 Introduction 1.1 ESPRIT algorithm and central limit error scaling 1.2 Contribution 1.3 Related work 1.4 Technical overview and 1.5 Organization 2 Proof of the central limit error scaling 3 Proof of the optimal error scaling 4 Second-order eigenvector perturbation theory 5 Strong eigenvector comparison 5.1 Construction of the “good” P 5.2 Taylor expansion with respect to the error terms 5.3 Error cancellation in the Taylor expansion 5.4 Proof of Theorem 5.1 A Preliminaries B Vandermonde matrice C Deferred proofs for Section 2 D Deferred proofs for Section 4 E Deferred proofs for Section 5 F Lower bound for spectral estimation References 1.2 Contribution The main contribution of this paper is a novel analysis of the ESPRIT algorithm, which demonstrates the optimal error scaling of ESPRIT. These findings are summarized in the following theorem: Remark 1.5 (The assumption α > 1). In the above theorem, we assume α > 1 to simplify the form of our bounds. Note that Theorem 1.4 still yields results for the ESPRIT algorithm under small noise and zero noise Ej ≡ 0 since α is only required to be an upper bound on the sub-Gaussian rate of tail decay. Our proof can be extended to provide sharper estimates for small values of α. We present the proof of Theorem 1.6 in Appendix F. This result is similar to the Cramér–Rao bound in signal processing (cf. [SN89]). Due to the different settings, we provide a self-contained proof using the total variation distance between Gaussian distributions. This paper is available on arxiv under CC BY 4.0 DEED license. Authors:
(1) Zhiyan Ding, Department of Mathematics, University of California, Berkeley;
(2) Ethan N. Epperly, Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA;
(3) Lin Lin, Department of Mathematics, University of California, Berkeley, Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, and Challenge Institute for Quantum Computation, University of California, Berkeley;
(4) Ruizhe Zhang, Simons Institute for the Theory of Computing. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 ESPRIT algorithm and central limit error scaling 1.1 ESPRIT algorithm and central limit error scaling 1.2 Contribution 1.2 Contribution 1.3 Related work 1.3 Related work 1.4 Technical overview and 1.5 Organization 1.4 Technical overview and 1.5 Organization 2 Proof of the central limit error scaling 2 Proof of the central limit error scaling 3 Proof of the optimal error scaling 3 Proof of the optimal error scaling 4 Second-order eigenvector perturbation theory 4 Second-order eigenvector perturbation theory 5 Strong eigenvector comparison 5 Strong eigenvector comparison 5.1 Construction of the “good” P 5.1 Construction of the “good” P 5.2 Taylor expansion with respect to the error terms 5.2 Taylor expansion with respect to the error terms 5.3 Error cancellation in the Taylor expansion 5.3 Error cancellation in the Taylor expansion 5.4 Proof of Theorem 5.1 5.4 Proof of Theorem 5.1 A Preliminaries A Preliminaries B Vandermonde matrice B Vandermonde matrice C Deferred proofs for Section 2 C Deferred proofs for Section 2 D Deferred proofs for Section 4 D Deferred proofs for Section 4 E Deferred proofs for Section 5 E Deferred proofs for Section 5 F Lower bound for spectral estimation F Lower bound for spectral estimation References References 1.2 Contribution The main contribution of this paper is a novel analysis of the ESPRIT algorithm, which demonstrates the optimal error scaling of ESPRIT. These findings are summarized in the following theorem: Remark 1.5 (The assumption α > 1). In the above theorem, we assume α > 1 to simplify the form of our bounds. Note that Theorem 1.4 still yields results for the ESPRIT algorithm under small noise and zero noise Ej ≡ 0 since α is only required to be an upper bound on the sub-Gaussian rate of tail decay. Our proof can be extended to provide sharper estimates for small values of α. Remark We present the proof of Theorem 1.6 in Appendix F. This result is similar to the Cramér–Rao bound in signal processing (cf. [SN89]). Due to the different settings, we provide a self-contained proof using the total variation distance between Gaussian distributions. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Authors: (1) Zhiyan Ding, Department of Mathematics, University of California, Berkeley; (2) Ethan N. Epperly, Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA; (3) Lin Lin, Department of Mathematics, University of California, Berkeley, Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, and Challenge Institute for Quantum Computation, University of California, Berkeley; (4) Ruizhe Zhang, Simons Institute for the Theory of Computing. Authors: Authors: (1) Zhiyan Ding, Department of Mathematics, University of California, Berkeley; (2) Ethan N. Epperly, Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA; (3) Lin Lin, Department of Mathematics, University of California, Berkeley, Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, and Challenge Institute for Quantum Computation, University of California, Berkeley; (4) Ruizhe Zhang, Simons Institute for the Theory of Computing.

Optimal Error Scaling of ESPRIT Algorithm Demonstrated

Table of Links Abstract and 1 Introduction 1.1 ESPRIT algorithm and central limit error scaling 1.2 Contribution 1.3 Related work 1.4 Technical overview and 1.5 Organization 2 Proof of the central limit error scaling 3 Proof of the optimal error scaling 4 Second-order eigenvector perturbation theory 5 Strong eigenvector comparison 5.1 Construction of the “good” P 5.2 Taylor expansion with respect to the error terms 5.3 Error cancellation in the Taylor expansion 5.4 Proof of Theorem 5.1 A Preliminaries B Vandermonde matrice C Deferred proofs for Section 2 D Deferred proofs for Section 4 E Deferred proofs for Section 5 F Lower bound for spectral estimation References Abstract 1 Introduction Spectral estimation is a fundamental problem in statistical signal processing. The goal of spectral estimation is to reconstruct fine details of a signal from noisy measurements. Algorithms for spectral estimation have broad applications spanning image/audio processing and compression [GNHB08], acoustics and electromagnetics [KV96, Sch86], geophysics [SSF90], inverse problems [Fan10, FSY10], time series analysis [SM08], spectrometry [SCGM96], machine learning [HBH+18, QGR+22], and quantum computing [Som19, LT22]. [HBH+18, QGR+22], and quantum computing [Som19, LT22]. In this paper, we will consider the following formulation of the spectral estimation problem: Consider a positive spectral measure A. Separation of locations. All dominant locations are separated from each other and from non-dominant locations: It is important to note that separation between non-dominant locations is not required. B. Separation of intensities. We assume the intensities are positive and ordered We assume the cumulative intensity of non-dominant locations is bounded: This setup is directly motivated by the eigenvalue estimation problem on quantum computers, which has received significant attention recently [Som19, SHT22, DTO22, LT22, WFZ+23, WBC22, LNY23, SCS+23, DL23]. We also expect that this setup could be applicable in many fields, such as communication, sensing, and audio processing. In this paper, we focus on the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm [PRK86]. Since its introduction in 1986, the ESPRIT algorithm has proven one of the most effective methods for spectral estimation in practice. The main question of this paper is as follows: The main contribution of this paper is a significant improvement over the central limit error scaling outlined in Theorem 1.2. We establish that ESPRIT achieves the optimal error scaling for estimating dominant locations and dominant intensities (Theorem 1.4). To our knowledge, this is the first demonstration of a noisy super-resolution scaling (in the sense of Definition 1.1) for spectral estimation under comparable assumptions. This paper is available on arxiv under CC BY 4.0 DEED license. [2] Our use of the term “noisy super-resolution scaling” differs from that in [Moi15], where the term refers to classical super-resolution scaling in the presence of small noise. Authors:
(1) Zhiyan Ding, Department of Mathematics, University of California, Berkeley;
(2) Ethan N. Epperly, Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA;
(3) Lin Lin, Department of Mathematics, University of California, Berkeley, Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, and Challenge Institute for Quantum Computation, University of California, Berkeley;
(4) Ruizhe Zhang, Simons Institute for the Theory of Computing. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 ESPRIT algorithm and central limit error scaling 1.1 ESPRIT algorithm and central limit error scaling 1.2 Contribution 1.2 Contribution 1.3 Related work 1.3 Related work 1.4 Technical overview and 1.5 Organization 1.4 Technical overview and 1.5 Organization 2 Proof of the central limit error scaling 2 Proof of the central limit error scaling 3 Proof of the optimal error scaling 3 Proof of the optimal error scaling 4 Second-order eigenvector perturbation theory 4 Second-order eigenvector perturbation theory 5 Strong eigenvector comparison 5 Strong eigenvector comparison 5.1 Construction of the “good” P 5.1 Construction of the “good” P 5.2 Taylor expansion with respect to the error terms 5.2 Taylor expansion with respect to the error terms 5.3 Error cancellation in the Taylor expansion 5.3 Error cancellation in the Taylor expansion 5.4 Proof of Theorem 5.1 5.4 Proof of Theorem 5.1 A Preliminaries A Preliminaries B Vandermonde matrice B Vandermonde matrice C Deferred proofs for Section 2 C Deferred proofs for Section 2 D Deferred proofs for Section 4 D Deferred proofs for Section 4 E Deferred proofs for Section 5 E Deferred proofs for Section 5 F Lower bound for spectral estimation F Lower bound for spectral estimation References References Abstract 1 Introduction Spectral estimation is a fundamental problem in statistical signal processing. The goal of spectral estimation is to reconstruct fine details of a signal from noisy measurements. Algorithms for spectral estimation have broad applications spanning image/audio processing and compression [GNHB08], acoustics and electromagnetics [KV96, Sch86], geophysics [SSF90], inverse problems [Fan10, FSY10], time series analysis [SM08], spectrometry [SCGM96], machine learning [HBH+18, QGR+22], and quantum computing [Som19, LT22]. [HBH+18, QGR+22], and quantum computing [Som19, LT22]. In this paper, we will consider the following formulation of the spectral estimation problem: Consider a positive spectral measure A. Separation of locations . All dominant locations are separated from each other and from non-dominant locations: Separation of locations It is important to note that separation between non-dominant locations is not required. B. Separation of intensities. We assume the intensities are positive and ordered Separation of intensities. We assume the cumulative intensity of non-dominant locations is bounded: This setup is directly motivated by the eigenvalue estimation problem on quantum computers, which has received significant attention recently [Som19, SHT22, DTO22, LT22, WFZ+23, WBC22, LNY23, SCS+23, DL23]. We also expect that this setup could be applicable in many fields, such as communication, sensing, and audio processing. In this paper, we focus on the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm [PRK86]. Since its introduction in 1986, the ESPRIT algorithm has proven one of the most effective methods for spectral estimation in practice. The main question of this paper is as follows: The main contribution of this paper is a significant improvement over the central limit error scaling outlined in Theorem 1.2. We establish that ESPRIT achieves the optimal error scaling for estimating dominant locations and dominant intensities (Theorem 1.4). To our knowledge, this is the first demonstration of a noisy super-resolution scaling (in the sense of Definition 1.1) for spectral estimation under comparable assumptions. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv [2] Our use of the term “noisy super-resolution scaling” differs from that in [Moi15], where the term refers to classical super-resolution scaling in the presence of small noise. Authors: (1) Zhiyan Ding, Department of Mathematics, University of California, Berkeley; (2) Ethan N. Epperly, Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA; (3) Lin Lin, Department of Mathematics, University of California, Berkeley, Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, and Challenge Institute for Quantum Computation, University of California, Berkeley; (4) Ruizhe Zhang, Simons Institute for the Theory of Computing. Authors: Authors: (1) Zhiyan Ding, Department of Mathematics, University of California, Berkeley; (2) Ethan N. Epperly, Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA; (3) Lin Lin, Department of Mathematics, University of California, Berkeley, Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, and Challenge Institute for Quantum Computation, University of California, Berkeley; (4) Ruizhe Zhang, Simons Institute for the Theory of Computing.

Optimal Error Scaling for ESPRIT in Noisy Spectral Estimation

Table of Links Abstract and 1 Introduction 2 Preliminaries 3 Temporal Hierarchies 4 Rating Maps 5 Optimal Imprints for TL(AT) 6 Conclusion and References Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3 Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5 5. Optimal imprints for TL(AT) This paper is available on arxiv under CC BY 4.0 DEED license. Authors:
(1) Thomas Place;
(2) Marc Zaitoun. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminaries 2 Preliminaries 3 Temporal Hierarchies 3 Temporal Hierarchies 4 Rating Maps 4 Rating Maps 5 Optimal Imprints for TL(AT) 5 Optimal Imprints for TL(AT) 6 Conclusion and References 6 Conclusion and References Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3 Appendix A. Appendix to Section 2 & Appendix B. Appendix to Section 3 Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5 Appendix C. Appendix to Section 4 & Appendix D. Appendix to Section 5 5. Optimal imprints for TL(AT) This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Authors: (1) Thomas Place; (2) Marc Zaitoun. Authors: Authors: (1) Thomas Place; (2) Marc Zaitoun.

Optimal Imprints for TL(AT): Explaining Our Main Theorem

Authors:
(1) Jongmin Lee, Department of Mathematical Science, Seoul National University;
(2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Abstract and 1 Introduction 1.1 Notations and preliminaries 1.2 Prior works 2 Anchored Value Iteration 2.1 Accelerated rate for Bellman consistency operator 2.2 Accelerated rate for Bellman optimality opera 3 Convergence when y=1 4 Complexity lower bound 5 Approximate Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 7 Conclusion, Acknowledgments and Disclosure of Funding and References A Preliminaries B Omitted proofs in Section 2 C Omitted proofs in Section 3 D Omitted proofs in Section 4 E Omitted proofs in Section 5 F Omitted proofs in Section 6 G Broader Impacts H Limitations A Preliminaries For notational unity, we use the symbol U when both V and Q can be used. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Jongmin Lee, Department of Mathematical Science, Seoul National University; (2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Authors: Authors: (1) Jongmin Lee, Department of Mathematical Science, Seoul National University; (2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Abstract and 1 Introduction Abstract and 1 Introduction 1.1 Notations and preliminaries 1.1 Notations and preliminaries 1.2 Prior works 1.2 Prior works 2 Anchored Value Iteration 2 Anchored Value Iteration 2.1 Accelerated rate for Bellman consistency operator 2.1 Accelerated rate for Bellman consistency operator 2.2 Accelerated rate for Bellman optimality opera 2.2 Accelerated rate for Bellman optimality opera 3 Convergence when y=1 3 Convergence when y=1 4 Complexity lower bound 4 Complexity lower bound 5 Approximate Anchored Value Iteration 5 Approximate Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 7 Conclusion, Acknowledgments and Disclosure of Funding and References 7 Conclusion, Acknowledgments and Disclosure of Funding and References A Preliminaries A Preliminaries B Omitted proofs in Section 2 B Omitted proofs in Section 2 C Omitted proofs in Section 3 C Omitted proofs in Section 3 D Omitted proofs in Section 4 D Omitted proofs in Section 4 E Omitted proofs in Section 5 E Omitted proofs in Section 5 F Omitted proofs in Section 6 F Omitted proofs in Section 6 G Broader Impacts G Broader Impacts H Limitations H Limitations A Preliminaries For notational unity, we use the symbol U when both V and Q can be used. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Foundational Lemmas for Bellman Optimality and Anti-Optimality Operators

This is Part 6 of a 7-part series of the research paper “Towards Usability Guidelines for the Design of Effective Arabic Websites”. Use the table of links below to navigate to the other parts. Table of Links Abstract and I. Introduction II. Previous Works III. Research Methodology IV. Font and Image Inspection Studies V. Results VI. Discussion VII. Conclusions, Future Research Directions, and References VI. DISCUSSION Our comprehensive inspection of current practices concerning text formatting and image usage within the top visited Arabic websites in Saudi Arabia revealed various interesting findings. The usability inspection was performed to gather insights about the font types and sizes used to style Arabic text, especially that they could directly influence the legibility of text on websites. Let us review and attempt to answer the research questions posited in the introduction. What font types and font sizes are used by major Arabic websites? On the positive side, the selection of the Arabic websites included a large pool of websites, which were selected objectively by relying on Alexa Internet latest ranking. In effect, the inspection covered a total of 73 top viewed websites (52 local and 21 international) in Saudi Arabia. The websites' ranking is based on a set of metrics, primarily the average viewing time for the last three months. Typically, a font has several characteristics including type, size, width, weight, and serifs [37], [46]. However, for the purpose of this research, we directed our focus towards two key font characteristics namely type and size. On the web, font type usually refers to the font family that represents a certain typeface, including all sizes and styles, for example, Arial and Verdana. To style a particular family type, the size of text may be increased or decreased using the font size attribute. There are different ways to resize the font including Ems (em), pixels (px), points (pt), and percent (%). In our font size analysis, we used pixels (px) as the sole measurement for the font since it is the main measurement unit for media shown on computer screens. Moreover, web designers usually use pixels as their key measurement unit for font size. Points, however, are used to measure font size for print media. It is noted that more than half of the Arabic websites breached the principle of consistency when applying font types to the text. This means that multiple font types were used on the same web pages. Previous research, however, has already emphasized the role of consistency of user interface elements, such as font, in the overall usability of websites [35], [36]. On that basis, it is recommended that designers should adhere to the same font family to create attractive and consistent Arabic web designs. Moreover, there was no evident consensus amongst the websites in favoring a particular font type in order to display the Arabic text. More than 35 font types were used by the selected 73 websites. There are various potential theories that could explain this staggering variability; firstly, the lack of research and thereby guidelines with respect to styling Arabic text to guide the effective design of Arabic websites; secondly, the weak conformance to international design practices and recommendations; thirdly, the lack of awareness that text formatting might ultimately impact text legibility and thereby readers' mental load and overall satisfaction. Nonetheless, a pattern of font usage emerged within the Arabic websites; six font types recurred several times in the formatting of titles/headline, menus, and paragraphs. These font types were used differently according to the type of text element. For titles and menus, Droid Arabic Kufi, Tahoma, Helvetica, and Arial, emerged as favorites. It is no surprise that Droid Arabic Kufi came out as a winner for it is a unique font that reflects the Arabic calligraphy. This font gives a true feeling of the traditional Arabic scripts (Kufic scripts) that date back to the 7th century. For paragraphs, Tahoma, Helvetica, Arial and Times New Roman were used more frequently. International recommendations suggest using font type Verdana since it is one of the most legible fonts as rated by online readers of English websites [38]. Nonetheless, this font did not appear frequently in the Arabic websites that we tested. Moreover, other famous fonts that were found earlier to be legible or attractive by English readers, such as Georgia [18] and Courier [10], did not rank well in the Arabic websites. However, Arial emerged as one of the top six font types used by the Arabic websites. Previous research confirmed that font type Arial was read significantly faster and was perceived as legible [10]. Design guidelines have indicated that font size 12 points (~ 16 pixels) is the most suited size for displaying text on English websites [39]. This is somewhat disputable as subsequent research suggested the use of font size 14 points (~19 pixels) instead. For displaying the Arabic text, recommendations are inconclusive yet. For example, [17] proposed to use font 14 points (~19 pixels) whereas [4] argued that font size 16 (22 pixels) to 18 points (~24 pixels) must be used for better readability. These suggestions are in strike contrast with the current practices of the Arabic web design. More than 60% of the inspected websites used font size ranging between 12 pixels to 16 pixels (i.e. 9 points to 12 points) to style the paragraphs. Similarly, menu items are sized from 13 pixels to 14 pixels, which is still considerably illegible [17]. In respect to using images in web design, we aimed at answering the following question. What types of images are incorporated within the Saudi websites given the Saudi culture? All Arabic websites included images as part of their content. Authors in [40] and [41] suggested that graphics improve the overall attractiveness of websites. Moreover, 90% of the Arabic websites we tested included human images. This is known to improve user engagement, trust, and satisfaction [21]. However, 80% of the Arabic websites used conservative images that respect the Islamic identity and cultural values of the Saudi people. Research studies agree that web designs should adhere to the cultural values of its users to achieve success and acceptance [42], [43], [44]. Two-thirds of the websites used images of both genders. The remaining third, however, used images of men only. This could be explained by the masculine dimension of the Saudi society as indicated by Hofstede scores [45]. More than 80% of the websites included images containing groups of people. Again, this can be justified by the collectivism dimension of the Saudi society [45]. Finally, there was high usage of animations, possibly, in an attempt to engage web visitors and improve the aesthetics of the websites. Evidently, the high presence of images in these websites contributed to lower performance scores and increased site loading time. Optimization of images‟ size, using the appropriate image compression techniques, is therefore recommended as a result of our analysis. Authors:
(1) Abdallah Namoun, Department of Information Systems, Faculty of Computer and Information Systems, Madinah, Saudi Arabia;
(2) Ahmad B. Alkhodre, Department of Information Technology, Faculty of Computer and Information Systems, Madinah, Saudi Arabia. This paper is available on arxiv under CC BY 4.0 DEED license. This is Part 6 of a 7-part series of the research paper “Towards Usability Guidelines for the Design of Effective Arabic Websites”. Use the table of links below to navigate to the other parts. This is Part 6 of a 7-part series of the research paper “ Towards Usability Guidelines for the Design of Effective Arabic Websites ”. Use the table of links below to navigate to the other parts. Part 6 Towards Usability Guidelines for the Design of Effective Arabic Websites Table of Links Abstract and I. Introduction Abstract and I. Introduction II. Previous Works II. Previous Works III. Research Methodology III. Research Methodology IV. Font and Image Inspection Studies IV. Font and Image Inspection Studies V. Results V. Results VI. Discussion VI. Discussion VII. Conclusions, Future Research Directions, and References VII. Conclusions, Future Research Directions, and References VI. DISCUSSION Our comprehensive inspection of current practices concerning text formatting and image usage within the top visited Arabic websites in Saudi Arabia revealed various interesting findings. The usability inspection was performed to gather insights about the font types and sizes used to style Arabic text, especially that they could directly influence the legibility of text on websites. Let us review and attempt to answer the research questions posited in the introduction. What font types and font sizes are used by major Arabic websites? On the positive side, the selection of the Arabic websites included a large pool of websites, which were selected objectively by relying on Alexa Internet latest ranking. In effect, the inspection covered a total of 73 top viewed websites (52 local and 21 international) in Saudi Arabia. The websites' ranking is based on a set of metrics, primarily the average viewing time for the last three months. Typically, a font has several characteristics including type, size, width, weight, and serifs [37], [46]. However, for the purpose of this research, we directed our focus towards two key font characteristics namely type and size. On the web, font type usually refers to the font family that represents a certain typeface, including all sizes and styles, for example, Arial and Verdana. To style a particular family type, the size of text may be increased or decreased using the font size attribute. There are different ways to resize the font including Ems (em), pixels (px), points (pt), and percent (%). In our font size analysis, we used pixels (px) as the sole measurement for the font since it is the main measurement unit for media shown on computer screens. Moreover, web designers usually use pixels as their key measurement unit for font size. Points, however, are used to measure font size for print media. It is noted that more than half of the Arabic websites breached the principle of consistency when applying font types to the text. This means that multiple font types were used on the same web pages. Previous research, however, has already emphasized the role of consistency of user interface elements, such as font, in the overall usability of websites [35], [36]. On that basis, it is recommended that designers should adhere to the same font family to create attractive and consistent Arabic web designs. Moreover, there was no evident consensus amongst the websites in favoring a particular font type in order to display the Arabic text. More than 35 font types were used by the selected 73 websites. There are various potential theories that could explain this staggering variability; firstly, the lack of research and thereby guidelines with respect to styling Arabic text to guide the effective design of Arabic websites; secondly, the weak conformance to international design practices and recommendations; thirdly, the lack of awareness that text formatting might ultimately impact text legibility and thereby readers' mental load and overall satisfaction. Nonetheless, a pattern of font usage emerged within the Arabic websites; six font types recurred several times in the formatting of titles/headline, menus, and paragraphs. These font types were used differently according to the type of text element. For titles and menus, Droid Arabic Kufi, Tahoma, Helvetica, and Arial, emerged as favorites. It is no surprise that Droid Arabic Kufi came out as a winner for it is a unique font that reflects the Arabic calligraphy. This font gives a true feeling of the traditional Arabic scripts (Kufic scripts) that date back to the 7th century. For paragraphs, Tahoma, Helvetica, Arial and Times New Roman were used more frequently. International recommendations suggest using font type Verdana since it is one of the most legible fonts as rated by online readers of English websites [38]. Nonetheless, this font did not appear frequently in the Arabic websites that we tested. Moreover, other famous fonts that were found earlier to be legible or attractive by English readers, such as Georgia [18] and Courier [10], did not rank well in the Arabic websites. However, Arial emerged as one of the top six font types used by the Arabic websites. Previous research confirmed that font type Arial was read significantly faster and was perceived as legible [10]. Design guidelines have indicated that font size 12 points (~ 16 pixels) is the most suited size for displaying text on English websites [39]. This is somewhat disputable as subsequent research suggested the use of font size 14 points (~19 pixels) instead. For displaying the Arabic text, recommendations are inconclusive yet. For example, [17] proposed to use font 14 points (~19 pixels) whereas [4] argued that font size 16 (22 pixels) to 18 points (~24 pixels) must be used for better readability. These suggestions are in strike contrast with the current practices of the Arabic web design. More than 60% of the inspected websites used font size ranging between 12 pixels to 16 pixels (i.e. 9 points to 12 points) to style the paragraphs. Similarly, menu items are sized from 13 pixels to 14 pixels, which is still considerably illegible [17]. In respect to using images in web design, we aimed at answering the following question. What types of images are incorporated within the Saudi websites given the Saudi culture? All Arabic websites included images as part of their content. Authors in [40] and [41] suggested that graphics improve the overall attractiveness of websites. Moreover, 90% of the Arabic websites we tested included human images. This is known to improve user engagement, trust, and satisfaction [21]. However, 80% of the Arabic websites used conservative images that respect the Islamic identity and cultural values of the Saudi people. Research studies agree that web designs should adhere to the cultural values of its users to achieve success and acceptance [42], [43], [44]. Two-thirds of the websites used images of both genders. The remaining third, however, used images of men only. This could be explained by the masculine dimension of the Saudi society as indicated by Hofstede scores [45]. More than 80% of the websites included images containing groups of people. Again, this can be justified by the collectivism dimension of the Saudi society [45]. Finally, there was high usage of animations, possibly, in an attempt to engage web visitors and improve the aesthetics of the websites. Evidently, the high presence of images in these websites contributed to lower performance scores and increased site loading time. Optimization of images‟ size, using the appropriate image compression techniques, is therefore recommended as a result of our analysis. Authors: (1) Abdallah Namoun, Department of Information Systems, Faculty of Computer and Information Systems, Madinah, Saudi Arabia; (2) Ahmad B. Alkhodre, Department of Information Technology, Faculty of Computer and Information Systems, Madinah, Saudi Arabia. Authors : Authors (1) Abdallah Namoun, Department of Information Systems, Faculty of Computer and Information Systems, Madinah, Saudi Arabia; (2) Ahmad B. Alkhodre, Department of Information Technology, Faculty of Computer and Information Systems, Madinah, Saudi Arabia. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv

Design Arabic Websites with Consistent Fonts, Optimal Sizes, and Culturally Aligned Images

Authors:
(1) Jongmin Lee, Department of Mathematical Science, Seoul National University;
(2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Abstract and 1 Introduction 1.1 Notations and preliminaries 1.2 Prior works 2 Anchored Value Iteration 2.1 Accelerated rate for Bellman consistency operator 2.2 Accelerated rate for Bellman optimality opera 3 Convergence when y=1 4 Complexity lower bound 5 Approximate Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 7 Conclusion, Acknowledgments and Disclosure of Funding and References A Preliminaries B Omitted proofs in Section 2 C Omitted proofs in Section 3 D Omitted proofs in Section 4 E Omitted proofs in Section 5 F Omitted proofs in Section 6 G Broader Impacts H Limitations 2.2 Accelerated rate for Bellman optimality opera We now present the accelerated convergence rate of Anc-VI for the Bellman optimality operator. Our analysis uses what we call the Bellman anti-optimality operator, define Anc-VI with the Bellman optimality operator exhibits the same accelerated convergence rate as Anc-VI with the Bellman consistency operator. As in Theorem 1, the rate of Theorem 2 also becomes O(1/k) when γ ≈ 1, while VI has a O(1)-rate. Proof outline of Theorem 2. The key technical challenge of the proof comes from the fact that the Bellman optimality operator is non-linear. Similar to the Bellman consistency operator case, we have This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Jongmin Lee, Department of Mathematical Science, Seoul National University; (2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Authors: Authors: (1) Jongmin Lee, Department of Mathematical Science, Seoul National University; (2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Abstract and 1 Introduction Abstract and 1 Introduction 1.1 Notations and preliminaries 1.1 Notations and preliminaries 1.2 Prior works 1.2 Prior works 2 Anchored Value Iteration 2 Anchored Value Iteration 2.1 Accelerated rate for Bellman consistency operator 2.1 Accelerated rate for Bellman consistency operator 2.2 Accelerated rate for Bellman optimality opera 2.2 Accelerated rate for Bellman optimality opera 3 Convergence when y=1 3 Convergence when y=1 4 Complexity lower bound 4 Complexity lower bound 5 Approximate Anchored Value Iteration 5 Approximate Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 7 Conclusion, Acknowledgments and Disclosure of Funding and References 7 Conclusion, Acknowledgments and Disclosure of Funding and References A Preliminaries A Preliminaries B Omitted proofs in Section 2 B Omitted proofs in Section 2 C Omitted proofs in Section 3 C Omitted proofs in Section 3 D Omitted proofs in Section 4 D Omitted proofs in Section 4 E Omitted proofs in Section 5 E Omitted proofs in Section 5 F Omitted proofs in Section 6 F Omitted proofs in Section 6 G Broader Impacts G Broader Impacts H Limitations H Limitations 2.2 Accelerated rate for Bellman optimality opera We now present the accelerated convergence rate of Anc-VI for the Bellman optimality operator. Our analysis uses what we call the Bellman anti-optimality operator, define Anc-VI with the Bellman optimality operator exhibits the same accelerated convergence rate as Anc-VI with the Bellman consistency operator. As in Theorem 1, the rate of Theorem 2 also becomes O(1/k) when γ ≈ 1, while VI has a O(1)-rate. Proof outline of Theorem 2. The key technical challenge of the proof comes from the fact that the Bellman optimality operator is non-linear. Similar to the Bellman consistency operator case, we have This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

How Anc-VI Helps AI Learn Faster with Optimality Operators

Table of Links Abstract and 1. Introduction 2. Preliminaries and 2.1. Blind deconvolution 2.2. Quadratic neural networks 3. Methodology 3.1. Time domain quadratic convolutional filter 3.2. Superiority of cyclic features extraction by QCNN 3.3. Frequency domain linear filter with envelope spectrum objective function 3.4. Integral optimization with uncertainty-aware weighing scheme 4. Computational experiments 4.1. Experimental configurations 4.2. Case study 1: PU dataset 4.3. Case study 2: JNU dataset 4.4. Case study 3: HIT dataset 5. Computational experiments 5.1. Comparison of BD methods 5.2. Classification results on various noise conditions 5.3. Employing ClassBD to deep learning classifiers 5.4. Employing ClassBD to machine learning classifiers 5.5. Feature extraction ability of quadratic and conventional networks 5.6. Comparison of ClassBD filters 6. Conclusions Appendix and References 5.6. Comparison of ClassBD filters Given that ClassBD comprises two filters, we explore their respective contributions in this experiment. Here we test four combinations: a standalone time domain filter (T-filter), a standalone frequency domain filter (F-filter), an F-filter followed by a T-filter, and our proposed scheme (a T-filter followed by an F-filter). The results are presented in Table 15. Primarily, our scheme exhibits superior performance, indicating that both filters contribute significantly to the classification. Secondly, when comparing the single T-filter and F-filter, their performances are found to be dataset-dependent. On the PU and HIT datasets, T-filters outperform F-filters, whereas their performance is inferior to F-filters on the JNU dataset. Authors:
(1) Jing-Xiao Liao, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China and School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China;
(2) Chao He, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China;
(3) Jipu Li, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China;
(4) Jinwei Sun, School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China;
(5) Shiping Zhang (Corresponding author), School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China;
(6) Xiaoge Zhang (Corresponding author), Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Preliminaries and 2.1. Blind deconvolution 2. Preliminaries and 2.1. Blind deconvolution 2.2. Quadratic neural networks 2.2. Quadratic neural networks 3. Methodology 3. Methodology 3.1. Time domain quadratic convolutional filter 3.1. Time domain quadratic convolutional filter 3.2. Superiority of cyclic features extraction by QCNN 3.2. Superiority of cyclic features extraction by QCNN 3.3. Frequency domain linear filter with envelope spectrum objective function 3.3. Frequency domain linear filter with envelope spectrum objective function 3.4. Integral optimization with uncertainty-aware weighing scheme 3.4. Integral optimization with uncertainty-aware weighing scheme 4. Computational experiments 4.1. Experimental configurations 4.1. Experimental configurations 4.2. Case study 1: PU dataset 4.2. Case study 1: PU dataset 4.3. Case study 2: JNU dataset 4.3. Case study 2: JNU dataset 4.4. Case study 3: HIT dataset 4.4. Case study 3: HIT dataset 5. Computational experiments 5.1. Comparison of BD methods 5.1. Comparison of BD methods 5.2. Classification results on various noise conditions 5.2. Classification results on various noise conditions 5.3. Employing ClassBD to deep learning classifiers 5.3. Employing ClassBD to deep learning classifiers 5.4. Employing ClassBD to machine learning classifiers 5.4. Employing ClassBD to machine learning classifiers 5.5. Feature extraction ability of quadratic and conventional networks 5.5. Feature extraction ability of quadratic and conventional networks 5.6. Comparison of ClassBD filters 5.6. Comparison of ClassBD filters 6. Conclusions 6. Conclusions Appendix and References Appendix and References 5.6. Comparison of ClassBD filters Given that ClassBD comprises two filters, we explore their respective contributions in this experiment. Here we test four combinations: a standalone time domain filter (T-filter), a standalone frequency domain filter (F-filter), an F-filter followed by a T-filter, and our proposed scheme (a T-filter followed by an F-filter). The results are presented in Table 15. Primarily, our scheme exhibits superior performance, indicating that both filters contribute significantly to the classification. Secondly, when comparing the single T-filter and F-filter, their performances are found to be dataset-dependent. On the PU and HIT datasets, T-filters outperform F-filters, whereas their performance is inferior to F-filters on the JNU dataset. Authors: (1) Jing-Xiao Liao, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China and School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China; (2) Chao He, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China; (3) Jipu Li, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China; (4) Jinwei Sun, School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China; (5) Shiping Zhang (Corresponding author), School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China; (6) Xiaoge Zhang (Corresponding author), Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China. Authors: Authors: (1) Jing-Xiao Liao, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China and School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China; (2) Chao He, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China; (3) Jipu Li, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China; (4) Jinwei Sun, School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China; (5) Shiping Zhang (Corresponding author), School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin, China; (6) Xiaoge Zhang (Corresponding author), Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong, Special Administrative Region of China. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv

Stories, News & Mentions about OPTIMAL

Categories

Trending Topics

Stories, News & Mentions about OPTIMAL

Optimal’s Stories on HackerNoon

Categories

Trending Topics