INVESTIGATION OF FOURIER FEATURES IN NEURAL NETWORKS AND AN APPLICATION TO STEERING IN MESH NETWORKS by BULUT KUŞKONMAZ Submitted to the Graduate School of Social Sciences in partial fulfilment of the requirements for the degree of Master of Electronics Engineering Sabancı University September 2020 THESIS AUTHOR 2020 c All Rights Reserved ABSTRACT INVESTIGATION OF FOURIER FEATURES IN NEURAL NETWORKS AND AN APPLICATION TO STEERING IN MESH NETWORKS BULUT KUŞKONMAZ Electronics Engineering M.Sc. Thesis, September 2020 Thesis Advisor: Assist. Prof. Dr. Hüseyin Özkan Co-advisor: Prof. Dr. Özgür Gürbüz Keywords: Fourier features, Neural networks, SLFN, Classification, Kernel, Steering, Mesh networks Random Fourier features provide one of the most prominent ways to classify large- scale data sets when the classification is nonlinear. However, Fourier features, in its original proposal, are randomly drawn from a certain distribution and are not opti- mized. In this thesis, we investigate the use of Fourier features by a single hidden layer feedforward neural network (SLFN) and optimize those features (instead of drawing randomly) with several gradient-descent based approaches. The optimized Fourier features are deduced from the radial basis function (RBF kernel), and im- plemented in the hidden layer of the SLFN which is followed by the output layer. The resulting classification accuracy is compared with the results of SVM with RBF kernel. Particularly, (1) we tune the parameters such as the hidden layer size and RBF kernel bandwidth, and (2) test with ten different classification data sets. The introduced SLFN provides substantial computational gains with similar accuracy figures compared to the ones of SVM. We also test our SLFN for steering in wireless mesh networks and observe promising smart steering capabilities. iv ÖZET FOURIER ÖZNİTELİKLERİNİN SİNİR AĞLARI İLE İNCELENMESİ VE ÖRGÜ AĞLARDA BAĞLANTI YÖNLENDİRMEYE UYGULANMASI BULUT KUŞKONMAZ ELEKTRONİK MÜHENDİSLİĞİ YÜKSEK LİSANS TEZİ, EYLÜL 2020 Tez Danışmanı: Assist. Prof. Dr. Hüseyin Özkan İkinci Danışman: Prof. Dr. Özgür Gürbüz Anahtar Kelimeler: Fourier öznitelikleri, Sinir ağları, SLFN, Sınıflandırma, Çekirdek, Bağlantı yönlendirme, Örgü ağları Rastgele Fourier öznitelikleri, sınıflandırma doğrusal olmadığında büyük ölçekli veri kümelerini sınıflandırmanın en belirgin yollarından birini sağlar. Bununla birlikte, orjinal önerisinde Fourier öznitelikleri, belirli bir dağıtımdan rastgele çekilir ve opti- mize edilmez. Bu tezde, Fourier özniteliklerinin tek gizli katmanlı ileri beslemeli sinir ağı (SLFN) ile kullanımını araştırıyor ve bu öznitelikleri (rastgele seçim yerine) çeşitli gradyan-inişi tabanlı yaklaşımlarla optimize ediyoruz. Optimize edilmiş Fourier öznitelikleri, radyal bazlı fonksiyondan (RBF çekirdeği) çıkarılır ve çıkış katmanının takip ettiği SLFN’nin gizli katmanında uygulanır. Ortaya çıkan sınıflandırma doğru- luğu, RBF çekirdeği ile SVM’nin sonuçlarıyla karşılaştırılır. Özellikle, (1) gizli kat- man boyutu ve RBF çekirdek bant genişliği gibi parametreleri ayarlıyoruz ve (2) on farklı sınıflandırma veri seti ile test ediyoruz. Sunulan SLFN, SVM’ye kıyasla benzer doğruluk rakamlarına sahip önemli hesaplama kazançları sağlar. Ayrıca kablosuz ağ ağlarında bağlantı yönlendirme için SLFN’mizi test ediyor ve gelecek vaat eden akıllı bağlantı yönlendirme kabiliyetlerini gözlemliyoruz. v ACKNOWLEDGEMENTS I would like to dedicate my sincere appreciation to my advisor Assist. Prof. Dr. Hüseyin Özkan for his endless support and brilliant suggestions to complete this thesis. I gained the ability to be an engineer thanks to him. I also would like to thank my co-advisor, Prof. Dr. Özgür Gürbüz, for her immense help in both academic life and civil life. She has always been there to give me the right advice whenever I needed it since my undergraduate education. I would like to thank Prof. Dr. Albert Levi, Assist. Prof. Dr. Öznur Taştan, and Assist. Prof. Dr. Erdem Akagündüz for their meticulous evaluation of my thesis. I am grateful to my family, İbrahim, Ayten, and Güneş for their love and support throughout my whole education. I am grateful to my lab mates, Ali, Kutay, Mehmet, and Sandra who make my graduate life at Sabancı University easier and more enjoyable. Special thanks go to my ’Japanese Friends’, who gave me true friendship and love. I also want to thank Berke, who started his university journey with me as a roommate and continued as one of my true friends. Thanks to my close friend Oğuz, whom I studied Electronics Engineering with. I want to thank Volkan for being an amazing roommate and true friend who I can really trust. Finally, heartfelt thanks to the people of Kocaeli Doğa Sporları Kulübü (KODOSK), for making me proud of being a member of KODOSK. This work was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under Contract 118E268. vi Dedicated to my family and friends vii TABLE OF CONTENTS LIST OF TABLES ........................................................... x LIST OF FIGURES ......................................................... xi 1. INTRODUCTION........................................................ 1 1.1. Thesis Contributions...................................................4 1.2. Thesis Organization...................................................5 2. RELATED WORK ....................................................... 6 3. LEARNING OF FOURIER FEATURES WITH A NEURAL NET- WORK ..................................................................... 11 3.1. Random Kernel Expansion............................................ 13 3.2. Learning Fourier Features............................................. 16 3.3. Various Training Approaches for Learning Fourier Features........... 18 3.3.1. Single Layer Learning (SL)..................................... 18 3.3.2. Fourier Feature Selection (FFS)................................ 18 3.3.3. Two Layer Learning (TL)...................................... 19 3.3.4. Batch-Based Two Layer Learning (TL-B)...................... 19 3.3.5. Epoch-Based Two Layer Learning (TL-E)..................... 19 3.4. Experiments........................................................... 19 4. AN APPLICATION OF THE PROPOSED APPROACH: SMART STEERING FOR WIRELESS MESH NETWORKS................ 25 4.1. Introduction........................................................... 25 4.1.1. Related Work.................................................. 28 4.1.2. Chapter Organization.......................................... 29 4.2. Problem Description................................................... 29 4.3. The Proposed Classification Approach for Smart Steering............ 31 4.3.1. Classification Analysis in the Batch Setting................... 33 4.3.2. Online Classification for Real-time Smart Steering............ 35 viii 4.3.2.1. Perceptron in the Randomized Kernel Space: Online Kernel Perceptron.................................... 36 4.4. Experimental Results.................................................. 38 4.4.1. Steering Data.................................................. 39 4.4.2. Results of the Batch Analysis.................................. 40 4.4.3. Results of Online Classification................................ 45 4.5. Discussion.............................................................. 49 5. CONCLUSION ........................................................... 51 BIBLIOGRAPHY............................................................ 52 6. Bibliography............................................................... 52 ix LIST OF TABLES Table 3.1. Benchmark details as provided in [1]............................ 20 Table 3.2. Cross validation results: average bandwidth parameter g (up- per) and number D of units in the hidden layer (lower) with the corresponding standard deviations in each case....................... 21 Table 3.3. Benchmark results of TL, SL, TL-E and TL-B algorithms with CE/MSE loss and minibatch/SGD optimizers on ten different data sets 22 Table 3.4. Comparison of TL algorithm (CE and mini batch) with SVM (rbf kernel) in terms of classification accuracy........................ 23 Table 3.5. Comparison of two layer learning approach, single layer learning approach, single layer learning with chosen Fourier features (FFS) and linear SVM with Fourier features in terms of the mean and standard deviation of accuracy.................................................. 24 Table 4.1. Accuracy results of nonlinear SVM in single AP scenario....... 41 Table 4.2. Accuracy results of nonlinear SVM in the multi AP scenario... 44 Table 4.3. Error rate results of online kernel perceptron in the multi AP scenario................................................................ 48 Table 4.4. Accuracy results of TL learning algorithm in the multi access point (AP) scenario.................................................... 49 x LIST OF FIGURES Figure 3.1. An example of linear classification in (a) and nonlinear classi- fication in (b).......................................................... 11 Figure 3.2. Visual interpretation of kernel trick that allows nonlinear clas- sification with linear techniques....................................... 12 Figure 3.3. Visual representation of our single hidden layer feedforward neural network (SLFN), where d is dimension of the input data in- stance x and D is the size of the hidden layer implementing the map- ping with random Fourier features. The hidden layer activation is si- T D nusoidal producing the Fourier features {cos(wr xt +br)}r=1. Initially, D