Application of Locally Linear Embedding to the Spaces of Social
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Application of Locally Linear Embedding to the Spaces of Social Interactions and Signed Distance Functions Kush R. Varshney Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.881: Representation and Modeling for Image Analysis May 5, 2005 1. Introduction When dealing with high-dimensional data, dimensionality reduction is an important operation to allow the discovery of simple relationships between data points and simple modes in which the data varies. The input to a dimensionality reduction procedure is data in high dimension and the output is a mapping of that data to a low-dimensional manifold. Dimensionality reduction should preserve neighborhoods, i.e. points close to each other in the high-dimensional space should remain close in the computed low-dimensional space. Additionally, it is desirable for a dimensionality reduction method to allow nonlinear low-dimensional manifolds. Nonlinear dimensionality reduction by locally linear embedding (LLE) proposed by Roweis and Saul [1] is such a procedure that is intuitive, simple to implement, and does not involve local minima in its optimization. If the output low-dimensional manifold has two dimensions, intuitively this algorithm connects data points in high-dimensional space to neighboring data points with springs and then presses the spring structure between two glass plates [2]. Another informal description of the algorithm is that it cuts out locally linear swatches of the data in high dimension with scissors and places the swatches down in low dimension in such a way that preserves angles between data points close to each other [3], e.g. deconstructing a soccer ball into its constituent pentagonal and hexagonal patches and laying them out on a table after possibly deforming them. In [1], LLE is applied to documents and to face images, but the technique is restricted neither to a particular application domain nor to a particular domain of data representation within an application. Of key significance is the wide ranging applicability of LLE, because analysis of multivariate data is a universal problem in the sciences. In the work presented here, two application areas will be explored through LLE. After reviewing the algorithm, first document analysis with a social network interpretation will be considered using data from an epic poem. Second, the space of signed distance function (SDF) representations of images, often used in segmentation, will be looked at through topographic and bathymetric data. Some conclusions will also be given. 2. Locally Linear Embedding Algorithm In this section, the details of the simple LLE algorithm will be given, following from [3]. We start with N data points represented as D-dimensional, real-valued vectors xi. The first step of the algorithm is to identify the K nearest neighbors using Euclidean distance in the D-dimensional space for each data point xi. The second step is to determine coefficients wi,j that best reconstruct each xi using a linear combination of the K neighbors xj, where best is in a squared distance sense. This is the ‘locally linear’ part of the algorithm. Specifically, the following cost function is minimized with respect to all wi,j: N K i =1 j =1 J W = ∑ x i − ∑ wi , j x j 2 (1) under the constraint that for each i, the sum of the K weights wi,j be unity. The problem is of the constrained least squares variety and has a simple closed form solution. 1 Once these weights are obtained, the final step of the algorithm can be performed. The goal of dimensionality reduction is to map the D-dimensional xi to d-dimensional yi, where d < D. Thus with the calculated weights fixed, a cost function of the same form as (1) is minimized over the low-dimensional coordinates for yi: N K i =1 j =1 J Y = ∑ y i − ∑ wi , j y j 2 (2) with the following constraints. The value of the above cost function (2) remains unchanged for any translation of the entire set of vectors yi, so one constraint is that the coordinates be centered at the origin. The cost function can be minimized by setting all of the yi to equal the zero vector. To avoid this degenerate solution, another constraint is imposed: the sample covariance of the vectors yi is set to unity. The optimal low-dimensional mapping can then be calculated in closed form by solving an eigenvalue problem for eigenvectors corresponding to the smallest d + 1 eigenvalues. The LLE algorithm is no more complicated than these three steps. 3. Mahābhārata Dimension 4 Documents of text are not simply a ‘bag of words,’ but have much structure at various resolutions, including at the sentence level, the paragraph level, and the section level. When the document is a narrative, with different sections corresponding to different scenes or events, the co-presence of two characters Pujani indicates social interaction 10 Vitahavya between them. Thus, when a Brahmadatta document is a narrative with Pratarddana Divodasa numerous characters, analysis of Manipura the text may also lead to a Vabhruvahana Sulabha 5 Trita Nakula Ulupi Vasus Yavakri Satyasena Chitrangada characterization of the social Raivya Gritsamada Arvavasu Paravasu Puru Dharma Ganga Rama Putrika Upadhyayas Ks hatra Vikrita KalingasSanta Vitahavyas Bhrigu network among the characters, Jataveda Sila Sushena Pramati Shoma Varishtha Ravana Kapila Nimi Nahusha Vibhandaka Swasti Susharma Kasi Jatakarma Candradeva Sudeva Sagara Susruta Sudarsana Rukmi Indra Virupa Arundhati Gayatri Satrunjaya Haihaya Skanda Acharyas Acharya Sraddhas Chitravahana Rudra Aniruddha Puruhuta Ekata Viswamitra Nala Upadhyaya Prakasa Apava Galava Om Nila Mavellak as Haryyaswa Raksha Vasuhoma Nivritti Panchasikha Shakra Sarasvata Haihayas Damayanti Dyu Saryati Vrishadarbha Kumvakarna Samsaptak Oghavati as Suvarna Pitris Kumbhak arna Sakuntala Sachi Yautuka Gotra Brahmanas Bhagiratha Pandu 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Maghavat Kundajathara Srutasravas Namuchi Karandavas Kinkaras Adhvaryus Daivata Charvaka Kalakuta Yayavaras Tittri Suvahu Sadanira Daradas Vahlikas Saumadatti Uddras Kank Paradas as Satakumbha Pannagas Lalitika Madhus Margasirsha Sarasa Gandivas Bhuti Kirtika Madiraksha Kukura Susarmans Dundubhis Bheris Malava Malavi Stana Maninaga Nis Bhisma Galavas Satyakis Drighalochana Kundavedhin Vrihannala Utpalas Dussasana chira Gotras Akshata Yajamana Plakshavatarana Kuhu Vyamas Chekitanas Prahladas Nishkas Vajras Dandas Yugandhara Kas Jambha Swastivachana Saikhavatya Govardhana Chanura Vrindavana Jivala Dhenuk Himavati ya a Angara Dis Utkalas Nivara Kanakas Sighra Pavitra Dharana Surathas Yakshi Gunakesi Bhadrasakha Garbhadhana Skandas Tridiva Vedavati Chitravaha a Rasa Valis Vipulas Tomaras Kashyapa Varuthini Kambhojas Gopali Hiranyapura Vahu Vighasa Yatis Thousand Atman As Gulmas Swastikas Jharjharas Hiranwati Mandala Nagapura Vainyas Mahameru Abhimana Atharvana ta Vallakis Sudarsanas Jayarata Gotamas Vâsudeva Niyamas Uddalaki Udavasa Priyadatta Dasharhas Asma Asmas Vrishala Varhi Bhaikshya Vanaprashtha Vodhya Gargasrota Subhumika Brahmasatris Madhuka Yajaka Mandalas Kuni Satvata Shrutakarma Shatanika Samba Havi Prativindya Pippala wamedha Hara Graha Chitravarman Salabha Amitaujas Duhshala Kukshi Vrihat Kalanemi Matimat Arishta Darada Kshema Gomuk Karus Udvaha hakas ha Rati Madravati Vasishthas Mati Siddhi Ekapada Ekaksha Vrishas Sudeshna Pati Bhartri Durmada Mamata Pradweshi Bhanumati Padmanabha Saha pati Ajaikapat Us anas Yavanas Medha Viswachi Sakuntalas Pushti Vanayu Purvachitti Subhaga Sahajanya Vinasana Gopati Bhadra Durjaya Agneya Jyaishtha Suyodhana Rik Srutakirti Panchali Savaras Agnivesya Gada Kanika Vrihanta Parasaras Durmarshana Patalas Pandava Narmada Duhsasana Uttara Kadamvas Valhika Jaya Satyaka Paripatra Kampana Satakratu Vabhru Brahmars his Mok sha Vichitravirya Dhundhumara Surabhi Purna Maheswara Ghritachi Dhanwantari Puloma Apsaras Garuda Jaratkaru Pushkara Sauri Samvartak Ks Kuthara Samas hemaka a Kas hmira Vankhu Rohitaka Avanti Tuk Kamboja haras Ris Maheswari hikulya Kaliya Mridangas Adhyaryu Satyavans Yuyudhanas Jyotishka Matalis Kundalin Kauravyas Adhidaivata Pramathas Dhanin Kalpa Sloka Salwakas Tumburu Kartaviryas Gandharas Krodhavasas Anjana Vadaris Piyalas Praswapa Akampana Sairindhri Kotika Kapi Sindhusauviras Mek Ghatotkachas Suviras Nirmochana Surashtras alas Vaiswadeva Talajanghas Ditis Tryamvak Sthunakarna Suparsva Hotravahana Prithus Karanas Tamraliptak Chatak Anjalika as Viradyumna Barbaras Kayavya Mahapurus ha Sarmin As Vaisakha win ani Kaunteya Vahurupa Manki Sautramani Plak Karavapana Paurnamasa Sarvamedha Citra shapras ravana Vrishalas Munjakesa Mayura Salya Kaleyas Krodhavasa Virata Arani Viswakarman Mlechchhas Virupak Ambalika sha Sank ara Adhyatma Manohara Purus Atiratha Naks Alamvus hatras ha Prishata Chaidya Supratik Pritivindhya Vrikodara a Savitri Kichakas Viswamitras Kimpuna Kanchis Paundras Uddhava Charudes hna Guha Gana Daitya Salas Sata Suktimati Upanis Saumya Saivya hadas Varanavata Parjanya Prabhasa Govinda Mandara Vishnus Brahman Indras Kartavirya Kunjara Tapana Riks Prabhakara Rishikas Purushottama Chins Kuk Kamandalus Nipas Parshni kuras Saugandhika Subha Apaga Adhibhuta Panavas Sudeshnas Chaitra Vivaswats Tamraliptas Kampanas Champaka Harahunas Gorochana Uk Ashtavakras Jatasuras Riddhi tha watthas Parthas Avantis Vatadhanas Pausha Vinas Madhusudana Pundarikak Ayu Parighas Kovidara Pinak as sha Sula Chandrak Asuri Mansa Vrishakapi Manwantara Pukkasa Naramedha Citrasena Dvaraka Karttaviryya Sahadeva Durdharsha Bhimaratha Vasusena Upanandaka Srutakarman As Kurujangala maka Pinaki Ahivradhna Tumvuru Saddhyas Vira Samvara Vidyutparna Suryavarchas Anavadya Sugandha Angiras Saktri Kiriti Kanyakuvja Sudasa Upayaja Chinas Matsyas Indraprastha Puranas Vak ulas Andhak a Devarata Adityas Kirmira Prajapati Jaimini Hidimva Surya Vinatas Kuntis Tripura Eraka Hayagriva Padma Pragjyotis Uraga Uluk Sudaman Utsava as Sakamvari Gauri Sahya Nalakuveras Anakas Bhagadattas Vrindaraka Prakritis Vilwas Padmas Vrikodaras Aranyakas Gangadwara Saraswatas Mandakas Sarvatobhadra Vahula Cholas Brahmacharyya Jeshthya Vakas Anjanaparvan Jiva Vyasanas Shakraprastha Vajradatta Visakha Ajak Aswagriva Suhma Tuhunda a Aswa Ganesa Dahana Sarpa Huhu Haha Kavya Viswa Vashk Chandrahantri ala Uragas Kalmashapada Bhurisravas Yaja Virochanas Kaurava Mats ya Ahicchatra Vajapeya Devasarman Sudharma Dhananjayas Nivatak Ekachak Kichaka Panchala avachas ra Aswatthamans Karushas Dussasanas Durvasa Lokapalas Ugrasena Srutasena Vadri Kalakeya Kundodara Taraka Romak Dadhicha Sankhyas Ruchiparvan Dhavas Plaks Akritavrana Somadattas has Trinavindu Sakras Kukuras Nalas Vrishabha Vaidehaka Pippalada Indravarman Devas thana Shudra Vimala Sivas Dridharatha Viswas Manojava Vrishaparvan Kunda Amaravati Suvarchas Ganas Nyaya Bhargavas Kausiki Dhumra Viprachitti Krodhahantri Bharadwajas Ramas Marichis Sutasoma Kaveri Hunas Haimavati Vrishni Mantra Yuyutsu Santanu Saunaka Pakshas Richika Asokas Guhyaka Maitreya Draupadis Purochana Sthanu Uddalaka Mahapadma Dwaravati Susarman Vangas Tunda Senapati Krittikas Kasmiras Kins Bhutilaya Vaidic Vas ukas atis Karishini Samangas Vallavas Amvashta Pindas Ruchi Muchukunda Mahendras Pattis Harita Mleccha Vais Yogin es hika Ativala Devala Rajasuya Ek alavya Kundadhara Kapali Somas Chandra Vadava Mahabhutas Pratardana Matanga Suchandra Vikshara Ranku Partha Kundala Kusika Sakas Satyavati Sena Pritana Janardana Kalak eyas Sauti Atirathas Maudgalya Bhumi Sakti Sarabha Katha Kokila Mok Upasunda Sunda shadharma Daruk a Pingala Dhananjaya Dasarnas Praghasa Sarva Kritavarmans Abhiras Himalaya Slokas Kahoda Ramayana Mahishakas Parigha Tanu Himvat Ayogava Rajan Krittimat Anakadundubhi Valhikas Bhishma Naigameya Sakha Nairitas Devika Nagadatta Nandaka Sulochana Asrama Sangraha Jatasura Patti Gulma Chamu Mark andeya Karatoya Paulomas Ghoshayatra Ravi Nara Su Lohita Jayasena Somaka Raja Siva Danavas Maruts As ramas Ugra Sannyasa Urvasi Krakachas Plaksha Jamvuman Hridikas Vaidehas Agrahayana Vaidehi Ravanas Venas Karira Pilu Nupuras Visala Chitra Pradyumna Utathya Kacha Aswasiras Kratu Sinhika Danayu Nikumbha Bhava Keralas Vik arna Yudishthira Diti Yaksha Khandava Panchalas Jaigishavya Prithu Dalvya Dhrishtadyumnas Jata Pratismriti Alambusha Kardama Praruja Kamvala Magadhas Parvana Dhartarashtra Sattwata Aratta Koshalas Chakravakas Sanjaya Sukra Salwas Gandhara Andhakas Sama Marichi Pulaha Kanka Sanatsujata Prajagara Uparichara Duhsaha Vahini Ajagara Dirghalochana Satwata Viduras Jalasandha Kumuda Tanganas Mahanadi Uttamaujas Yudhamanyu Sudakshinas Pishacas Sampati Harsha Dushpradharsha Parasara Sindhu Dwaraka Kiratas Adrisyanti Chitraratha Parasurama Sisupala Sugriva Swetaketu Sarama Kautsa Durvishaha Mandakini Dhumraksha Ananga Ingudas Vetravati Ursa Vivindhya Kshemavriddhi Bhurivala Jaitra Srutanta Gauda Visalaksha Tamra Viravahu Sastras Satadru Pisacha Vaisampayana Punnagas Bhimas Tarkhya Bhimasenas Tapati Satyaki Pitharak a Vaka Pampa Pramesthi Kuntalas Maghavan Rishyamuka Aishika Stri Mausala Saubha Sala Dirghavahu Pramatha Angas Samvarana Chandrabhaga Charmanwati Hridika Jamadagnya Dhrishtadyumna Rishi Vajra Mani Dundubhas Kadru Samva Avindhya Paramatma Vamadevas Kak utsthas Ahank ara Suchimukha Baladeva Naimishas Pandavas Sarduli Matangi Bhadramana Mrigamanda Mrigi Kak Dhritarashtri iSubala Vahvasin Sandilya Arushi Somadatta Kshatriya Vipula Yuyudhana Munis Kohala Suparna Hastinapura Apta Pinjaraka Vamana Nishada Vegavat Sahasrapat Svarga Vritta Kshetrajna Mushkas Raghava Surenu Kanchanakshi Suprava Sarasvatis text. Performing this sort of Sakra Vinda Vali Vitasta Iravati Purana Sree Vrishaka Indrajits Sujata Duhshas ana Gandharvi Salwa Suhasta Raudrakarman Abhaya Anuvinda Angirass Manorama Parnasa Draupadi Elapatra Laks hmi Karavira Kis kindhya Prahrada Vasishta Shakuni Palita Santi Suvarman Sambhava Aranya Pradha Pris hatas Asoka Mahabharata Naimisha Jantu Aruna Kark Dundubha otaka Adi Mahodara Mankanaka Shudras Vasudeva Karna Bhojas Krodha Danu Baka Digvijaya Krimira Visalya Madra Jambu Amba Vandi Dwaita Aswinis Vyasas Upaplavya Ayoda Aryak Aswamedhik Anusasana a a Maushala Trijata Prajna Panasa Tala Visoka Pravriti Vahika Drona Adityaketu Kali Vrihadratha Vinata Indrajit Achala Tilaka Vadari Surasa Kundabhedin Niriti Dhruva Pratyus Dhara ha Swayamvara Kritavarman Virajas Sakala Srutarvan Saraswat Madraka Maya Treta Artha Aurva Purohita Mushik Kasyapas ada Sarasvati Sapta Syeni Suki Adharma Aha Magadha Vena Abhimanyu Aswatthaman Srutayudha Kamyaka Sivi Asramvasika Aswatara Mainda Auddalaka Prahasta Pramohana Arjunaka Iks hvakus Satyajit Dwaipayana Tamas Arjunas Krita Dronas Dwivida Vibhishanas Lakshmanas Ashtaka Kama Sans aptakas Parvas Arattas Panditaka Harivansa Sugrivas Durvimochana Anala Dwapara Akshauhinis Dhaumya Vipasa Atreya Duryodhanas Vrishnis Aparajita Anila Udyoga Kala Aranyaka Vedas Gavakshya Achamana Karma Gautami Yudhis hthiras Mahaprasthanika Dadhimukha Subhadra Arjuna Sattwa Madrakas Pauloma Balarama Mahadeva Akshauhini Krishnas Tara Hidimba Sauptika Dala Samana Vyana Rajas Vyasa Parana analysis, however, makes sense Udana Madras Samanta Sindhus Airavata Aruni Apana Kansa Ghatotkacha Yadavas Prana Alayudha Jayadratha Matali Dimvaka Mathura Vami Kavachas Nivata only if there are hundreds or Shalya Vahikas Hansa thousands of characters. In Vamadeva Parva Taks haka Veda -5 actual networks of social Aswins Janamejaya interaction, the requirement of Paushya having a large number of people is met without difficulty, but -10 there are few texts with Upamanyu arasandha hundreds, let alone thousands, of -5 0 5 10 15 characters. The Mahābhārata, an Dimension 2 Figure 1: Rāmāyaõa characters in the low-dimensional manifold. epic poem popularly touted as the longest poem in the world with about 110,000 verses, does have thousands of characters and will be used as a source of data. It will be treated as a surrogate for actual social interaction data. 2 Pujani 10 Vitahavya Pratarddana Divodasa Brahmadatta Manipura Vabhruvahana Sulabha TritaNakula Ulupi Vasus Yavakri Satyasena Chitrangada Raivya Grits amada Arvavasu Paravasu Dharma Rama Puru Putrika Upadhyayas Ganga Kshatra Vikrita Kalingas Santa Bhrigu Vitahavyas Jataveda Sila Varishtha Shoma Pramati Sushena Nimi Swasti Vibhandaka Ravana Nahusha Kapila Susharma Candradeva Jatakarma Kas iSagara Susruta Sudarsana Sudeva Indra Satrunjaya Skanda Virupa Rukmi Gayatri Arundhati Ekata Puruhuta Acharyas Acharya Rudra Sraddhas Haihaya Chitravahana Prakasa Vis wamitra Nala Om Sarasvata Vasuhoma Shakra Nivritti Panchasik ha Raksha Galava Haryyaswa Mavellakas Upadhyaya Nila Aniruddha Apava Saryati Vrishadarbha Kumvakarna Samsaptakas Oghavati Haihayas Damayanti Dyu Dwita Pandu Brahmanas Gotra Pitris Prisni Yautuk a Bhagiratha Kumbhakarna Sakuntala Sachi Suvarna Mitradeva Vagindra Tama Sravas Vitatya Vihavya Sutejas Mitravarman Vrihadyumna Iravat Nilas Sthulakesa Iks hvaku Mrityu Marutta Saryatis Sanat Samvarta Mahishmati Samanas Agni Dhupas Dhritarashtra Parama Aryyaman Arvudas Dhundhu Gopalas Varsha Garhapatya Lomasa Ratha Vaisyas Devavridha Vahnni Nidhi Yayati Devas Jajali Dasaratha Kuvalas wa Ajamidhas Prayaga Drahyu Turvasu Anu Dushmanta Surendra Aniruddhas Raka Pushpotkata Surpanakha Pragjyotish Srutayus Agastya Yama Sringin Vis warupa Nandini Narada Bhimasena Chitragupta Valikhilya Prisnigarbha Atris Vidya Seduka Sank ritis Samsaptaka Mandapala Tundikeras Vrihaspati Homa Mithila Dadhica Narayana Yakshas Vats Chandala a Nagas Yadu Visrava Kshupa Krisa Devaraja Vudvuda Swaha Krama Kalinga Prachina Niyama Pesi Kalala Oghavat Pravritti Kulinda Pushpak a Kritavirya Dasarathas Payoshni Talajangha Kiritin Uchchaisravas Varshas Yuvaraja Satyakarman Satyeshu Soma Satyaratha Sudras Amravati Bhojakata Satya Lakshmana Vaisravana Sruti Bhuvana Vaishampayana Damana Sita Satrughna Mada Yuvanaswa Dushana Pundarikas Avidya Ashtavakra Vrishaparva Hotri Jarita Yudhisthira Parvata Suk anya Vahuka Hiranyavindu Kulindas Ayodhya Udapana Rakshasas Sastra Sudha Daks Manu ha Namuci Srimat Bahlika Mitras Hiranyahas Satadyumna Ghatotkaca Gandiva aha Rak shas a Vandin Sthulasiras Sankriti Rituparna Dama Ansuman Dravida Sannyasins Phalgunas Srutayus Gosava ha ta Sraddha Swadha Vrihadaswa Chyavana Kanwa Rathas Dadhichi Vivaswat Meru Yojana Saudasa Hari Chayana Valakaswa Ilwala Upanishad Ekaratra Kaikeya Dattatreya Kirata Amvas htha Indradyumna Prasthala Jyotis htoma Saurashtra Rukmin Amrita Ritwik Jamvaka Kasyapa Menaka Arsha Paisacha Surasenas Aditi Nirukta Vashat Ayutayush Alambus Sruvavati Brahma Chaturhotra Chandrayana Ansumat Tuladhara Syumarasmi Maharatha Sukratu Manthara Kandarpa Tashtri Sachis Krishnâ Lanka Chhandas Abhis hahas Brahmacharya Gandhamadana Rishyasringa Bhagadatta Rig Sudymna Vaiswanara Purujit Kasa Gokarna Vatsabhumi Pramadvara Vibhishana Malini Naraka Viswedevas Chekitana Kratha Brahmas Yuga Modakas Agama Apas adas Munjaprishtha Karttika Agastyas Nakulas Achyutayus Sumitras Promathin Vajravega Swaitya Krathas Vajapeyas Hemakuta Brihaspati Vatsas Jamvudwipa Shik handi Sarnga Medhatithi Puruk Vrishasena utsa Lapita Brahmans Raibhya Sunaka Satasringa Tittiri Rajasuyas Akriti Kausikas Yajurveda Valadeva Ambika Trisiras Saras wati Mahat Rigveda Madri Satayupa Sahasrachitya Nasatya Sipivishta Satyavat Shalva Pancala Vishwak Vidhi Bhurishrava sena Yaudheyas Mlechchha Suhotra Prithudaka Janakas Naishadha Mahismati Vasatiya Sravastha Chaturmasyas Khara Kaikaya Hiranyak Sakadwipa a Janaka Mahisha Jahnavi Devasena Aks Gopas Pavana hara Surparaka Kuruks hetra Prakriti Kriti India Prativindhyas Jamadagnis Hastinapore Kaikeyas Karusha Chedi Kinnaras Nalak Saubhadra Hiranyakasipu Nishadha Aswapati Bhishmak uvera Suvira Hutasana Dyutimat Yajamanas Dandadhara Satyadhriti Jishnu Phalguna Madayanti Agnishtoma Kanchana Kuru Samantapanchaka Candramas Jatavedas Syamaka Kshara Pradhana Bhima Samadhi Chitta Gandharvas Bhagas Krisara Ishana Deva Dhritavarman Vidura Vaishnavi Prachetas as Lik Dasra hita Sharadvata Mlechechas Anima Kalki Kakutstha Jamvu Dhannushaksha Medhavi Rikshavat Nrigas Surashtra Ujjayanta Charu Ketumala Jamadagni Dundubhi Danta Sagaras Sishupala Kapataroman Kaikeyi Makaras Anarta Venna Nishadhas Kalamra Adhirathas Amvashthas Patala Sakradeva Kshemadhurti Sandili Adbhuta Kamandalu Siks Pragjyotishas Karnas Bhishmakas ha Aus Dandakas inari Dussala Agniswattas Sumati Sudra Kusa Brahmacharins Havyavaha Pavak Dilipa Nishatha Hrishik Manasa Chitrabhanu Harivarsha es a Viswakrit Ajas Muru Charanas Pandyas Chedis Rohini Mahasena Kalika Ritwijas Kusasthali Mahendra Prasenajit Rukmini Varchas Nanda Vardhakshemi Adhiratha Suprabha Vis hvamitra Somapa Vadarapachana Rishis Vaiswanaras Sabha Prajapatis Gandhari Duryodhana Kurus Mann Upas Virincha Savyasaci ruti Sudyumna Vaikartana Shantanu Rantidevas Amurtarayas Srinjaya Namas Brahmanic Nishadas Chamasodbheda Kalanjara Dandaka Narayanas Tarantuk a Vrishasenas Kins Adri Sasada Asamanjas uk a Vriddhakshatra Elavrita Pitha Mura Chitras enas Bhagirathi Sattwatas Mandhata Daityasena Drupada Kuvera Madiraswa Ars htishena Vyaghradatta Harischandra Dwilipa Haya Bhuridyumna Prishadaswa Paundrayas Airavatas Manas Kshudrakas Kalmasha Suhmas Navarashtra Sahadevas Durvasas Viswakarma Sunitha Anenas Viswagaswa Ayus Mandhatri Bhrigus Havyavahana Samichi Lata Divya Vudha Mainaka Viswadevas Dasarhas Suratha Satyavatis Santanus Chakravaka Girivraja Vrihatkshatra Dantavakra Atharvan Yojanas Danda Mahabhisha Manimat Palas Valarama Sank Himalayan as arshana Somakas Sasin Prabha Mahes hvara Kraunca Vrishi Hiranyagarbha Maheshwara Harinaswa Dhundumara Bhumisaya Amurtarya Ik Pritha shwak Vas is htha Pancharatra Niruk tas Baranasi Laghima Tarakaksha Maharshis Sumana Suta Rantideva Mlechcha Hayi Raghu Anaranya Dhrishtaketu Viradhanwan Vahuda Maricha Valadhi Medhavis Maniman Arantuka Atiratra Samans Idhmavaha 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Naimishas Rishi Pandavas Mausala Angas Aishika Stri Ahankara Paramatma Vamadevas Jamadagnya Kakutsthas Avindhya Kshatriya Kshetrajna Suchimukha Dhrishtadyumna Samva Svarga Has tinapura Vajra Baladeva Mani Kohala Samvarana Nishada Hridika Vamana Chandrabhaga Suparna Sala Dundubhas Dirghavahu Kadru Sarduli Matangi Bhadramana Mrigamanda Mrigi Saubha Somadatta Arus Kaki Dhritarashtri Vahvasin hi Charmanwati Vipula Pramatha Palita Sakra Draupadi Munis Purana Mushkas Duhshas ana Yuyudhana Sujata Raghava Subala Indrajits Vas ishta Vrishaka Apta Vritta Pinjaraka Vegavat Vinda Vitasta Parnas Vali Iravati aala Sree Gandharvi Lakshmi Sahasrapat Angirass Sandilya Mahabharata Naimisha Santi Sambhava Aranya Shudras Prahrada Salwa Shakuni Mankanak a Kisk indhya Jantu Adi Karavira Mahodara Elapatra Karkotaka Anuvinda Asoka Aruna Dundubha Manorama Pradha Suhasta Raudrakarman Abhaya Suvarman Prishatas Vas udeva Karna Drona Pravriti Madra Vahika Baka Digvijaya Krimira Tala Trijata Prajna Panasa Jambu Amba Vandi Aswinis Upaplavya Vyasas Dwaita Vadari Achala Anus Aswamedhika Maushala Visoka Ayoda asana Bhojas Aryaka Vrihadratha Visalya Vinata Danu Krodha Adityaketu Saraswat Sapta Artha Kritavarman Sak Treta Srutarvan Swayamvara Kali Indrajit Tilaka Virajas Surasa Dhruva Dhara Pratyusha Niriti Kundabhedin Maya Arjunaka Magadha Sarasvati Madrak a Abhimanyu Aswatthaman Vena Auddalaka Prahasta Pramohana Sivi Purohita Aurva Asramvas Kamyaka ik ahthiras Mushikada Aswatara Kasyapas Mainda Srutayudha Aha Adharma Syeni Suki Dwaipayana Laks Vibhishanas hmanas Arjunas Tamas Dwivida Ikshvakus Satyajit Dronas Kama Sansaptakas Arattas Parvas Krita As htaka Akshauhinis Dwapara Sugrivas Dhaumya Harivansa Anala Panditak a Durvimochana Karma Gavakshya Achamana Duryodhanas Vrishnis Aparajita Atreya Anila Kala Vipasa Vedas Udyoga Aranyaka Mahapras Yudhis thanika Dadhimukha Gautami Subhadra Arjuna Madrakas Sattwa Balarama Pauloma Tara Hidimba Akshauhini Mahadeva Sauptika Krishnas Dala Samana Rajas Vyana Vyasa Parana Udana Madras Samanta Aruni Airavata Sindhus Ghatotkacha Apana Kansa Alayudha Prana Yadavas Matali Jayadratha Dimvaka Mathura Kavachas VamiShalya Nivata Vahikas Hansa Parva Veda Taks hakaVamadeva Aswins Janamejaya Paushya 5 0 Dimension 4 The Mahābhārata is in the Sanskrit language, but an English translation [4] is used in this work for two primary reasons. First, it is significantly easier to automatically identify proper nouns in English than Sanskrit because English has capital and lowercase letters. Second, nouns in English are not declined based on their function within a sentence, remaining invariant to usage. In this work, names of individual people, of groups of people, and of places will be considered. -5 -10 Jarasandha Upamanyu -15 -20 -25 Utanka Dimension 4 The epic is divided into eight-20 -15 -10 -5 0 5 10 15 20 Dimension 3 een books, each subdivided Figure 2: Two sets of groups in the low-dimensional manifold. into varying numbers of sections. Overall, there are 2,110 sections, which will be treated as the events of social interaction. In the English translation, there are 10,854 distinct capitalized words, of which 7,296 were identified as proper 0 nouns. Of the 7,296 proper Dhaumya nouns, 3,980 appear just once Aruni and 3,316 appear more than Veda -5 once. These 3,316 names are Aswins the data points that will be used Janamejaya Paushya in a 2,110 dimensional space. For each dimension (section), -10 the data is a count of how Upamanyu many times the name appears in that section. This type of -15 data is very similar to the text example with encyclopedia articles in [1]. Some results are -20 presented below with the number of neighbors K = 24. -25 Utanka -30 -15 -10 -5 0 Dimension 3 5 10 Figure 3: Characters related to the narration of the Mahābhārata. 3 15 In Fig. 1, the data points, the names, are plotted as they have been mapped to dimensions two and four of the computed In Fig. 2, dimensions two and three are plotted, with two groups of names highlighted. One set is shown in detail in Fig. 3 and the other in Fig. 4. The names in Fig. 3, in green, are not characters in the main Mahābhārata story either, but lead to the story being narrated. Dhaumya is a teacher with students Aruni, Veda, and Upamanyu. Dhaumya tests Upamanyu with the help of the Aswins. Utanka is a student of Veda who is tested with the help of Paushya. Utanka goes on to prompt the telling of the 1 Mlechchha Mlechechas Mlechcha Mlecchas 0 Mlechchas Mlechchhas Mleccha Aratta Pishacas Shakuni Salwa Vahika Sakala Madraka -1 Dimension 4 Arattas Madrakas -2 Madras -3 Shalya -4 Vahikas -5 -1 0 1 2 Dimension 3 3 4 5 Figure 4: Places, groups, and people from the west. 15 Rama Utanka Kumbhakarna Lakshmana Ravana 10 Sita Vitahavya Rakshasa 5 Dimension 2 low-dimensional space. The figure illustrates that LLE is neighborhood preserving and maintains global properties as follows. The names highlighted in red, Rama, Ravana, Kumbhakarna, Lakshmana, Sita, Sugriva, Surpanakha, etc. are all characters of another epic poem, the Rāmāyaõa, that precedes the Mahābhārata chronologically and is recounted at various points by various characters, but is not part of the actual Mahābhārata story. Not surprisingly, all of these names are kept close together in the low-dimensional space. A main character in the Rāmāyaõa, Hanuman, is immortal and is encountered by the protagonists of the Mahābhārata in one instance. Correspondingly, the Hanuman data point, highlighted in green, is closer to the central mass of points, which are connected with the central narrative. 0 Yayati Puru -5 Pratarddana Divodasa Dushana Dasaratha Sugriva Raghu Pavana Vibhishana Upamanyu Sumitras Sumitra Videha Pus Promathin Vajravega hpaka Paushya Lanka Kaikeyi Surpanak Jatayu Vasu ha Kumvakarna Aswins Ayodhya Rakshasas Gritsamada Kakutstha Kishkindhya Khara Kurukshetra Nala Satrughna Sulabha Hanuman Parva Kausalya Visrava Raivya Bhrigu Janamejaya Mahodara Paravasu Yavakri Raghus Arvavas u Vitahavyas Manthara Yojanas Mithila Sudhanwan Sitas Raka Pushpotkata Shatrughna Matali Prithudaka Kapalamochana Tak shak a Maricha Nivata Vaisravana Vahuk a Agnis htoma Haihaya Dhumrak Nandigrama sha Virochana Sudeva Dhundhu Kausalyas Raks ha Arjuna Rushangu Kuvalaswa Kasi Dandaka Panchasik ha Kavachas Sthuna Damayanti Krauncha Aruni Atiratra Nalakuvera Varishtha Haryyaswa Janak a Haihayas Drupada Sarabhanga VarshaArundhati Nagas Kauravya 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Asuras Kans aa Rishis Prithu Adhibhuta Twenty Devala Dushmanta Yadavas Dimvak Ulupi Udana Sakuntala Adhyatma Vyana Kapila Ansuman Siva Yajnavalkya Bhava Mathura Vabhruvahana Maheswara Purusha Prak riti Rudra Narayana Brahma Apana Prana Manipura Hansa Tandi Brahmadatta Pujani Mahadeva Sagara -10 Jarasandha -15 -15 -10 -5 0 Dimension 1 5 10 Figure 5: First two dimensions of the low-dimensional manifold. 4 15 Mahābhārata to Janamejaya. These names are also separated from the main conglomeration and fall along one axis. The English translation is inconsistent in the transliteration of Mleccha, a group of people from the west in Persia and Greece, but nevertheless, all of the different instances fall in the lowdimensional manifold near each other and near others originating in the west in Fig. 4 in red. Salwa and Shakuni are both kings from the west. The Pishacas, Arattas, Vahikas, and Madrakas are all people from the west, from places such as Afghanistan. The Madrakas are from the kingdom of Madra, with capital Sakala and king Shalya. This grouping occurs despite the fact that Salwa, Shakuni, and Shalya have extremely different roles in the story and are not copresent much of the time. Thus LLE is able to discover modes of variation in the data that are not readily apparent otherwise, in this case, direction of origin. All of these names are important to the story and thus are part of the central mass of points. An interesting feature of the computed low-dimensional space is the data point mapped closest to the origin in the first two dimensions shown in Fig. 5, Mahameru. It is a mountain that is purported to be the cosmic axis or the center of the physical and metaphysical universe. It is mapped close to the origin in the other dimensions as well. Thus, this dataset of the Mahābhārata text, as a surrogate for social interaction data, illustrates that the use of LLE preserves both local and global structure. Even when the form of the structure is not known in advance, by viewing the low-dimensional space, it is possible to find properties within the dataset that group data points and separate data points. 4. Earth Relief The signed distance function is a powerful way to implicitly represent a curve, often used in segmentation algorithms [5]. (A curve partitions a plane into two regions, with one designated the inside and the other the outside.) One intuitive way to describe signed distance functions is through analogy with Earth elevation, specifically the Hawaiian Islands [6]. Elevation data 8000 6000 1000 4000 2000 2000 0 3000 -2000 -4000 4000 -6000 5000 -8000 1000 2000 3000 4000 5000 6000 Figure 6: 2-Minute Gridded Global Relief Data (meters) 5 7000 8000 9000 10000 -10000 above sea level is known as topography and depth data below sea level is known as bathymetry; collectively, these data are known as relief. In the case of Earth relief, the two regions are the above water portion – the inside, and the underwater portion – the outside. Thus in Hawaii, an island that sticks up out of the water is the inside region. By convention, the SDF is negative inside and positive outside, counter to relief data. The space of signed distance functions is not linear because the sum of two signed distance functions is not a signed distance function [7]. In this section, low-dimensional manifolds will be determined using nonlinear dimensionality reduction by LLE for relief and for negated signed distance functions using data points consisting of small patches in coastal areas of the Earth. The dataset used for Earth relief is the 2-Minute Gridded Global Relief Data [8], which has resolution of two minutes in both latitude and longitude. There are sixty minutes per degree and 360 total degrees of longitude on the earth, giving 10,800 columns of pixels. There are 180 total degrees of latitude and thus 5,400 rows in the dataset. Elevation is given in meters, as shown in Fig. 6. From this data, 2000 data points – 48 pixel × 48 pixel patches – were selected in the manner now described. A row and a column were selected uniformly and the 48 pixel × 48 pixel subimage with that row and column as its bottom left corner was examined. Patches not having at least 20% above water and at least 20% underwater pixels were rejected. The process was continued until 2000 points were accepted. The resulting data points are shown in Fig. 7, with Figure 7: Southwest corners of 2,000 data points marked as +. the crosses indicating the southwest corner of the patch. Examples of 48 × 48 patches are shown in the left column of Fig. 8. The top row is from a data point in Hawaii, the center row is from a data point in Antarctica highlighted with a red circle in Fig. 7, and the bottom row is from a data point in Southeast Asia highlighted with a magenta circle in Fig. 7. The relief data was treated as 48·48 = 2,304 dimensional for the purposes of LLE. For each relief data point, an SDF representation was created as follows using a fast marching method [9]. In order to reduce boundary effects, first a larger patch of relief was taken for each 6 10 2000 0 0 -10 -2000 -20 -4000 -30 20 1000 10 500 0 0 -10 2000 20 10 1000 0 0 -10 Figure 8: Relief data points (left column), binary image (center column), and SDF data points (right column). data point. Specifically, 96 × 96 patches concentric with the 48 × 48 data points were used. For each larger patch, the relief was converted to a binary image demarcating the inside region and outside region. From the binary image, an SDF was generated using an implementation of the fast marching method [10]. Finally, the SDF image was clipped to be 48 × 48 and negated. Example 48 × 48 binary images and 48 × 48 negated signed distance functions are shown in the second and third columns of Fig. 8, respectively. Comparing the true relief with the SDF, one can see that the two are not dissimilar, but there are differences. By definition, the SDF has constant slope, whereas in the true relief, the slope may be variable. Also, any ridges and valleys within a region that the true relief may have will not be captured by the SDF, as seen in the bottom row of Fig. 8. When the relief is primarily linear, then correspondence with the SDF is fairly good, as seen in the first two examples of Fig. 8. Now, the low-dimensional manifolds will be compared as computed for K = 6. The first two dimensions of the relief data manifold are plotted in Fig. 9a and the first two dimensions of the SDF manifold are plotted in Fig. 9b as scatter plots. The colors in the relief data manifold are simply assigned linearly based on the dimension 1 coordinate. The color for each data point is transferred to the corresponding data point in the SDF manifold. It is apparent that there is a correlation between the two manifolds in the first dimension – as the first dimension coordinate 7 (a) (b) Figure 9: Low-dimensional manifolds for (a) relief data and (b) SDF data. in the relief manifold increases, the first dimension coordinate in the SDF manifold decreases. This inverse relationship is also apparent in Fig. 10, which plots the sorted dimension one coordinates of the relief manifold against the dimension one coordinates of the corresponding data points in the SDF manifold. Thus, despite certain features not being captured by the SDF representation, the primary mode of variation in both sets of data is similar. The relationship in the second dimension of the low-dimensional manifolds is not as strong as in the first direction, but exists nonetheless. Fig. 11 shows this relationship in the same manner as Fig. 9. The distribution of colors in Fig. 11b, the SDF manifold, is certainly not random, but the pattern is more difficult to discern. Another way to investigate the similarity of the two manifolds is to consider pairwise distances between the mappings of the data points. There may be metrics more or less appropriate for use in manifolds determined by LLE, but standard Euclidean distance was used in this case on the first three dimensions of each set. In the ideal case, pairwise distance in the relief manifold would be proportional to pairwise distance in the SDF manifold, so plotting them against each other would yield a straight line. Doing so with the actual pairwise distances, shown in Fig. 12 8 An attempt will now be made to discover the underlying properties in the data that are the principal modes of variation. A variety of different local descriptors were investigated to determine whether they are 3 Corresponding Dimension 1 Coordinate for Signed Distance Function with the least-squares fit cubic function overlaid, does not give a straight line, but encouragingly, does give a monotonic trend. The conclusion from this analysis is that data points that are mapped close to each other in the relief manifold are also mapped close to each other in the SDF manifold. 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -3 -2 -1 0 1 2 3 Dimension 1 Coordinate for Relief Data (sorted) 4 5 Figure 10: Correspondence between dimension 1 of relief and SDF low-dimensional manifolds. (a) (b) Figure 11: Low-dimensional manifolds for (a) relief data and (b) SDF data. 9 associated with the dimensions of the manifold. Two local descriptors were notable: the percentage of above water pixels in the image patch and the orientation of the gradients within the image patch. Distance between corresponding pairs in signed distance function representation As shown in Fig. 13, the second and third dimensions of the SDF manifold are extremely correlated to the proportion of pixels that are above water. The coloration in the figure shows the percentage of above water pixels in each data point. Red data points are 80% above water and blue data points are 20% above water. This relationship is very obvious from the figure. As demonstrated above, the connection between the SDF manifold and the relief manifold in dimension two exists but is not very strong. Consequently, the effect of the ratio of underwater pixels to above water pixels on the 10 second and third dimension 9 coordinates in the relief manifold is much less. As seen in Fig. 14, 8 above water proportion is 7 associated with how the data points get mapped to the manifold 6 but it does not seem to be the 5 principal effect. 4 The second local descriptor found to have a strong relationship with the low-dimensional manifolds is 2 the orientation of the gradients. A 1 contour map for an example data point is shown for both the relief 0 0 2 4 6 8 10 12 14 16 18 Pairwise distances with true relief representation (sorted) data and for the SDF in the first Figure 12: Correspondence between pairwise distances. column of Fig. 15. The center column illustrates the gradient in an enlarged part of the image and the right column shows a histogram of the gradient directions with bins of width 30 degrees. The distribution of orientations in the relief image and the SDF image has the same general shape, but the distribution in the SDF image is peakier, due to the lack of ridges and valleys within a region that exist in relief. The full histogram provides a rich collection of local descriptors, but in the analysis below, the feature that is considered is the orientation bin with maximal count. 3 Fig. 16 demonstrates that the first dimension of the SDF manifold is related to the gradient orientation. The data points in the scatter plots are colored according to the maximum histogram bin. As dimension one in the SDF manifold increases, the maximum orientation completes a full cycle of angles counterclockwise. As the first dimensions of the SDF manifold and relief manifold are inversely proportional, the relief manifold’s first dimension changes with gradient angle as well, but clockwise as the first dimension coordinate increases. This trend is shown in Fig. 17 using the same color scheme as Fig. 16. Thus we have seen that two simple properties in the image patches can be used to label the axes in the low-dimensional manifolds. The signed distance function has many fewer degrees of 10 Figure 13: SDF manifold as a function of the proportion of above water pixels. 11 Figure 14: Relief manifold as a function of the proportion of above water pixels. freedom than relief data, so the correlation with the maximum gradient angle and especially with the number of above water pixels is much stronger. However, despite the differences in representation, the same properties principally affect how the data points get mapped to lower 12 dimension. In some sense these properties are like scaling and rotation, and do not say much about the actual shape. That these two properties dominate is not surprising, but is counter to the intuition of what the space of shape should be. That is not to say that the actual form of the shape plays no role. Examining the histogram in ways other than looking at the maximal bin, for 500 22 10 400 24 20 300 26 30 28 200 40 30 100 10 20 30 40 30 35 40 0 -165-105-45 15 75 135 degrees (y direction reversed in images) 1200 10 20 22 1000 24 800 26 30 28 40 30 10 20 30 40 30 600 400 200 35 40 0 -165-105-45 15 75 135 degrees (y direction reversed in images) Figure 15: Examples of gradient angle histogram for relief data (top row) and SDF data (bottom row). example, will give more insight to that role. In future work, it would be interesting to normalize the proportion of underwater pixels and to rotate all image patches to a common maximum angle before applying LLE. It may also be interesting to perform LLE on a scale invariant feature transform [11] of the image patches. 5. Conclusion Nonlinear dimensionality reduction by locally linear embedding has been applied to social interaction data and to a space of signed distance function images along with relief images. Local and global structure in the low-dimensional space of social interactions coincides with what is expected intuitively. Additionally, modes of variation in the data are illuminated that would not have otherwise been seen. The dimensionality reduction approach seems like a 13 Figure 16: SDF manifold as a function of the maximum gradient angle. 14 Figure 17: Relief manifold as a function of the maximum gradient angle. 15 promising way to construct or study social networks as an alternative to using graphs that connect two characters if they have an interaction [12]. Dimensionality reduction and specifically nonlinear dimensionality reduction by locally linear embedding is a versatile technique applicable to all forms of high-dimensional data analysis. In comparing the low-dimensional manifolds produced by LLE for the relief data and the SDF data, it was shown that there is a good amount of correspondence. Thus, the analogy of equating the Hawaiian Islands to signed distance functions is fairly accurate. In addition, two simple local descriptors were found that are very closely related to the different axes in the low-dimensional manifolds. These two descriptors are tied to scaling and rotation; thus, it seems as though the structure of the space of signed distance functions is defined in large part by scaling and rotation, which are shape preserving, rather than by features more descriptive of shape. If manifolds arranged by shape properties are to be obtained through dimensionality reduction on the space of signed distance functions, it seems that scale and orientation will have to first be normalized. References [1] Sam T. Roweis and Lawrence K. Saul, “Nonlinear dimensionality reduction by locally linear embedding,” Science, vol. 290, pp. 2323-2326, 22 Dec. 2000. [2] Sam T. Roweis, 6.881 guest lecture, Cambridge, Massachusetts, Feb. 22, 2005. [3] Lawrence K. Saul and Sam T. Roweis, “An introduction to locally linear embedding,” unpublished. Available at: http://www.cs.toronto.edu/~roweis/lle/publications.html. [4] Kisari Mohan Ganguli, The Mahabharata of Krishna-Dwaipayana Vyasa, Calcutta: Bharata Karyalaya Press, 1883-1896. [5] Vicent Caselles, Ron Kimmel, and Guillermo Sapiro, “Geodesic active contours,” International Journal of Computer Vision, vol. 22, pp. 61-79, 1997. [6] Alan S. Willsky, personal conversations, Sept. 2004 – present. [7] Müjdat Çetin, Junmo Kim, and Alan S. Willsky, personal conversations, Sept. 2004 – present. [8] National Geophysical Data Center, “2-minute gridded global relief data (ETOPO2).” Available at http://www.ngdc.noaa.gov/mgg/global/. [9] James A. Sethian, “A fast marching level set method for monotonically advancing fronts,” Proceedings of the National Academy of Sciences, vol. 93, pp. 1591-1595, 1996. [10] Walter Sun and Venkat Chandar, “FMMgenDistMap.” [11] David G. Lowe, “Distinctive image features from scale-invariant keypoints,” International Journal of Computer Vision, vol. 60, pp. 91-110, 2004. [12] Duncan J. Watts, Small Worlds: The Dynamics of Networks between Order and Randomness, Princeton: Princeton University Press, 1999. 16
