Clustering with Same-Cluster Queries
|
|
|
- Isabel Conley
- 5 years ago
- Views:
Transcription
1 Clustering with Same-Cluster Queries Hassan Ashtiani, Shai Ben-David, Shrinu Kushagra David Cheriton School of Computer Science University of Waterloo Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
2 Motivation Clustering is the task of automatically partitioning a set of instances into meaningful (?!) subsets Well-separated subsets Coherent subsets... In contrast to supervised learning, the goal is not clear/unique! Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
3 Motivation Clustering is the task of automatically partitioning a set of instances into meaningful (?!) subsets Well-separated subsets Coherent subsets... In contrast to supervised learning, the goal is not clear/unique! Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
4 Motivation How does your fancy clustering method perform here? Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
5 Motivation How does your fancy clustering method perform here? Under-specificity is a real issue in clustering We need domain knowledge Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
6 Challenges of Clustering Under-specificity Various modelling choices... with contradicting outcomes A systematic approach to incorporate domain knowledge? High Computational Complexity Optimizing K-means, K-median, K-center,... objective functions is NP-hard Seemingly orthogonal, but actually related! Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
7 Acquiring Domain Knowledge Trial-and-error/Intuitions Off-line Models Constrained Clustering (Wagstaff et. al (2000)) Demonstration-based Clustering (Ashtiani and Ben-David (2015)) Interactive Clustering Merge/split Model (Balcan and Blum (2008)) Our approach Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
8 Our Approach Learner interacts with an expert/oracle to get advice Same-Cluster Query Do x 1 and x 2 belong to the same cluster? A natural/user-friendly form of query Record De-Duplication Assisted Troubleshooting... Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
9 Key Takeaways Interactive clustering in the form of same-cluster queries can help us deal with Under-specificity Computational complexity (?!!) Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
10 Key Takeaways Interactive clustering in the form of same-cluster queries can help us deal with Under-specificity Computational complexity (?!!) The use of a few queries can make an otherwise NP-hard clustering problem tractable! Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
11 Problem Setting Input is X = {x} n i=1 Rd Learner can ask same-cluster queries from the oracle The goal is to recover the oracle s clustering C = (C1,..., C k ) n and d could be very large Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
12 Problem Setting Input is X = {x} n i=1 Rd Learner can ask same-cluster queries from the oracle The goal is to recover the oracle s clustering C = (C1,..., C k ) n and d could be very large Still need more structure/inductive-bias... No-free-lunch in clustering Need to ask Ω(n) queries! Target C is nice Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
13 γ-margin Property C = {C 1,..., C k } with centers {µ 1, µ 2,..., µ k } satisfies the γ-margin property if for all x C i and y X \ C i, (1 + γ)d(x, µ i ) < d(y, µ i ) Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
14 γ-margin Property Query complexity? Computational complexity? Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
15 Computational Complexity Special case: oracle s clustering is the solution of K-means Learner just wants to find K-means optimal solution min {µ} k j=1 x X min j x µ j 2 2 Fixed objective, no under-specificity issue Additional structure: γ-margin property Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
16 Computational Complexity Special case: oracle s clustering is the solution of K-means Learner just wants to find K-means optimal solution min {µ} k j=1 x X min j x µ j 2 2 Fixed objective, no under-specificity issue Additional structure: γ-margin property Theorem Solving Euclidean K-means clustering with no queries is NP-hard if γ Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
17 Computational Complexity Without queries: NP-hard for realistic values of γ Tractable only for unrealistically large values of γ Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
18 Positive Result (with Queries) Theorem There is an algorithm that finds C for any γ > 0 (with constant probability) which Runs in O(knd) Asks O(k 2 log k + k log n) queries Works for any nice target Query complexity is Dimension-independent! Only logarithmic in n Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
19 Surprising Conclusion For the realistic situation 0 γ 0.84 Clustering is NP-hard without queries BUT tractable with a small number of queries Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
20 Algorithm x R Algorithm s Idea 1. Estimate a center Compute distance of all points to the green point. Sort them. = = = =... Query uniformly till we have enough points from one cluster Use binary search to find the separation. 2. Prune points belonging to that cluster Binary search to find the effective radius 3. Repeat for the other clusters Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
21 Algorithm x R Compute distance of all points to the green point. Sort them. = = = =... No need to know k or γ in advance Use binary search to find the separation. Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
22 Hardness Result Euclidean k-means is NP-hard even when the optimal solution satisfies the γ-margin property for γ < True even with O(log n) same-cluster queries Reduction from Exact Cover by 3-Sets Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
23 Summary Clustering is an under-specified problem Domain knowledge can be conveyed using interaction An efficient algorithm for interactive clustering Same-cluster queries can also reduce computational complexity An NP-hard clustering problem became tractable using a few queries Can we exploit same-cluster queries in other settings? Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
24 Thank You! Ashtiani, Ben-David, Kushagra Interactive Clustering December 5, / 20
Representation Learning for Clustering: A Statistical Framework
Representation Learning for Clustering: A Statistical Framework Hassan Ashtiani School of Computer Science University of Waterloo mhzokaei@uwaterloo.ca Shai Ben-David School of Computer Science University
Relaxed Oracles for Semi-Supervised Clustering
Relaxed Oracles for Semi-Supervised Clustering Taewan Kim The University of Texas at Austin twankim@utexas.edu Joydeep Ghosh The University of Texas at Austin jghosh@utexas.edu Abstract Pairwise same-cluster
Approximate Correlation Clustering using Same-Cluster Queries
Approximate Correlation Clustering using Same-Cluster Queries CSE, IIT Delhi LATIN Talk, April 19, 2018 [Joint work with Nir Ailon (Technion) and Anup Bhattacharya (IITD)] Clustering Clustering is the
Approximate Clustering with Same-Cluster Queries
Approximate Clustering with Same-Cluster Queries Nir Ailon 1, Anup Bhattacharya 2, Ragesh Jaiswal 3, and Amit Kumar 4 1 Technion, Haifa, Israel nailon@cs.technion.ac.il 2 Department of Computer Science
John Oliver from The Daily Show. Supporting worthy causes at the G20 Pittsburgh Summit: Bayesians Against Discrimination. Ban Genetic Algorithms
John Oliver from The Daily Show Supporting worthy causes at the G20 Pittsburgh Summit: Bayesians Against Discrimination Ban Genetic Algorithms Support Vector Machines Watch out for the protests tonight
Lecture 5 Finding meaningful clusters in data. 5.1 Kleinberg s axiomatic framework for clustering
CSE 291: Unsupervised learning Spring 2008 Lecture 5 Finding meaningful clusters in data So far we ve been in the vector quantization mindset, where we want to approximate a data set by a small number
CS229 Final Project: k-means Algorithm
CS229 Final Project: k-means Algorithm Colin Wei and Alfred Xue SUNet ID: colinwei axue December 11, 2014 1 Notes This project was done in conjuction with a similar project that explored k-means in CS
Measures of Clustering Quality: A Working Set of Axioms for Clustering
Measures of Clustering Quality: A Working Set of Axioms for Clustering Margareta Ackerman and Shai Ben-David School of Computer Science University of Waterloo, Canada Abstract Aiming towards the development
Theoretical Foundations of Clustering few results, many challenges
Theoretical Foundations of Clustering few results, many challenges Shai Ben-David University of Waterloo MLSS, Beijing, June 2014 High level view of (Statistical) Machine Learning The purpose of science
Testing Bipartiteness of Geometric Intersection Graphs David Eppstein
Testing Bipartiteness of Geometric Intersection Graphs David Eppstein Univ. of California, Irvine School of Information and Computer Science Intersection Graphs Given arrangement of geometric objects,
Module 8: Range-Searching in Dictionaries for Points
Module 8: Range-Searching in Dictionaries for Points CS 240 Data Structures and Data Management T. Biedl K. Lanctot M. Sepehri S. Wild Based on lecture notes by many previous cs240 instructors David R.
Security Control Methods for Statistical Database
Security Control Methods for Statistical Database Li Xiong CS573 Data Privacy and Security Statistical Database A statistical database is a database which provides statistics on subsets of records OLAP
CS264: Beyond Worst-Case Analysis Lecture #6: Perturbation-Stable Clustering
CS264: Beyond Worst-Case Analysis Lecture #6: Perturbation-Stable Clustering Tim Roughgarden January 26, 2017 1 Clustering Is Hard Only When It Doesn t Matter In some optimization problems, the objective
CS264: Homework #4. Due by midnight on Wednesday, October 22, 2014
CS264: Homework #4 Due by midnight on Wednesday, October 22, 2014 Instructions: (1) Form a group of 1-3 students. You should turn in only one write-up for your entire group. (2) Turn in your solutions
arxiv: v1 [cs.cg] 8 Jan 2018
![arxiv: v1 [cs.cg] 8 Jan 2018](/thumbs/76/73575871.jpg "arxiv: v1 [cs.cg] 8 Jan 2018")
Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon Eunjin Oh Hee-Kap Ahn arxiv:1801.02292v1 [cs.cg] 8 Jan 2018 Abstract Given a set of sites in a simple polygon, a geodesic Voronoi diagram
On the Max Coloring Problem
On the Max Coloring Problem Leah Epstein Asaf Levin May 22, 2010 Abstract We consider max coloring on hereditary graph classes. The problem is defined as follows. Given a graph G = (V, E) and positive
Using the Filter, Field Sets, and Sort Order Menus
Using the Filter, Field Sets, and Sort Order Menus The Filter menu is used to determine the group of records that appear on the page, such as all active students or former students The Field Sets menu
Geometric data structures:
Geometric data structures: Machine Learning for Big Data CSE547/STAT548, University of Washington Sham Kakade Sham Kakade 2017 1 Announcements: HW3 posted Today: Review: LSH for Euclidean distance Other
Efficient Case Based Feature Construction
Efficient Case Based Feature Construction Ingo Mierswa and Michael Wurst Artificial Intelligence Unit,Department of Computer Science, University of Dortmund, Germany {mierswa, wurst}@ls8.cs.uni-dortmund.de
Computing intersections in a set of line segments: the Bentley-Ottmann algorithm
Computing intersections in a set of line segments: the Bentley-Ottmann algorithm Michiel Smid October 14, 2003 1 Introduction In these notes, we introduce a powerful technique for solving geometric problems.
Ranking Clustered Data with Pairwise Comparisons
Ranking Clustered Data with Pairwise Comparisons Alisa Maas ajmaas@cs.wisc.edu 1. INTRODUCTION 1.1 Background Machine learning often relies heavily on being able to rank the relative fitness of instances
A Framework for Statistical Clustering with a Constant Time Approximation Algorithms for K-Median Clustering
A Framework for Statistical Clustering with a Constant Time Approximation Algorithms for K-Median Clustering Shai Ben-David Department of Computer Science Technion, Haifa 32000, Israel and School of ECE
Semi- Supervised Learning
Semi- Supervised Learning Aarti Singh Machine Learning 10-601 Dec 1, 2011 Slides Courtesy: Jerry Zhu 1 Supervised Learning Feature Space Label Space Goal: Optimal predictor (Bayes Rule) depends on unknown
Semi-supervised learning and active learning
Semi-supervised learning and active learning Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Combining classifiers Ensemble learning: a machine learning paradigm where multiple learners
K-Nearest Neighbour (Continued) Dr. Xiaowei Huang
K-Nearest Neighbour (Continued) Dr. Xiaowei Huang https://cgi.csc.liv.ac.uk/~xiaowei/ A few things: No lectures on Week 7 (i.e., the week starting from Monday 5 th November), and Week 11 (i.e., the week
26 The closest pair problem
The closest pair problem 1 26 The closest pair problem Sweep algorithms solve many kinds of proximity problems efficiently. We present a simple sweep that solves the two-dimensional closest pair problem
LEARNING CIRCUITS AND NETWORKS BY INJECTING VALUES. Lev Reyzin Yale University
LEARNING CIRCUITS AND NETWORKS BY INJECTING VALUES Lev Reyzin Yale University PAPERS Dana Angluin, James Aspnes, Jiang Chen, and Yinghua Wu. Learning a Circuit by Injecting Values. In Proceedings of the
Randomized Algorithms: Selection
Randomized Algorithms: Selection CSE21 Winter 2017, Day 25 (B00), Day 16 (A00) March 15, 2017 http://vlsicad.ucsd.edu/courses/cse21-w17 Selection Problem: WHAT Given list of distinct integers a 1, a 2,,
Clustering. Unsupervised Learning
Clustering. Unsupervised Learning Maria-Florina Balcan 03/02/2016 Clustering, Informal Goals Goal: Automatically partition unlabeled data into groups of similar datapoints. Question: When and why would
Clustering. So far in the course. Clustering. Clustering. Subhransu Maji. CMPSCI 689: Machine Learning. dist(x, y) = x y 2 2
So far in the course Clustering Subhransu Maji : Machine Learning 2 April 2015 7 April 2015 Supervised learning: learning with a teacher You had training data which was (feature, label) pairs and the goal
Constructing Hidden Units using Examples and Queries
Constructing Hidden Units using Examples and Queries Eric B. Baum Kevin J. Lang NEC Research Institute 4 Independence Way Princeton, NJ 08540 ABSTRACT While the network loading problem for 2-layer threshold
Computational Geometry
Range searching and kd-trees 1 Database queries 1D range trees Databases Databases store records or objects Personnel database: Each employee has a name, id code, date of birth, function, salary, start
Clustering Algorithms. Margareta Ackerman
Clustering Algorithms Margareta Ackerman A sea of algorithms As we discussed last class, there are MANY clustering algorithms, and new ones are proposed all the time. They are very different from each
The Parameterized Complexity of Finding Point Sets with Hereditary Properties
The Parameterized Complexity of Finding Point Sets with Hereditary Properties David Eppstein University of California, Irvine Daniel Lokshtanov University of Bergen University of California, Santa Barbara
Microsoft Office Access 2007: Intermediate Course 01 Relational Databases
Microsoft Office Access 2007: Intermediate Course 01 Relational Databases Slide 1 Relational Databases Course objectives Normalize tables Set relationships between tables Implement referential integrity
A Computational Theory of Clustering
A Computational Theory of Clustering Avrim Blum Carnegie Mellon University Based on work joint with Nina Balcan, Anupam Gupta, and Santosh Vempala Point of this talk A new way to theoretically analyze
Clustering. Subhransu Maji. CMPSCI 689: Machine Learning. 2 April April 2015
Clustering Subhransu Maji CMPSCI 689: Machine Learning 2 April 2015 7 April 2015 So far in the course Supervised learning: learning with a teacher You had training data which was (feature, label) pairs
Clustering. SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic
Clustering SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic Clustering is one of the fundamental and ubiquitous tasks in exploratory data analysis a first intuition about the
Lecture 2 The k-means clustering problem
CSE 29: Unsupervised learning Spring 2008 Lecture 2 The -means clustering problem 2. The -means cost function Last time we saw the -center problem, in which the input is a set S of data points and the
KD-Tree Algorithm for Propensity Score Matching PhD Qualifying Exam Defense
KD-Tree Algorithm for Propensity Score Matching PhD Qualifying Exam Defense John R Hott University of Virginia May 11, 2012 1 / 62 Motivation Epidemiology: Clinical Trials Phase II and III pre-market trials
Clustering. Chapter 10 in Introduction to statistical learning
Clustering Chapter 10 in Introduction to statistical learning 16 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 1 Clustering ² Clustering is the art of finding groups in data (Kaufman and Rousseeuw, 1990). ² What
Computational Geometry
Windowing queries Windowing Windowing queries Zoom in; re-center and zoom in; select by outlining Windowing Windowing queries Windowing Windowing queries Given a set of n axis-parallel line segments, preprocess
Clustering Stable Instances of Euclidean k-means
Clustering Stable Instances of Euclidean k-means Abhratanu Dutta Northwestern University adutta@u.northwestern.edu Aravindan Vijayaraghavan Northwestern University aravindv@northwestern.edu Alex Wang Carnegie
Instance-Optimal Geometric Algorithms
Instance-Optimal Geometric Algorithms Timothy Chan School of CS U. Waterloo joint work with Peyman Afshani (MADALGO, Aarhus U) Jérémy Barbay (U. Chile) Theme Beyond worst-case analysis Adaptive algorithms
Introduction to Randomized Algorithms
Introduction to Randomized Algorithms Gopinath Mishra Advanced Computing and Microelectronics Unit Indian Statistical Institute Kolkata 700108, India. Organization 1 Introduction 2 Some basic ideas from
Shuheng Zhou. Annotated Bibliography
Shuheng Zhou Annotated Bibliography High-dimensional Statistical Inference S. Zhou, J. Lafferty and L. Wasserman, Compressed Regression, in Advances in Neural Information Processing Systems 20 (NIPS 2007).
9.1 Cook-Levin Theorem
CS787: Advanced Algorithms Scribe: Shijin Kong and David Malec Lecturer: Shuchi Chawla Topic: NP-Completeness, Approximation Algorithms Date: 10/1/2007 As we ve already seen in the preceding lecture, two
6.856 Randomized Algorithms
6.856 Randomized Algorithms David Karger Handout #4, September 21, 2002 Homework 1 Solutions Problem 1 MR 1.8. (a) The min-cut algorithm given in class works because at each step it is very unlikely (probability
Lecture and notes by: Nate Chenette, Brent Myers, Hari Prasad November 8, Property Testing
Property Testing 1 Introduction Broadly, property testing is the study of the following class of problems: Given the ability to perform (local) queries concerning a particular object (e.g., a function,
Fast Hierarchical Clustering via Dynamic Closest Pairs
Fast Hierarchical Clustering via Dynamic Closest Pairs David Eppstein Dept. Information and Computer Science Univ. of California, Irvine http://www.ics.uci.edu/ eppstein/ 1 My Interest In Clustering What
A Dendrogram. Bioinformatics (Lec 17)
A Dendrogram 3/15/05 1 Hierarchical Clustering [Johnson, SC, 1967] Given n points in R d, compute the distance between every pair of points While (not done) Pick closest pair of points s i and s j and
3. Data Preprocessing. 3.1 Introduction
3. Data Preprocessing Contents of this Chapter 3.1 Introduction 3.2 Data cleaning 3.3 Data integration 3.4 Data transformation 3.5 Data reduction SFU, CMPT 740, 03-3, Martin Ester 84 3.1 Introduction Motivation
Ranking Clustered Data with Pairwise Comparisons
Ranking Clustered Data with Pairwise Comparisons Kevin Kowalski nargle@cs.wisc.edu 1. INTRODUCTION Background. Machine learning often relies heavily on being able to rank instances in a large set of data
2. Data Preprocessing
2. Data Preprocessing Contents of this Chapter 2.1 Introduction 2.2 Data cleaning 2.3 Data integration 2.4 Data transformation 2.5 Data reduction Reference: [Han and Kamber 2006, Chapter 2] SFU, CMPT 459
Introduction to Clustering
Introduction to Clustering Ref: Chengkai Li, Department of Computer Science and Engineering, University of Texas at Arlington (Slides courtesy of Vipin Kumar) What is Cluster Analysis? Finding groups of
Niceness Assumptions for Learning Algorithms
Niceness Assumptions for Learning Algorithms by Shrinu Kushagra A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer
Logic and Discrete Mathematics. Section 2.5 Equivalence relations and partitions
Logic and Discrete Mathematics Section 2.5 Equivalence relations and partitions Slides version: January 2015 Equivalence relations Let X be a set and R X X a binary relation on X. We call R an equivalence
Towards Robust Indexing for Ranked Queries
Towards Robust Indexing for Ranked Queries Dong Xin Chen Chen Jiawei Han Department of Computer Science University of Illinois at Urbana-Champaign Urbana, IL, 61801 {dongxin, cchen37, hanj}@uiuc.edu ABSTRACT
Compressing Social Networks
Compressing Social Networks The Minimum Logarithmic Arrangement Problem Chad Waters School of Computing Clemson University cgwater@clemson.edu March 4, 2013 Motivation Determine the extent to which social
Introduction to spectral clustering
Introduction to spectral clustering Vasileios Zografos zografos@isy.liu.se Klas Nordberg klas@isy.liu.se What this course is Basic introduction into the core ideas of spectral clustering Sufficient to
Computational Geometry
Windowing queries Windowing Windowing queries Zoom in; re-center and zoom in; select by outlining Windowing Windowing queries Windowing Windowing queries Given a set of n axis-parallel line segments, preprocess
Clustering. Unsupervised Learning
Clustering. Unsupervised Learning Maria-Florina Balcan 11/05/2018 Clustering, Informal Goals Goal: Automatically partition unlabeled data into groups of similar datapoints. Question: When and why would
Locality- Sensitive Hashing Random Projections for NN Search
Case Study 2: Document Retrieval Locality- Sensitive Hashing Random Projections for NN Search Machine Learning for Big Data CSE547/STAT548, University of Washington Sham Kakade April 18, 2017 Sham Kakade
1 Case study of SVM (Rob)
DRAFT a final version will be posted shortly COS 424: Interacting with Data Lecturer: Rob Schapire and David Blei Lecture # 8 Scribe: Indraneel Mukherjee March 1, 2007 In the previous lecture we saw how
An improvement in the build algorithm for Kd-trees using mathematical mean
An improvement in the build algorithm for Kd-trees using mathematical mean Priyank Trivedi, Abhinandan Patni, Zeon Trevor Fernando and Tejaswi Agarwal School of Computing Sciences and Engineering, VIT
High Dimensional Indexing by Clustering
Yufei Tao ITEE University of Queensland Recall that, our discussion so far has assumed that the dimensionality d is moderately high, such that it can be regarded as a constant. This means that d should
Efficient k-coverage Algorithms
School of Computing Science Simon Fraser University, Canada Efficient k-coverage Algorithms for Wireless Sensor Networks (joint work with Majid Bagheri) INFOCOM Minisymposium 07 7 May 2007 1 Motivations
Approximability Results for the p-center Problem
Approximability Results for the p-center Problem Stefan Buettcher Course Project Algorithm Design and Analysis Prof. Timothy Chan University of Waterloo, Spring 2004 The p-center
An Optimal Algorithm for the Euclidean Bottleneck Full Steiner Tree Problem
An Optimal Algorithm for the Euclidean Bottleneck Full Steiner Tree Problem Ahmad Biniaz Anil Maheshwari Michiel Smid September 30, 2013 Abstract Let P and S be two disjoint sets of n and m points in the
Machine Learning : Clustering, Self-Organizing Maps
Machine Learning Clustering, Self-Organizing Maps 12/12/2013 Machine Learning : Clustering, Self-Organizing Maps Clustering The task: partition a set of objects into meaningful subsets (clusters). The
Summary of Last Chapter. Course Content. Chapter 3 Objectives. Chapter 3: Data Preprocessing. Dr. Osmar R. Zaïane. University of Alberta 4
Principles of Knowledge Discovery in Data Fall 2004 Chapter 3: Data Preprocessing Dr. Osmar R. Zaïane University of Alberta Summary of Last Chapter What is a data warehouse and what is it for? What is
Clustering: Centroid-Based Partitioning
Clustering: Centroid-Based Partitioning Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong 1 / 29 Y Tao Clustering: Centroid-Based Partitioning In this lecture, we
Randomized algorithms have several advantages over deterministic ones. We discuss them here:
CS787: Advanced Algorithms Lecture 6: Randomized Algorithms In this lecture we introduce randomized algorithms. We will begin by motivating the use of randomized algorithms through a few examples. Then
Clustering. CE-717: Machine Learning Sharif University of Technology Spring Soleymani
Clustering CE-717: Machine Learning Sharif University of Technology Spring 2016 Soleymani Outline Clustering Definition Clustering main approaches Partitional (flat) Hierarchical Clustering validation
A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation [Wen,Yin,Goldfarb,Zhang 2009]
![A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation [Wen,Yin,Goldfarb,Zhang 2009]](/thumbs/81/83358288.jpg "A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation [Wen,Yin,Goldfarb,Zhang 2009]")
A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation [Wen,Yin,Goldfarb,Zhang 2009] Yongjia Song University of Wisconsin-Madison April 22, 2010 Yongjia Song
CS 261: Data Structures. Skip Lists
CS 261: Data Structures Skip Lists 1 Complexity Lists and Arrays OPERATIONS ORDINARY LISTS AND ARRAYS SORTED ARRAYS SORTED LISTS Add O(1) O(n) O(n) Remove O(n) O(n) O(n) Contains O(n) O(log n) O(n) 2 Sorted
Chapter 4. Greedy Algorithms. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 4 Greedy Algorithms Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 4.5 Minimum Spanning Tree Minimum Spanning Tree Minimum spanning tree. Given a connected
Notes. Some of these slides are based on a slide set provided by Ulf Leser. CS 640 Query Processing Winter / 30. Notes
uery Processing Olaf Hartig David R. Cheriton School of Computer Science University of Waterloo CS 640 Principles of Database Management and Use Winter 2013 Some of these slides are based on a slide set
Special Topic: Missing Values. Missing Can Mean Many Things. Missing Values Common in Real Data
Special Topic: Missing Values Missing Values Common in Real Data Pneumonia: 6.3% of attribute values are missing one attribute is missing in 61% of cases C-Section: only about 1/2% of attribute values
Advanced Algorithm Design and Analysis (Lecture 12) SW5 fall 2005 Simonas Šaltenis E1-215b
Advanced Algorithm Design and Analysis (Lecture 12) SW5 fall 2005 Simonas Šaltenis E1-215b simas@cs.aau.dk Range Searching in 2D Main goals of the lecture: to understand and to be able to analyze the kd-trees
Parametrizing the easiness of machine learning problems. Sanjoy Dasgupta, UC San Diego
Parametrizing the easiness of machine learning problems Sanjoy Dasgupta, UC San Diego Outline Linear separators Mixture models Nonparametric clustering Nonparametric classification and regression Nearest
Chapter 4. Greedy Algorithms. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 4 Greedy Algorithms Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 4.5 Minimum Spanning Tree Minimum Spanning Tree Minimum spanning tree. Given a connected
Fast Clustering using MapReduce
Fast Clustering using MapReduce Alina Ene Sungjin Im Benjamin Moseley September 6, 2011 Abstract Clustering problems have numerous applications and are becoming more challenging as the size of the data
On communication complexity of classification
On communication complexity of classification Shay Moran (Princeton) Joint with Daniel Kane (UCSD), Roi Livni (TAU), and Amir Yehudayoff (Technion) Warmup (i): convex set disjointness Alice s input: n
Distributed Relationship Schemes for Trees
Distributed Relationship Schemes for Trees Cyril Gavoille Arnaud Labourel University of Bordeaux, France December 17-21, 2007, Sendai Distributed bounded distance oracle Problem Assign a piece of information
Machine Learning Department School of Computer Science Carnegie Mellon University. K- Means + GMMs
10-601 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University K- Means + GMMs Clustering Readings: Murphy 25.5 Bishop 12.1, 12.3 HTF 14.3.0 Mitchell
Randomized Algorithms 2017A - Lecture 10 Metric Embeddings into Random Trees
Randomized Algorithms 2017A - Lecture 10 Metric Embeddings into Random Trees Lior Kamma 1 Introduction Embeddings and Distortion An embedding of a metric space (X, d X ) into a metric space (Y, d Y ) is
Module 3: Sorting and Randomized Algorithms. Selection vs. Sorting. Crucial Subroutines. CS Data Structures and Data Management
Module 3: Sorting and Randomized Algorithms CS 240 - Data Structures and Data Management Sajed Haque Veronika Irvine Taylor Smith Based on lecture notes by many previous cs240 instructors David R. Cheriton
Incorporating Known Pathways into Gene Clustering Algorithms for Genetic Expression Data
Incorporating Known Pathways into Gene Clustering Algorithms for Genetic Expression Data Ryan Atallah, John Ryan, David Aeschlimann December 14, 2013 Abstract In this project, we study the problem of classifying
EPG-representations with small grid-size
EPG-representations with small grid-size Martin Derka mderka@uwaterloo.ca David R. Cheriton School of Computer Science University of Waterloo GD 2017 September 26, 2017 Joint work with T. Biedl, V. Dujmovic,
Module 3: Sorting and Randomized Algorithms
Module 3: Sorting and Randomized Algorithms CS 240 - Data Structures and Data Management Sajed Haque Veronika Irvine Taylor Smith Based on lecture notes by many previous cs240 instructors David R. Cheriton
A Course in Machine Learning
A Course in Machine Learning Hal Daumé III 13 UNSUPERVISED LEARNING If you have access to labeled training data, you know what to do. This is the supervised setting, in which you have a teacher telling
Exact Algorithms Lecture 7: FPT Hardness and the ETH
Exact Algorithms Lecture 7: FPT Hardness and the ETH February 12, 2016 Lecturer: Michael Lampis 1 Reminder: FPT algorithms Definition 1. A parameterized problem is a function from (χ, k) {0, 1} N to {0,
Faster Algorithms for the Constrained k-means Problem
Faster Algorithms for the Constrained k-means Problem CSE, IIT Delhi June 16, 2015 [Joint work with Anup Bhattacharya (IITD) and Amit Kumar (IITD)] k-means Clustering Problem Problem (k-means) Given n
Big Data Infrastructure CS 489/698 Big Data Infrastructure (Winter 2016)
Big Data Infrastructure CS 489/698 Big Data Infrastructure (Winter 2016) Week 9: Data Mining (3/4) March 8, 2016 Jimmy Lin David R. Cheriton School of Computer Science University of Waterloo These slides
Towards Theoretical Foundations of Clustering
Towards Theoretical Foundations of Clustering by Margareta Ackerman A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Computer
R & G Chapter 13. Implementation of single Relational Operations Choices depend on indexes, memory, stats, Joins Blocked nested loops:
Relational Query Optimization R & G Chapter 13 Review Implementation of single Relational Operations Choices depend on indexes, memory, stats, Joins Blocked nested loops: simple, exploits extra memory
An Introduction to Machine Learning
TRIPODS Summer Boorcamp: Topology and Machine Learning August 6, 2018 General Set-up Introduction Set-up and Goal Suppose we have X 1,X 2,...,X n data samples. Can we predict properites about any given
Range Reporting. Range Reporting. Range Reporting Problem. Applications
Philip Bille Problem problem. Preprocess at set of points P R 2 to support report(x1, y1, x2, y2): Return the set of points in R P, where R is rectangle given by (x1, y1) and (x2, y2). Applications Relational
Dynamic Collision Detection
Distance Computation Between Non-Convex Polyhedra June 17, 2002 Applications Dynamic Collision Detection Applications Dynamic Collision Detection Evaluating Safety Tolerances Applications Dynamic Collision