大规模数据挖掘：项目课程与网络分析

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xvi CONTENTS

10.7.2 An Algorithm for Finding Triangles . . . . . . . . . . . . 381

10.7.3 Optimality of the Triangle-Finding Algorithm . . . . . . . 382

10.7.4 Finding Triangles Using MapReduce . . . . . . . . . . . . 383

10.7.5 Using Fewer Reduce Tasks . . . . . . . . . . . . . . . . . . 384

10.7.6 Exercises for Section 10.7 . . . . . . . . . . . . . . . . . . 385

10.8 Neighborhood Properties of Graphs . . . . . . . . . . . . . . . . . 386

10.8.1 Directed Graphs and Neighborhoods . . . . . . . . . . . . 386

10.8.2 The Diameter of a Graph . . . . . . . . . . . . . . . . . . 388

10.8.3 Transitive Closure and Reachability . . . . . . . . . . . . 388

10.8.4 Transitive Closure Via MapReduce . . . . . . . . . . . . . 390

10.8.5 Smart Transitive Closure . . . . . . . . . . . . . . . . . . 392

10.8.6 Transitive Closure by Graph Reduction . . . . . . . . . . 394

10.8.7 Approximating the Sizes of Neighborhoods . . . . . . . . 395

10.8.8 Exercises for Section 10.8 . . . . . . . . . . . . . . . . . . 397

10.9 Summary of Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . 398

10.10References for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . 402

11 Dimensionality Reduction 405

11.1 Eigenvalues and Eigenvectors of Symmetric Matrices . . . . . . . 406

11.1.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . 406

11.1.2 Computing Eigenvalues and Eigenvectors . . . . . . . . . 407

11.1.3 Finding Eigenpairs by Power Iteration . . . . . . . . . . . 408

11.1.4 The Matrix of Eig e nvectors . . . . . . . . . . . . . . . . . 411

11.1.5 Exercises for Section 11.1 . . . . . . . . . . . . . . . . . . 411

11.2 Principal-Component Analysis . . . . . . . . . . . . . . . . . . . 412

11.2.1 An Illustrative Example . . . . . . . . . . . . . . . . . . . 413

11.2.2 Using Eige nvectors for Dimensiona lity Reduction . . . . . 416

11.2.3 The Matrix of Distances . . . . . . . . . . . . . . . . . . . 417

11.2.4 Exercises for Section 11.2 . . . . . . . . . . . . . . . . . . 418

11.3 Singular-Value Decomposition . . . . . . . . . . . . . . . . . . . . 418

11.3.1 Deﬁnition of SVD . . . . . . . . . . . . . . . . . . . . . . 418

11.3.2 Interpretation o f SVD . . . . . . . . . . . . . . . . . . . . 420

11.3.3 Dimensionality Reduction Using SVD . . . . . . . . . . . 422

11.3.4 Why Zeroing Low Singular Values Works . . . . . . . . . 423

11.3.5 Querying Using Concepts . . . . . . . . . . . . . . . . . . 425

11.3.6 Computing the SVD of a Matrix . . . . . . . . . . . . . . 426

11.3.7 Exercises for Section 11.3 . . . . . . . . . . . . . . . . . . 427

11.4 CUR Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 428

11.4.1 Deﬁnition of CUR . . . . . . . . . . . . . . . . . . . . . . 429

11.4.2 Choosing Rows and Columns Properly . . . . . . . . . . . 430

11.4.3 Constructing the Middle Matrix . . . . . . . . . . . . . . 431

11.4.4 The Complete CUR Decomposition . . . . . . . . . . . . 432

11.4.5 Eliminating Duplicate Rows and Columns . . . . . . . . . 433

11.4.6 Exercises for Section 11.4 . . . . . . . . . . . . . . . . . . 434

11.5 Summary of Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . 434

CONTENTS xvii

11.6 References for Chapter 11 . . . . . . . . . . . . . . . . . . . . . . 436

12 Large-Scale M achine Learning 439

12.1 The Ma chine-Learning Model . . . . . . . . . . . . . . . . . . . . 440

12.1.1 Training Sets . . . . . . . . . . . . . . . . . . . . . . . . . 440

12.1.2 Some Illustrative Examples . . . . . . . . . . . . . . . . . 440

12.1.3 Approaches to Machine Lea rning . . . . . . . . . . . . . . 443

12.1.4 Machine-Learning Architecture . . . . . . . . . . . . . . . 444

12.1.5 Exercises for Section 12.1 . . . . . . . . . . . . . . . . . . 447

12.2 Perceptrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

12.2.1 Training a Perceptron with Zero Threshold . . . . . . . . 447

12.2.2 Convergence of Perceptrons . . . . . . . . . . . . . . . . . 451

12.2.3 The Winnow Algorithm . . . . . . . . . . . . . . . . . . . 451

12.2.4 Allowing the Threshold to Vary . . . . . . . . . . . . . . . 453

12.2.5 Multiclass Perceptrons . . . . . . . . . . . . . . . . . . . . 455

12.2.6 Transforming the Training Set . . . . . . . . . . . . . . . 456

12.2.7 Problems With Perceptrons . . . . . . . . . . . . . . . . . 457

12.2.8 Parallel Implementation of Perceptrons . . . . . . . . . . 458

12.2.9 Exercises for Section 12.2 . . . . . . . . . . . . . . . . . . 459

12.3 Support-Vector Machines . . . . . . . . . . . . . . . . . . . . . . 461

12.3.1 The Mechanics of an SVM . . . . . . . . . . . . . . . . . . 461

12.3.2 Normalizing the Hyperplane . . . . . . . . . . . . . . . . . 462

12.3.3 Finding Optimal Approximate Separators . . . . . . . . . 464

12.3.4 SVM Solutions by Gradient Descent . . . . . . . . . . . . 467

12.3.5 Sto chastic Gradient Descent . . . . . . . . . . . . . . . . . 470

12.3.6 Parallel Implementation of SVM . . . . . . . . . . . . . . 471

12.3.7 Exercises for Section 12.3 . . . . . . . . . . . . . . . . . . 471

12.4 Learning from Nearest Neighbors . . . . . . . . . . . . . . . . . . 472

12.4.1 The Framework for Nearest-Neighbor Calculations . . . . 47 2

12.4.2 Learning with One Nearest Neighbor . . . . . . . . . . . . 473

12.4.3 Learning One-Dimensional Functions . . . . . . . . . . . . 474

12.4.4 Kernel Regression . . . . . . . . . . . . . . . . . . . . . . 476

12.4.5 Dealing with High-Dimensional Euclidean Data . . . . . . 477

12.4.6 Dealing with Non-Euclidean Distances . . . . . . . . . . . 478

12.4.7 Exercises for Section 12.4 . . . . . . . . . . . . . . . . . . 479

12.5 Comparison of Learning Methods . . . . . . . . . . . . . . . . . . 480

12.6 Summary of Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . 481

12.7 References for Chapter 12 . . . . . . . . . . . . . . . . . . . . . . 483

2 CHAPTER 1. DATA MINING

and standar d deviation of this Gaussian distribution completely characterize the

distribution and would become the model of the data. 2

1.1.2 Machine Learning

There are some who regard data mining as synonymous with machine learning.

There is no question that s ome data mining appropriately uses algorithms from

machine learning. Machine-learning practitioners use the data as a training s et,

to train an algorithm of one of the many types used by machine-learning prac -

titioners, such a s Bayes nets, support-vector machines, decision trees, hidden

Markov models, and many others.

There are situations where using data in this way makes se ns e. The typical

case where machine learning is a good approach is when we have little idea of

what we are looking for in the data . For exa mple, it is rather unclear what

it is about movies that makes certain movie-goers like or dislike it. Thus,

in answering the “Netﬂix challenge” to devise an algorithm that predicts the

ratings of movies by users, based on a sample of their responses, machine-

learning algorithms have proved quite successful. We shall discuss a simple

form of this type of algorithm in Section

9.4.

On the other hand, machine learning has not proved successful in situations

where we can describe the goals of the mining more directly. An interesting

case in point is the attempt by WhizBang! Labs

to use machine learning to

locate people’s resumes on the Web. It was not able to do better than algorithms

in the typical resume. Since everyone who has looked at or written a r esume has

a pretty good idea of what resumes contain, there was no mystery about what

makes a Web page a resume. Thus, there was no advantage to machine-learning

over the direct design of an algorithm to discover resumes.

1.1.3 Computational Approaches to Modeling

More recently, computer scientists have looked at data mining as an algorithmic

problem. In this case, the model of the data is simply the answer to a complex

query about it. For instance, given the set of numbers of Example

1.1, we mig ht

compute their average and standard deviation. No te that these values might

not be the parameters of the Ga ussian that best ﬁts the data, although they

will almost certainly be very close if the size of the data is large.

There are ma ny diﬀerent approaches to modeling data. We have already

mentioned the poss ibility of constructing a statistical process whereby the data

could have b e e n generated. Most other approaches to modeling can b e describe d

as either

1. Summarizing the data succinctly and approximately, or

This startup attempted to use machine learning to mine large-scale data, and hired many

of the top machine-learning people to do so. Unfortunately, it was not able to sur vive.

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