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21.1.3 Types of Image Query . . . . . . . .. .. ... .. .. ... . 630
21.1.4 What Users Do with Image Collections . . . . . . . . . . . . 631
21.2 Basic Technologies from Information Retrieval . . . . . . . . . . . . . 632
21.2.1 Word Counts .. .. ... .. .. ... .. ... .. .. ... . 632
21.2.2 Smoothing Word Counts . . . . . . . . . . . . . . . . . . . . . 633
21.2.3 Approximate Nearest Neighbors and Hashing . . . . . . . . . 634
21.2.4 Ranking Documents ... .. .. ... .. ... .. .. ... . 638
21.3 Images as Documents . .. ... .. .. ... .. ... .. .. ... . 639
21.3.1 Matching Without Quantization . .. .. ... .. .. ... . 640
21.3.2 Ranking Image Search Results . . . . . . . . . . . . . . . . . 641
21.3.3 Browsing and Layout .. .. .. ... .. ... .. .. ... . 643
21.3.4 Laying Out Images for Browsing ... .. ... .. .. ... . 644
21.4 Predicting Annotations for Pictures .. ... .. ... .. .. ... . 645
21.4.1 Annotations from Nearby Words ... .. ... .. .. ... . 646
21.4.2 Annotations from the Whole Image . . . . . . . . . . . . . . 646
21.4.3 Predicting Correlated Words with Classifiers . . . . . . . . . 648
21.4.4 Names and Faces . ... .. .. ... .. ... .. .. ... . 649
21.4.5 Generating Tags with Segments . ... .. ... .. .. ... . 651
21.5 The State of the Art of Word Prediction ... .. ... .. .. ... . 654
21.5.1 Resources .. .. .. ... .. .. ... .. ... .. .. ... . 655
21.5.2 Comparing Methods ... .. .. ... .. ... .. .. ... . 655
21.5.3 Open Problems . . . ... .. .. ... .. ... .. .. ... . 656
21.6 Notes . . . . . ... .. .. ... .. .. ... .. ... .. .. ... . 659

VII BACKGROUND MATERIAL 661

22 Optimization Techniques 663
22.1 Linear Least-Squares Methods . . . . . . . . . . . . . . . . . . . . . . 663
22.1.1 Normal Equations and the Pseudoinverse . . . . . . . . . . . 664
22.1.2 Homogeneous Systems and Eigenvalue Problems . . . . . . . 665
22.1.3 Generalized Eigenvalues Problems .. .. ... .. .. ... . 666
22.1.4 An Example: Fitting a Line to Points in a Plane . . . . . . . 666
22.1.5 Singular Value Decomposition . . ... .. ... .. .. ... . 667
22.2 Nonlinear Least-Squares Methods . . . . . . . . . . . . . . . . . . . . 669
22.2.1 Newton’s Method: Square Systems of Nonlinear Equations. . 670
22.2.2 Newton’s Method for Overconstrained Systems . . . . . . . . 670
22.2.3 The Gauss–Newton and Levenberg–Marquardt Algorithms . 671
22.3 Sparse Coding and Dictionary Learning ... .. ... .. .. ... . 672
22.3.1 Sparse Coding . . . ... .. .. ... .. ... .. .. ... . 672
22.3.2 Dictionary Learning ... .. .. ... .. ... .. .. ... . 673