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symmetric Laplacian. Physical Interpretation of Laplacian Eigenfunctions 0 10 20 30 40 50 60 70 80 90 100−0. 200. 2 0Max Min 0Max Min (c)/phi. 00 /phi. 01 /phi. 02 /phi. 03/phi. 00/phi. 00 /phi. 03/phi. 02/phi. 01 /phi. 01 /phi. 02 /phi. 03 (b) (a) /phi. 0 0 /phi. 0 /phi. 0 /phi. 0 1 /phi. 0 2 /phi. 0 /phi. 0 /phi. 0

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Rule classification organizes the rules according to their semantic interpretation. Three basic classes of rules are defined: •Traffic flow rules involve source and destination addresses. •Provided services rules consist of destination port (i. e. , the service) and destination address (i. e. , the service

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In practical evaluation, the rule base instead of single rule isused to test the performance of the IDS. Ten thousand runs of GP are executed and the average results are reported. The average value of FAR is 0. 41% and the average value of P Dis 0. 5714. The ROC shows P Dclose to 100% when the FAR is

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The study used the 1999 DARPA intrusion detection data set. To make the data set more realistic, the subset chosen con- sisted of 1% to 1. 5% attacks and 98. 5% to 99% normal traffic. The results from the Enhanced SVMs had 87. 74% accuracy, a10. 20% FP rate, and a 27. 27% FN rate. Those results were sub-

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relations. Facts observed in the KG are stored as a collection of triples IDþ¼fðh; r; tÞg. Each triple is composed of a head entity h2E, a tail entity t2E, and a relation r2IRbetween them, e. g. , ( AlfredHitchcock, DirectorOf, Psycho ). Here, Edenotes the set of entities, and IRthe set of relations.

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former uses the same sparse projection matrix MrðurÞfor each relation r, i. e. , h?¼MrðurÞh;t?¼MrðurÞt: The latter introduces two separate sparse projection matri- cesM1 rðu1 rÞandM2 rðu2 rÞfor that relation, one to project head entities, and the other tail entities, i. e. , h?¼M1 rðu1 rÞh;t?¼M2 rðu2

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such that að‘Þ¼Mð‘Þzð‘/C01Þþbð‘Þ;‘¼1;. . . ;L ; zð‘Þ¼ReLUðað‘ÞÞ;‘¼1;. . . ;L ; where Mð‘Þandbð‘Þrepresent the weight matrix and bias for the‘th layer respectively. After the feedforward process, the score is given by matching the output of the last hidden layer and the embedding of the tail entity, i. e. ,

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position of the projection matrices associated with all sub- categories of ci. Two types of composition operations are used, i. e. , addition : Mci¼b1Mcð1Þ iþ/C1/C1/C1þ b‘Mcð‘Þ i; multiplication : Mci¼Mcð1Þ i/C14/C1/C1/C1/C14 Mcð‘Þ i: Here cð1Þ i;. . . ;cð‘Þ iare sub-categories of ciin the hierarchy;

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attributes of entities (e. g. , ( AlfredHitchcock, Gender, Male )), but most KG embedding techniques do not explicitly distinguish between relations and attributes. Take the tensor factorization model RESCAL as an example. In this model, each KG relation is encoded as a slice of the tensor, no matter

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scheme which can multicast from the source to all the sinks. To simplify notation, we adopt the con-vention that for. At time, information transactions occur in the following order: T1. sends to, T2. sends to, , and, T3. sends to T4. sends to T5. sends to T6. sends to T7. sends to T8. sends to T9. decodes

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ARC Linkage International Grant LX045446 and the ARC Discovery Project Grant DP0453089. F. Scarselli, M. Gori, and G. Monfardini are with the Faculty of Informa- tion Engineering, University of Siena, Siena 53100, Italy (e-mail: franco@dii. unisi. it; marco@dii. unisi. it; monfardini@dii. unisi. it).

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that identifies the revenue/penalty region at the cloud server j Dcloud j≤Dj. We consider to tradeoff between the power consumption and computation delay in the cloud computing subsystem. That is, we have the the SP2 min yj, fj, nj, σj∑ j∈Mσjnj( Ajfp j+Bj)s. t. ⎧ ⎪⎨ ⎪⎩∑ j∈Myj=Y Dcloud j≤Dj∀j∈M (4)−(7).

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Definition 2: Objective function F(f, n, σ)and constraint functions G(y), H(y, f, n, σ) ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩F(f, n, σ)≜∑ j∈Mσjnj( Ajfp j+Bj) G(y)≜∑ j∈Myj−Y H(y, f, n, σ)≜[ h1, . . . , hj, . . . , hM]T hj≜σj[ C( nj, yjK/fj) njfj/K−yj+K fj] −Dj. Thus, MINLP SP2 is min y∈Y, f∈F, n∈N, σ∈/Sigma1F(f, n, σ) s. t. {G(y)=0

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3 Solve MPkby, e. g. , branch and bound; 4 iffeasible solution then 5 Obtain solution( nk, σk, LBk) ; 6 else if unbounded solution then 7 Choose arbitrary nk∈Nandσk∈/Sigma1; 8 SetLBk←− ∞ ; 9 end if 10 Solve SP( nk, σk) by, e. g. , dual decomposition; 11 iffeasible solution then 12 Obtain solution( yk, fk)

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and Lagrangian multiplier( λk, μk) ; 13 SetUBk←min{ UBk−1, F( fk, nk, σk)} ; 14 if⏐⏐UBk−LBk⏐⏐≤ϵthen/*Converged */ 15 return( yk, fk, nk, σk) ; 16 else/*Add feasible constraint */ 17 SetIk+1←Ik∪{k}, Jk+1←Jk; 18 end if 19 else if infeasible solution then 20 Solve SPF( nk, σk) by, e. g. , dual decomposition;

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21 Obtain solution( yk, fk) and Lagrangian multiplier( λk, μk) ; 22 SetUBk←UBk−1; /*Add infeasible constraint */ 23 SetIk+1←Ik, Jk+1←Jk∪{k}; 24 end if 25 Setk←k+1; 26end while Definition 4: SP(nk, σk) min y∈Y, f∈FF( f, nk, σk) s. t. {G(y)=0 H( y, f, nk, σk) ≤0. Definition 5: SP feasibility-check SPF (nk, σk) min

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For generality, we define the cost matrix to be the n×n matrix C≜⎡ ⎢⎣d11C11. . . d1nC1n. . . . . . . . . d n1Cn1··· dnnCnn⎤ ⎥⎦. An assignment is a set of nentry positions in the cost matrix, no two of which lie in the same row or column. The sum of the nentries of an assignment is its cost. An assign-

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mations are made mAP would remain at 59% (Section 3. 2). 2. 3. Training Supervised pre-training. We discriminatively pre-trained the CNN on a large auxiliary dataset (ILSVRC 2012) with image-level annotations (i. e. , no bounding box labels). Pre- training was performed using the open source Caffe CNN

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1. 0 1. 0 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 0. 9 1. 0 0. 9 0. 9 0. 8 0. 8 0. 8 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 6 0. 6 1. 0 0. 8 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 6 0. 6 1. 0 0. 9 0. 8 0. 8 0. 8 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7 0. 7

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0. 593R−CNN FT fc7: sensitivity and impact occ trn size asp view part00. 20. 40. 60. 8 0. 2110. 731 0. 5420. 676 0. 3850. 786 0. 4840. 709 0. 4530. 779 0. 3680. 720 0. 633R−CNN FT fc7 BB: sensitivity and impact occ trn size asp view part00. 20. 40. 60. 8 0. 1320. 339 0. 2160. 347 0. 0560. 487 0. 1260. 453 0. 1370. 391 0. 0940. 388

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The advantage of neural on-line learning rules is that the inputs can be used in the algorithm at once, thus enabling faster adaptation in a nonstationary environment. A resulting tradeoff, however, isthattheconvergenceisslow, anddepends on a good choice of the learning rate sequence, i. e. , the step

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HYV¨ARINEN: FAST AND ROBUST FIXED-POINT ALGORITHMS 633 the good statistical properties (e. g. , robustness) of the new contrast functions, and the good algorithmic properties of thefixed-point algorithm, a very appealing method for ICA wasobtained. Simulations as well as applications on real-life data

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186 IRE TRANSACTIONS ON X Y 6. 2020 2. 4986 6. 1104 2. 0853 10. 4136 4. 1818 8. 2045 3. 0911 8. 2147 4. 3144 8. 0390 4. 5017 8. 6096 3. 0127 7. 6243 1. 1825 11. 9780 11. 2824 10. 4118 6. 6854 7. 3278 2. 5620 12. 0662 8. 3889 5. 7356 0. 0540 TABLE I X 5. 7885 8. 2829 7. 0329 6. 7674 6. 2707 7. 7501 10. 6216

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novel system designs, i. e. , designs of relay weights and allo-cation of transmit power, that meet the following objectives:1) maximize the achievable secrecy rate subject to a totaltransmit power constraint, or 2) minimize the total transmitpower subject to a secrecy rate constraint. We should note

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a common receiver (i. e. , multiple access) in the presence ofan eavesdropper is considered, and the optimal transmit powerallocation policy is chosen to maximize the secrecy sum-rate. A user that is prevented from transmitting based on the ob-tained power allocation can help increase the secrecy rate

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a 4a X, 1 0. 4528 0. 9816 2 1. 5104 w TABLE III GAUSSIAN, Y = 8 a 4a X, 1 0. 245 1 0. 5006 2 0. 7560 1. 0500 3 1. 3439 1. 7480 4 2. 1520 co TABLE IV GAUSSXAN, ~ = 16 a 1 0. 7584 0. %82 2 0. 3880 0. 5224 3 0. 6568 0. 7996 4 0. 9423 1. 0993 5 1. 2562 1. 4371 6 1. 6181 1. 8435 7 2. 0690 2. 4008 8 2. 1326 co

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starts from the lowest level P2and gradually approaches P5as shown in Figure 1(b). From P2toP5, the spa- tial size is gradually down-sampled with factor 2. We use {N2, N3, N4, N5}to denote newly generated feature maps corresponding to {P2, P3, P4, P5}. Note that N2is simply P2, without any processing.

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00. 10. 20. 30. 40. 5 LEVEL 1 LEVEL 2 LEVEL 3 LEVEL 4FEATURE DISTRIBUTION level 1 level 2 level 3 level 4 Figure 3. Ratio of features pooled from different feature levels with adaptive feature pooling. Each line represents a set of pro- posals that should be assigned to the same feature level in FPN,

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Mask R-CNN [ 21] + FPN [ 35] 39. 8 62. 3 43. 4 22. 1 43. 2 51. 2 ResNeXt-101 PANet / PANet [ms-train] 41. 2 / 42. 5 60. 4 / 62. 3 44. 4 / 46. 4 22. 7 / 26. 3 44. 0 / 47. 0 54. 6 / 52. 3 ResNet-50 PANet / PANet [ms-train] 45. 0 / 47. 4 65. 0 / 67. 2 48. 6 / 51. 8 25. 4 / 30. 1 48. 6 / 51. 7 59. 1 / 60. 0 ResNeXt-101

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✓✓✓ 35. 7 / 37. 1 / 38. 9 57. 3 38. 0 18. 6 / 24. 2 / 25. 3 39. 4 / 42. 5 / 43. 6 51. 7 / 47. 1 / 49. 9 ✓✓✓✓ 36. 4 / 38. 0 / 39. 9 57. 8 39. 2 19. 3 / 23. 3 / 26. 2 39. 7 / 42. 9 / 44. 3 52. 6 / 49. 4 / 51. 3 ✓✓✓ ✓ 36. 3 / 37. 9 / 39. 6 58. 0 38. 9 19. 0 / 25. 4 / 26. 4 40. 1 / 43. 1 / 44. 9 52. 4 / 48. 6 / 50. 5

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Settings AP AP 50 AP75 APbbAPbb 50APbb 75 baseline 35. 7 57. 3 38 37. 1 58. 9 40. 0 fu. fc1fc2 35. 7 57. 2 38. 2 37. 3 59. 1 40. 1 fc1fu. fc2 36. 3 58. 0 38. 9 37. 9 60. 0 40. 7 MAX 36. 3 58. 0 38. 9 37. 9 60. 0 40. 7 SUM 36. 2 58. 0 38. 8 38. 0 59. 8 40. 7 Table 4. Ablation study on adaptive feature pooling on val-2017 in

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terms of mask AP and box ap APbbof the independently trained object detector. Settings AP AP 50 AP75 APS APM APL baseline 36. 9 58. 5 39. 7 19. 6 40. 7 53. 2 conv2 37. 5 59. 3 40. 1 20. 7 41. 2 54. 1 conv3 37. 6 59. 1 40. 6 20. 3 41. 3 53. 8 conv4 37. 2 58. 9 40. 0 19. 0 41. 2 52. 8 PROD 36. 9 58. 6 39. 7 20. 2 40. 8 52. 2

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bles 6and 7, compared with last year champion, we achieve 9. 1% absolute and 24% relative improvement on instance segmentation. While for object detection, 9. 4% absolute and23% relative improvement is yielded. APbbAPbb 50APbb 75APbb SAPbb MAPbb L Champion 2015 [ 23] 37. 4 59. 0 40. 2 18. 3 41. 7 52. 9

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adopted. The common testing tricks [ 23, 33, 10, 15, 39, 62], such as multi-scale testing, horizontal flip testing, mask vot- ing and box voting, are used too. For multi-scale testing, we set the longer edge to 1, 400 and let the other range from 600 to1, 200 with step 200. Only 4scales are used. Second,

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Mask R-CNN [fine-only] [ 21] 31. 5 26. 2 49. 9 30. 5 23. 7 46. 9 22. 8 32. 2 18. 6 19. 1 16. 0 SegNet - 29. 5 55. 6 29. 9 23. 4 43. 4 29. 8 41. 0 33. 3 18. 7 16. 7 Mask R-CNN [COCO] [ 21] 36. 4 32. 0 58. 1 34. 8 27. 0 49. 1 30. 1 40. 9 30. 9 24. 1 18. 7 PANet [fine-only] 36. 5 31. 8 57. 1 36. 8 30. 4 54. 8 27. 0 36. 3 25. 5 22. 6 20. 8

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category. Searching on Google with “Yan Mo Nobel Prize, ”resulted in 1, 050, 000 web pointers on the Internet (as of3 January 2013). “For all praises as well as criticisms, ” saidMo recently, “I am grateful. ” What types of praises andcriticisms has Mo actually received over his 31-year writingcareer?

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computers with a high-performance computing platform, with a data mining task being deployed by running someparallel programming tools, such as MapReduce orEnterprise Control Language (ECL), on a large number ofcomputing nodes (i. e. , clusters). The role of the softwarecomponent is to make sure that

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note that our approach is equally applicable to cases whereinformation releases refer to different kinds of respondents (e. g. , business establishments). In the following, we therefore use the terms individual and respondent interchangeably. Since removal of explicit identifiers is the first step to

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1;. . . ;An†, a set of attributes fAi;. . . ;AjgfA1;. . . ;Ang; and a tuple t2T, t‰Ai;. . . ;AjŠdenotes the sequence of the values ofAi;. . . ;Ajint, T‰Ai;. . . ;AjŠdenotes the projection, maintaining duplicate tuples, of attributes Ai;. . . ;AjinT. Also, jTjdenotesT's cardinality, that is, the number of tuples inT.

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We can now introduce the definition of k-anonymity for a table as follows: Definition 2. 2 ( k-anonymity). LetT…A 1;. . . ;An†be a table andQIbe a quasi-identifier associated with it. Tis said to satisfy k-anonymity wrt QIiff each sequence of values in T‰QIŠappears at least with k occurrences in T‰QIŠ.

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all domains in a generalization hierarchy. In the following, dom…Ai;T†denotes the domain of attribute Aiin tableT. We start by introducing the definition of generalized table as follows: Definition 3. 1 (Generalized Table). LetTi…A1;. . . , An†and Tj…A1;. . . ;An†be two tables defined on the same set of

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thus collapsing all tuples in Tto the same list of values, provides k-anonymity at the price of a strong generalization of the data. Such extreme generalization is not needed if a more specific table (i. e. , containing more specific values)exists which satisfies k-anonymity. This concept is captured

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Tjis said to be a k-minimal generalization of a table Tiiff: 1. Tjsatisfies k-anonymity enforcing minimal required suppression (Definitions 2. 2 and 4. 2). 2. jTijÿjTjjMaxSup: 3. 8Tz:TiTzandTzsatisfies Conditions 1 and 2 ):…DVi;z<DVi;j†. Intuitively, generalization Tjisk-minimal iff it satisfies k-

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search all the strategies. This process is clearly much toocostly, given the high number of strategies that should befollowed. The number of different strategies for a domain tupleDTˆhD 1;. . . ;Dniis…h1‡. . . ‡hn†! h1!. . . hn!, where each hiis the length of the path from Dito the top domain in DGHDi.

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1 Symmetric min g(i-l, j-l)+2d(i, j) [ g(i-l, j-2)+2d(i, j-l)+d(i, j) g(i-2, j-l)+2d(i-l, j)+d(i, j) g(i-l, j-2)+(d(i, j-l)+d(i, j)) g(i-2, j-l)+d(i-l, j)+d(i, j) 1 Asymmetric ~ Symmetrlc 1 mi: [(i-l, j-l)+2d(i, j) ], 1 g(i-2, j-3)+2d(i. -I, j-2)+2d(i, j-l)+d(i, j) g(i-3, j-2)+2d(j-2, j-1)+2d(i-l, j)+d(i, j)

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(IoU) ratio is less than 0. 3 to any ground-truth faces; 2) positives: IoU above 0. 65 to a ground truth face; 3) part faces: IoU between 0. 4 and 0. 65 to a ground truth face; and 4) landmark faces: faces labeled five landmarks’ positions. There is an unclear gap between part faces and negatives, and

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Traces; Generalization and Function Approximation; Planning andLearning; Dimensions of Reinforcement Learning; and Case Studies. Talking Nets: An Oral History of Neural Networks —James A. AndersonandEdwardRosenfeld, Eds. Cambridge, MA: MITPress, 1998, 433 pp. , soft cover, $39. 95. ISBN 0–262–01167–0. )

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R. Vinayakumar et al. : Deep Learning Approach for Intelligent IDS in Section IV. Section V includes information related to major shortcomings of IDS datasets, problem formulation and statistical measures. Section VI includes description of datasets. Section VII and Section VIII includes experimental

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attain considerable performance. It was found that the layer containing 1, 024 units had shown highest number of attack detection rates. When we increased the number of hidden units from 1, 024 to 2, 048, the performance in attack detection rate deteriorated. Hence, we decided to use 1, 024 units for

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optimalatanyscale. Theoptimaldetectorhasasimpleapproximate implementation inwhichedgesaremarkedatmaximaingradientmag- nitudeofaGaussian-smoothed image. Weextendthissimpledetector usingoperatorsofseveralwidthstocopewithdifferentsignal-to-noise ratiosintheimage. Wepresentageneralmethod, calledfeaturesyn-

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toaddathirdcriteriontocircumventthepossibilityof multipleresponsestoasingleedge. Usingnumericalop- timization, wederiveoptimaloperatorsforridgeandroof edges. Wewillthenspecializethecriteriaforstepedges andgiveaparametricclosedformforthesolution. Inthe processwewilldiscoverthatthereisanuncertaintyprin-

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LetHn(x)betheresponseofthefiltertonoiseonly, and HG(x)beitsresponsetotheedge, andsupposethereisa localmaximuminthetotalresponseatthepointx=xO. Thenwehave Hn(XO)+HG(x0)=0. (4) TheTaylorexpansionofH&(xo)abouttheorigingives H&(xo)=HG(O)+HG(0)x0+O(x0). (5) ByassumptionHG(0)=0, i. e. , theresponseofthefil-

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scaleswiththeoperatorwidth. Thatis, wefirstdefinean operatorf, whichistheresultofstretchingfbyafactor ofw, fw(x)=f(xlw). Thenaftersubstitutinginto(12)we findthattheintermaximum spacingforf, isx, , (fj)= wxzc(f). Therefore, ifafunctionfsatisfiesthemultiple responseconstraint(13)forfixedk, thenthefunctionf,

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haveoverthefullrange. Also, thisenablesthevalueof f'(0)tobesetasaboundarycondition, ratherthanex- pressedasanintegraloff". Iftheintegraltobemini- mizedsharesthesamelimitsastheconstraintintegrals, itispossibletoexploittheisoperimetric constraintcon- dition(see[6, p. 216]). Whenthisconditionisfulfilled,

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functionfattheorigin. Sincef(x)isasymmetric, wecan extendtheabovedefinition totherange [-W, W]using f(-x)=-f(x). Thefourboundaryconditionsenableus tosolveforthequantities a1througha4intermsofthe unknownconstants a, co, c, ands. Theboundarycondi- tionsmayberewrittena. +a4+c=0 a, easinw+a, eacosw+a3etsinX

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sionalspaceoffunctionstoanonlinearoptimization in threevariablesa, w, and3(notsurprisingly, thecom- binedcriteriondoesnotdependonc). Unfortunately the resultingcriterion, whichmuststillsatisfythemultiple responseconstraint, isprobablytoocomplextobesolved analytically, andnumericalmethodsmustbeusedtopro-

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whererisascloseaspossibleto1. Theperformance in- dexesandparametervaluesforseveralfiltersaregivenin Fig. 4. Theaicoefficientsforallthesefilterscanbefound from(37), byfixingcto, say, c=1. Unfortunately, the largestvalueofrthatcouldbeobtainedusingthecon- strainednumericaloptimization wasabout0. 576forfilter

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number6inthetable. Inourimplementation, wehaveFiltcrParameters nx, zE1Araw=_ 10. 154. 210. 21521. 595500. 1225063. 97566 20. 32. 870. 31312. 471200. 3828431. 26860 30. 52. 130. 4177. 858692. 6285618. 28800 40. 81. 570. 5155. 065002. 5677011. 06100 51. 01. 330. 5613. 455800. 07161 4. 80684 1. 21. 120. 5762. 052201. 569:392. 91540

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71410. 750. 4840. 002973. 503507. 47700 Fig. 4. Filterparametersandperformance measuresforthefiltersillus- tratedinFig. 5. approximated thisfilterusingthefirstderivativeofa Gaussianasdescribedinthenextsection. ThefirstderivativeofGaussianoperator, orevenfilter 6itself, shouldnotbetakenasthefinalwordinedge

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detectionfilters, evenwithrespecttothecriteriawehave used. Ifwearewillingtotolerateaslightreductionin multipleresponseperformance r, wecanobtainsignifi- cantimprovements intheothertwocriteria. Forexample, filters4and5bothhavesignificantly betterEAproduct thanfilter6, andonlyslightlylowerr. FromFig. 5we

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688 ~~~~IEEETRANSACTIONS ONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL. PAMI8, NO. 6, NOVEMBER 1986 8ze 40 60 ae. 6zZ. azz. 2. 8Z. o 320 380 -1. 3141194 1, 28S213 alaTI, Z2. 3" le as zzleZ'!eZ. qZW 3" 3ze 1. 1515[57 zeqoso80 149 3ee Z26Z49Zee 3qo 380qoo 355 0. 6200538 99ze 60 as IN 220Z45Z"Z" 3zv 3qe 350 3ae

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IEEETRANSACTIONS ONPATTERNANALYSISANDMACHINEINTELLIGENCE, VOL. PAMI-8, NO. 6, NOVEMBER1986 (a) (b) Fig. 7. (a)Partsimage, 576by454pixels. (b)Imagethesholded atT, . (c) Imagethresholded at2T, . (d)Imagethresholdedwithhysteresisusing boththethresholdsin(a)and(b). thresholdalongthelengthofthecontour. Suppose we

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CANNY:COMPUTATIONAL APPROACH TOEDGEDETECTION functionalignednormaltotheedgedirectionwithapro- jectionfunctionparalleltotheedgedirection. Asubstan- tialsavingsincomputational effortispossibleifthepro- jectionfunctionisaGaussianwiththesameaasthe(first derivativeofthe)Gaussianusedasthedetectionfunction.

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Ifthewindowfunctionisabruptlytruncated, e. g. , ifitis rectangular, thefilteredimagewillnotbesmoothbecause oftheveryhighbandwidthofthiswindow. Thiseffectis relatedtotheGibbsphenomenon inFouriertheorywhich occurswhenasignalistransformed overafinitewindow. Whennonmaximum suppression isappliedtothisrough

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resolution, i. e. , thereislesspossibilityofinterference fromneighboring edges. Thatargumentisalsoveryrel- evantinthepresentcontext, astodatetherehasbeenno consideration ofthepossibilityofmorethanoneedgein agivenoperatorsupport. Interestingly, Rosenfeldand Thurstonproposedexactlytheoppositecriterionin

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synthesisisappliedwefindthatredundant responsesof thelargeroperator areeliminatedleadingtoasharpedge map. Bycontrast, inFig. 9theedgesmarkedbythetwoop- eratorsareessentiallyindependent, anddirectsuperposi- tionoftheedgesgivesausefuledgemap. Whenweapply featuresynthesistothesesetsofedgeswefindthatmost

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aslargeaprojectionfunctionaspossible. Thereareprac- ticallimitations onthishowever, inparticularedgesinan imageareoflimitedextent, andfewareperfectlylinear. However, mostedgescontinueforsomedistance, infact muchfurtherthanthe3or4pixelsupportsofmostedge operators. Evencurvededgescanbeapproximated bylin-

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sampletheoutputofnonelongated maskswiththesame direction. Thisoutputissampledatregularintervalsina lineparalleltotheedgedirection. Ifthesamplesareclose together(lessthan2aapart), theresultingmaskisessen- tiallyflatovermostofitsrangeintheedgedirectionand fallssmoothlyofftozeroatitsends. Twocrosssections

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Wehavedescribed aprocedureforthedesignofedge detectorsforarbitraryedgeprofiles. Thedesignwasbased onthespecification ofdetectionandlocalizationcriteria inamathematical form. Itwasnecessarytoaugmentthe originaltwocriteriawithamultipleresponse measurein ordertofullycapturetheintuitionofgooddetection. A

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noiseintheimage, asdeterminedbyanoiseestimation scheme. Thisdetectormadeuseofseveraloperatorwidths tocopewithvaryingimagesignal-to-noise ratios, andop- eratoroutputswerecombinedusingamethodcalledfea- turesynthesis, wheretheresponsesofthesmalleropera- torswereusedtopredictthelargeoperatorresponses. If

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inanumberofapplicationareas. Digitalaudio, video, andpicturesare increasingly furnished with distinguishing but imperceptiblemarks, which may contain a hidden copyright notice or serialnumber or even help to prevent unauthorized copying directly. Military communications systems make increasing use of

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bridge area (c. 1550). At that time, watermarks were mainly used to identify the mill producing the paper—a means of guaranteeing quality. (CourtesyofDr. E. Leedham-Green, CambridgeUniversity Archives. Reproduction technique: beta radiography. ) reactive inks) and secondary features whose presence may

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(a) (b) (c) (d) Fig. 7. When applied to images, the distortions introduced by StirMark are almost unnoticeable. “Lena” (a) before and (b) after StirMarkwithdefaultparameters. (c), (d)Forcomparison, thesame distortions have been applied to a grid. available since November 1997. 3It applies a minor unno-

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letVdenote the set of buses and Efor the set of transmission lines, then an undirected graph (V, E)can represent a power system. For a subset of branches A⊂E, l e tg(A)denote the set of meters on A’s branches and adjacent buses. In the graph (V, E\A), l e th(A)denote the number of interconnected mod-

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j=1Glj(sj−dj)≤fmax l ∀l∈L( μmin l, μmaxl) smin j≤sj≤smax j ∀j∈N( νmin j, νmax j) (28) where sjis the power generation at bus j, cjis the correspond- ing generation cost, djis the forecasted load at bus j, Gljis the shift factor (with respect to the reference bus) from bus jto branch l, fmin l andfmax

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l are the power flow limits for transmis- sion line l, smin j andsmax j are the lower and upper bounds of the power generation at bus j, ands=[s1, s2, . . . , s n]⊺. The objec- tive function is to minimize the aggregated generation cost, and the constraints are supply-demand balance constraint, transmis-

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satisfied. Second, if define two sets L1≜{l:Glj1>G lj2}and L2≜{l:Glj2>G lj1}, t ol e t LMPEP j1−LMPEP j2=(Gj1−Gj2)T(ˆμmin−ˆμmax) =∑ l∈L1(Glj1−Glj2)( ˆμmin l−ˆμmax l) +∑ l∈L2(Glj2−Glj1)( ˆμmax l−ˆμmin l) >0 (37)heuristically, one sufficient condition is ˆfl<fmax l (i. e. , ˆμmax l= 0)f o r∀l∈L1andˆfl>fmin

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to problems (i. e. , how best can we relax the black-box nature of the algorithms and have them exploit some knowledgeconcerning the optimization problem)? In particular, whileserious optimization practitioners almost always perform such matching, it is usually on a heuristic basis; can such matching

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with smaller coefficients. Thus random hidden layer gener- ates weakly correlated hidden layer features, which allow for a solution with a small norm and a good generalization performance. A formal description of an ELM is following. Consider a set of Ndistinct training samples ( xi, ti), i∈J1, NKwith

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which correspond to the data samples of a class j. Alternatively, the correlation matrices can be computed for each class separately 1 h, 1 t, . . . , c h, c t. Then the weights are applied during the summation of the correlation matriceshandt: h=α11 h+. . . +αcc h, (16) t=α11 t+. . . +αcc t. (17)

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inputs, the optimal value of sdiffers from 1. standard deviation sby√ d, and generating weights as W=N(0, s/√ d) (see Figure 6). In the following experiments, ELM is used with automat- ically generated weights from W=N(0, s/√ d). The input data is normalized to zero mean and unit variance. Biases are

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andtheminimaxtheorem, "NumerischeMathematik, vol. 5, pp. 371-379, 1963. R. T. Rockafellar, "Dualityandstabilityinextremumprob-lemsinvolvingconvexfunctions, "PacificJ. Math. , vol. 21, pp. 167-187, 1967. P. Wolfe, "Adualitytheoremfornonlinearprogramming, "Q. Appl. Math. , vol. 19, pp. 239-244, 1961. R. T. Rockafellar, "Non

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normalspace, "Proc. IEEESystemsScienceandCyberneticsConf. (Boston, Mass. , October11-13, 1967). J. M. Danskin, "Thetheoryofmax-minwithapplications, "J. SIAM, vol. 14, pp. 641-665, July1966. 113]W. Fenchel, "Convexcones, sets, andfunctions, "mimeo- graphednotes, PrincetonUniversity, Princeton, N. J. , September

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plantcontrol, "ISATrans. , vol. 5, pp. 175-183, April1966. C. B. BrosilowandL. S. Lasdon, "Atwoleveloptimizationtechniqueforrecycleprocesses, "1965Proc. AICHE-Symp. onApplication ofMathematical ModelsinChemicalEngineering Re-search, Design, andProduction(London, England). L. S. Lasdon, "Dualityanddecomposition

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inmathematicalprogramming, " SystemsResearchCenter, CaseInstituteofTech-nology, Cleveland, Ohio, Rept. SRC119-C-67-52, 1967. A. V. FiaccoandG. P. McCormick, SequentialUnconstrainedMinimization TechniquesforNonlinearProgramming. NewYork:Wiley, 1968. 12]R. FoxandL. Schmit, "Advancesintheintegratedapproach

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PETERE. HART, MEMBER, IEEE, NILSJ. NILSSON, MEMBER, IEEE, ANDBERTRAM RAPHAEL Abstract-Although theproblemofdetermining theminimum costpaththroughagrapharisesnaturallyinanumberofinteresting applications, therehasbeennounderlyingtheorytoguidethe development ofefficientsearchprocedures. Moreover, thereisno

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ManuscriptreceivedNovember24, 1967. TheauthorsarewiththeArtificialIntelligenceGroupoftheAppliedPhysicsLaboratory, StanfordResearchInstitute, MenloPark, Calif. mechanicaltheorem-proving andproblem-solving. These problemshaveusuallybeenapproachedinoneoftwo ways, whichweshallcallthemathematical approachand

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successoroperatorP, definedonnil}, whosevalueforeach niisasetofpairs{(nj, cij)}. Inotherwords, applyingrto nodeniyieldsallthesuccessorsnjofniandthecostscij associatedwiththearcsfromnitothevariousnj. Applica- tionofrtothesourcenodes, totheirsuccessors, andso forthaslongasnewnodescanbegeneratedresultsinan

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Apathfromn, tonkisanorderedsetofnodes(n1, n2, . . . , nk)witheachni+asuccessorofni. Thereexistsapath fromnitonjifandonlyifnjisaccessiblefromni. Everypathhasacostwhichisobtainedbyaddingtheindividual costsofeacharc, ci, i+l, inthepath. Anoptimalpathfrom nitonjisapathhavingthesmallestcostoverthesetofall

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IEEETRANSACTIONS ONSYSTEMS SCIENCEANDCYBERNETICS, JULY1968 optimalpath, itwillsometimesfailtofindsuchapathand thusnotbeadmissible. Anefficientalgorithmobviously needssomewaytoevaluateavailablenodestodetermine whichoneshouldbeexpandednext. Supposesomeevalu- ationfunctionf(n)couldbecalculatedforanynoden.

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Asimpleexamplewillillustratethatthisestimateis easytocalculateasthealgorithmproceeds. ConsidertheFig. 1. subgraphshowninFig. 1. Itconsistsofastartnodesand threeothernodes, n3, n2, andn3. Thearcsareshownwith arrowheadsandcosts. LetustracehowalgorithmA*pro- ceededingeneratingthissubgraph. Startingwiths, we

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whichcompletestheproof. Wecannowproveourfirst theorem. Theorem1 Ifii(n)<h(n)foralln, thenA*isadmissible. Proof:Weprovethistheorembyassumingthecontrary, namelythatA*doesnotterminatebyfindinganoptimal pathtoapreferredgoalnodeofs. Therearethreecasesto consider:eitherthealgorithmterminatesatanongoalnode,

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proceduresforcomputingtheestimatesAalwaysleadto valuesthatsatisfy(5). Weshallcallthisassumptionthe consistencyassumption. NotethattheestimateA(n)=0 forallntriviallysatisfiestheconsistencyassumption. Intuitively, theconsistencyassumptionwillgenerallybe satisfiedbyacomputation ruleforhthatuniformlyuses

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supposethatnodenisclosedbyA*. Theng(n)=g(n). Proof:ConsiderthesubgraphGsjustbeforeclosingn, andsupposethecontrary, i. e. , supposeg(n)>g(n). NoW thereexistssomeoptimalpathPfromston. Since'(n)> g(n), A*didnotfindP. ByLemma1, thereexistsanopen noden'onPwithg(n')=g(n'). Ifn'=n, wehave provedthelemma. Otherwise, 104

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contradicting thefactthatA*selectednforexpansion whenn'wasavailableandthusprovingthelemma. Thenextlemmastatesthatfismonotonically nonde- creasingonthesequenceofnodesclosedbyA*. LemmaS Let(n1, n2, . . , n, )bethesequenceofnodesclosedby A*. Then, iftheconsistencyassumptionissatisfied, p. q impliesf(np, )<f(nq).

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f(n)<f(t*)=f(t) (Strictinequalityoccursbecausenotiesareallowed. ) ThereexistssomegraphGn, , 0eEOn, 3forwhichA(n)- h(n)bythedefinitioinofh. NowbyLemma2, 0(n)=g(n). ThenonthegraphGn, Oif(n)=f(n). SinceAisnomore informedthan. A*, Acouldnotruleouttheexistenceof Gn, O;butAdidnotexpandnbeforeterminationandis,

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therefore, notadmissible, contrarytoourassumptionand completingtheproof. UpondefiningN(A, Gs)tobethetotalnumberofnodes inG, expandedbythealgorithmA, thefollowingsimple corollaryisimmediate. Corollary UnderthepremisesofTheorem2, N(A*, Gs)<N(A, Gs) withequalityifandonlyifAexpandstheidenticalsetof nodesasA*.

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choosen'insteadofn. Byrepeatingtheaboveargument, weobtainforsomeianA*, ea*thatexpandsonlynodes thatarealsoexpandedbyA, completingtheproofofthe theorem. Corollary1 Supposethepremisesofthetheoremaresatisfied. Then foranyagraphG, thereexistsanA*ea*suchthatN(A*, GQ)AN(A, G, ), withequalityifandonlyifAexpandsthe

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HARTetal. :DETERMINATION OFMINIMUMCOSTPATHS Itisbeyondthescopeofthediscussiontoconsiderhow todefineasuccessoroperator Porassigncostscijsothat theresultinggraphrealistically reflectsthenatureofa specificproblemdomain. 2 B. TheHeuristicPoweroftheEstimate A ThealgorithmA*isactuallyafamilyofalgorithms;the

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3Exceptforpossiblecriticalties, asdiscussedinCorollary2of Theorem3. ample, chooseh(n)=(x+y). Since(x+y)< V/x2+y2, thealgorithmisstilladmissible. Sinceweare notusing"all"ourknowledgeoftheproblemdomain, a fewextranodesmaybeexpanded, buttotalcomputational effortmaybereduced;again, each"extra"nodemustalso

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