1867:
2051:
2434:
1415:
2821:
2256:
1568:
2338:
1958:
1613:
2333:
3173:
2475:
3242:
2877:
3117:
1692:
2180:
1953:
1304:
2705:
1185:
1471:
1781:
2973:
1807:
2112:
1743:
1498:
1266:
1295:
3024:
2901:
2571:
2519:
2208:
2136:
1049:
of all pairs of data in the feature space. This operation is often computationally cheaper than the explicit computation of the coordinates. This approach is called the "
3379:
2085:
2654:
2680:
1836:
3064:
3044:
2547:
2495:
877:
1212:
915:
3349:
3326:
3306:
3286:
3266:
2997:
2921:
2700:
2595:
2293:
1518:
1235:
1137:
3571:
Aizerman, M. A.; Braverman, Emmanuel M.; Rozonoer, L. I. (1964). "Theoretical foundations of the potential function method in pattern recognition learning".
872:
862:
3070:. Some cite this running time shortcut as the primary benefit. Researchers also use it to justify the meanings and properties of existing algorithms.
703:
2213:
1525:
910:
1023:. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the
3884:
867:
718:
2046:{\displaystyle k(\mathbf {x} ,\mathbf {y} )=\mathbf {x} \cdot \mathbf {y} +\left\|\mathbf {x} \right\|^{2}\left\|\mathbf {y} \right\|^{2}}
449:
950:
753:
2298:
3122:
2439:
1573:
3178:
2826:
1003:) in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into
829:
1119:: rather than learning some fixed set of parameters corresponding to the features of their inputs, they instead "remember" the
1096:
378:
3079:
3847:
2429:{\displaystyle k(\mathbf {x} ,\mathbf {x'} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle _{\mathcal {V}}.}
1625:
2605:
1027:. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing.
887:
650:
185:
905:
3894:
1873:
738:
713:
662:
3823:
3802:
3722:
3671:
3555:
786:
781:
434:
3483:
2141:
444:
82:
3470:
3046:. The linear interpretation gives us insight about the algorithm. Furthermore, there is often no need to compute
1073:
943:
839:
603:
424:
1142:
1423:
1069:
814:
516:
292:
3738:
Honarkhah, M.; Caers, J. (2010). "Stochastic
Simulation of Patterns Using Distance-Based Pattern Modeling".
2053:. The training points are mapped to a 3-dimensional space where a separating hyperplane can be easily found.
1214:. Prediction for unlabeled inputs, i.e., those not in the training set, is treated by the application of a
3454:
1751:
771:
708:
618:
596:
439:
429:
17:
2926:
1792:
1410:{\displaystyle {\hat {y}}=\operatorname {sgn} \sum _{i=1}^{n}w_{i}y_{i}k(\mathbf {x} _{i},\mathbf {x'} ),}
3688:
1100:
922:
834:
819:
280:
102:
1697:
3402:
1000:
979:(SVM). These methods involve using linear classifiers to solve nonlinear problems. The general task of
882:
809:
559:
454:
242:
175:
135:
2608:. In fact, Mercer's condition can be reduced to this simpler case. If we choose as our measure the
1271:
3609:
3488:
1851:
1041:
without ever computing the coordinates of the data in that space, but rather by simply computing the
936:
542:
310:
180:
3005:
2882:
2552:
2500:
2295:. The computation is made much simpler if the kernel can be written in the form of a "feature map"
2189:
2117:
3740:
3593:
3245:
2976:
2816:{\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}k(\mathbf {x} _{i},\mathbf {x} _{j})c_{i}c_{j}\geq 0.}
2090:
1476:
1244:
1031:
564:
484:
407:
325:
155:
117:
112:
72:
67:
3362:
2068:
3422:
2614:
1855:
1116:
511:
360:
260:
87:
2659:
2268:. The word "kernel" is used in mathematics to denote a weighting function for a weighted sum or
1812:
3889:
3588:
3418:
3067:
1847:
1099:
and are statistically well-founded. Typically, their statistical properties are analyzed using
976:
691:
667:
569:
330:
305:
265:
77:
3839:
Learning with
Kernels : Support Vector Machines, Regularization, Optimization, and Beyond
3049:
3029:
2532:
2480:
2275:
Certain problems in machine learning have more structure than an arbitrary weighting function
3465:
2598:
2574:
1104:
645:
467:
419:
275:
190:
62:
3833:
3749:
1190:
992:
574:
524:
8:
3498:
3493:
3352:
2526:
2522:
2058:
1215:
1092:
1081:
1046:
1024:
1016:
677:
613:
584:
489:
315:
248:
234:
220:
195:
145:
97:
57:
3753:
3026:
would, in fact, have a linear interpretation in a different setting: the range space of
3765:
3641:
3621:
3503:
3356:
3334:
3311:
3291:
3271:
3251:
2982:
2906:
2685:
2604:
Mercer's theorem is similar to a generalization of the result from linear algebra that
2580:
2278:
1503:
1220:
1122:
655:
579:
365:
160:
3843:
3819:
3798:
3790:
3786:
3718:
3667:
3551:
3449:
3406:
3382:
2062:
1843:
1298:
1061:
748:
591:
504:
300:
270:
215:
210:
165:
107:
3869:
3769:
3645:
3757:
3631:
2609:
1842:
Kernel classifiers were described as early as the 1960s, with the invention of the
1783:
are the weights for the training examples, as determined by the learning algorithm;
1077:
1065:
984:
980:
972:
964:
776:
529:
479:
389:
373:
343:
205:
200:
150:
140:
38:
2477:
must be a proper inner product. On the other hand, an explicit representation for
3837:
3813:
3444:
3414:
2264:
1085:
804:
608:
474:
414:
3522:
3636:
3410:
1020:
1004:
824:
355:
92:
3761:
3288:
at least approximates the intuitive idea of similarity. Regardless of whether
1473:
is the kernelized binary classifier's predicted label for the unlabeled input
3878:
3659:
3460:
3434:
3394:
2183:
1787:
1042:
1038:
743:
672:
554:
285:
170:
3439:
3002:
Some algorithms that depend on arbitrary relationships in the native space
2251:{\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }
1570:
is the kernel function that measures similarity between any pair of inputs
1563:{\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }
1054:
2057:
The kernel trick avoids the explicit mapping that is needed to get linear
3074:
996:
549:
43:
3268:
that do not satisfy Mercer's condition may still perform reasonably if
698:
394:
320:
3248:. Empirically, for machine learning heuristics, choices of a function
3626:
857:
638:
1866:
2269:
3608:
Hofmann, Thomas; Scholkopf, Bernhard; Smola, Alexander J. (2008).
1850:(SVM) in the 1990s, when the SVM was found to be competitive with
3863:
3398:
988:
633:
3585:
Automatic capacity tuning of very large VC-dimension classifiers
2702:, then the integral in Mercer's theorem reduces to a summation
2328:{\displaystyle \varphi \colon {\mathcal {X}}\to {\mathcal {V}}}
384:
3692:
3168:{\displaystyle \{\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}\}}
983:
is to find and study general types of relations (for example
628:
623:
350:
3713:
Rasmussen, Carl Edward; Williams, Christopher K. I. (2006).
3393:
Application areas of kernel methods are diverse and include
2470:{\displaystyle \langle \cdot ,\cdot \rangle _{\mathcal {V}}}
1608:{\displaystyle \mathbf {x} ,\mathbf {x'} \in {\mathcal {X}}}
1053:". Kernel functions have been introduced for sequence data,
1011:: in contrast, kernel methods require only a user-specified
3237:{\displaystyle K_{ij}=k(\mathbf {x} _{i},\mathbf {x} _{j})}
2872:{\displaystyle (\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n})}
2823:
If this summation holds for all finite sequences of points
2606:
associates an inner product to any positive-definite matrix
1846:. They rose to great prominence with the popularity of the
3658:
3570:
1060:
Algorithms capable of operating with kernels include the
3112:{\displaystyle \mathbf {K} \in \mathbb {R} ^{n\times n}}
916:
List of datasets in computer vision and image processing
3815:
Kernel
Adaptive Filtering: A Comprehensive Introduction
3785:
1687:{\displaystyle \{(\mathbf {x} _{i},y_{i})\}_{i=1}^{n}}
1110:
1034:, which enable them to operate in a high-dimensional,
3607:
3587:. Advances in neural information processing systems.
3583:
Guyon, Isabelle; Boser, B.; Vapnik, Vladimir (1993).
3365:
3337:
3314:
3294:
3274:
3254:
3181:
3125:
3082:
3052:
3032:
3008:
2985:
2929:
2909:
2885:
2829:
2708:
2688:
2662:
2617:
2583:
2555:
2535:
2503:
2483:
2442:
2341:
2301:
2281:
2216:
2192:
2144:
2120:
2093:
2071:
1961:
1876:
1815:
1795:
1754:
1700:
1628:
1576:
1528:
1506:
1479:
1426:
1307:
1274:
1247:
1223:
1193:
1145:
1125:
2682:, which counts the number of points inside the set
3832:
3373:
3343:
3320:
3300:
3280:
3260:
3236:
3167:
3111:
3058:
3038:
3018:
2991:
2967:
2915:
2895:
2871:
2815:
2694:
2674:
2648:
2589:
2565:
2541:
2513:
2489:
2469:
2428:
2327:
2287:
2250:
2202:
2174:
2130:
2106:
2079:
2045:
1947:
1830:
1801:
1775:
1737:
1686:
1607:
1562:
1512:
1492:
1465:
1409:
1301:typically computes a weighted sum of similarities
1289:
1260:
1229:
1206:
1179:
1131:
3811:
3712:
3662:; Rostamizadeh, Afshin; Talwalkar, Ameet (2012).
3175:(sometimes also called a "kernel matrix"), where
3066:directly during computation, as is the case with
1948:{\displaystyle \varphi ((a,b))=(a,b,a^{2}+b^{2})}
3876:
3582:
1861:
1809:determines whether the predicted classification
3737:
911:List of datasets for machine-learning research
3689:"Support Vector Machines: Mercer's Condition"
2175:{\displaystyle k(\mathbf {x} ,\mathbf {x'} )}
1030:Kernel methods owe their name to the use of
1019:over all pairs of data points computed using
944:
3162:
3126:
2456:
2443:
2412:
2372:
1732:
1714:
1664:
1629:
1460:
1442:
3870:onlineprediction.net Kernel Methods Article
3545:
3812:Liu, W.; Principe, J.; Haykin, S. (2010).
951:
937:
3691:. Support Vector Machines. Archived from
3635:
3625:
3592:
3093:
2244:
1769:
1556:
3328:may still be referred to as a "kernel".
1865:
1187:and learn for it a corresponding weight
1180:{\displaystyle (\mathbf {x} _{i},y_{i})}
27:Class of algorithms for pattern analysis
3715:Gaussian Processes for Machine Learning
1694:in the classifier's training set, with
1466:{\displaystyle {\hat {y}}\in \{-1,+1\}}
14:
3877:
1776:{\displaystyle w_{i}\in \mathbb {R} }
1007:representations via a user-specified
3610:"Kernel Methods in Machine Learning"
2968:{\displaystyle (c_{1},\dots ,c_{n})}
1802:{\displaystyle \operatorname {sgn} }
1115:Kernel methods can be thought of as
1091:Most kernel algorithms are based on
1057:, text, images, as well as vectors.
3885:Kernel methods for machine learning
3795:Kernel Methods for Pattern Analysis
1111:Motivation and informal explanation
906:Glossary of artificial intelligence
24:
3779:
3686:
3428:
3011:
2888:
2558:
2506:
2461:
2417:
2320:
2310:
2235:
2225:
2195:
2123:
1738:{\displaystyle y_{i}\in \{-1,+1\}}
1600:
1547:
1537:
25:
3906:
3857:
3836:; Smola, A. J.; Bach, F. (2018).
2529:: an implicitly defined function
2061:to learn a nonlinear function or
1064:, support-vector machines (SVM),
975:, whose best known member is the
3666:. US, Massachusetts: MIT Press.
3484:Kernel methods for vector output
3367:
3221:
3206:
3152:
3131:
3084:
2856:
2835:
2774:
2759:
2573:can be equipped with a suitable
2525:. The alternative follows from
2400:
2382:
2358:
2349:
2161:
2152:
2096:
2073:
2029:
2009:
1996:
1988:
1977:
1969:
1637:
1587:
1578:
1482:
1393:
1378:
1290:{\displaystyle \mathbf {x} _{i}}
1277:
1268:and each of the training inputs
1250:
1151:
3664:Foundations of Machine Learning
3471:Neural network Gaussian process
3388:
1838:comes out positive or negative.
3797:. Cambridge University Press.
3731:
3706:
3680:
3652:
3601:
3564:
3539:
3515:
3231:
3201:
3019:{\displaystyle {\mathcal {X}}}
2999:satisfies Mercer's condition.
2962:
2930:
2896:{\displaystyle {\mathcal {X}}}
2866:
2830:
2784:
2754:
2642:
2634:
2627:
2621:
2566:{\displaystyle {\mathcal {X}}}
2514:{\displaystyle {\mathcal {V}}}
2408:
2395:
2386:
2378:
2366:
2345:
2315:
2240:
2203:{\displaystyle {\mathcal {V}}}
2169:
2148:
2131:{\displaystyle {\mathcal {X}}}
2033:
2025:
2013:
2005:
1981:
1965:
1942:
1904:
1898:
1895:
1883:
1880:
1822:
1660:
1632:
1552:
1433:
1401:
1373:
1314:
1297:. For instance, a kernelized
1241:, between the unlabeled input
1174:
1146:
1074:canonical correlation analysis
971:are a class of algorithms for
326:Relevance vector machine (RVM)
13:
1:
3573:Automation and Remote Control
3550:. Elsevier B.V. p. 203.
3546:Theodoridis, Sergios (2008).
3509:
2497:is not necessary, as long as
2107:{\displaystyle \mathbf {x'} }
1862:Mathematics: the kernel trick
1493:{\displaystyle \mathbf {x'} }
1261:{\displaystyle \mathbf {x'} }
1070:principal components analysis
815:Computational learning theory
379:Expectation–maximization (EM)
3455:Radial basis function kernel
3374:{\displaystyle \mathbf {K} }
3246:positive semi-definite (PSD)
2436:The key restriction is that
2080:{\displaystyle \mathbf {x} }
772:Coefficient of determination
619:Convolutional neural network
331:Support vector machine (SVM)
7:
3477:
2649:{\displaystyle \mu (T)=|T|}
1101:statistical learning theory
923:Outline of machine learning
820:Empirical risk minimization
10:
3911:
3637:10.1214/009053607000000677
3403:inverse distance weighting
2675:{\displaystyle T\subset X}
2549:exists whenever the space
2258:is often referred to as a
1831:{\displaystyle {\hat {y}}}
560:Feedforward neural network
311:Artificial neural networks
3895:Classification algorithms
3762:10.1007/s11004-010-9276-7
3489:Kernel density estimation
2923:real-valued coefficients
1870:SVM with kernel given by
543:Artificial neural network
3741:Mathematical Geosciences
3614:The Annals of Statistics
3059:{\displaystyle \varphi }
3039:{\displaystyle \varphi }
2977:positive definite kernel
2542:{\displaystyle \varphi }
2490:{\displaystyle \varphi }
1618:the sum ranges over the
1500:whose hidden true label
852:Journals and conferences
799:Mathematical foundations
709:Temporal difference (TD)
565:Recurrent neural network
485:Conditional random field
408:Dimensionality reduction
156:Dimensionality reduction
118:Quantum machine learning
113:Neuromorphic engineering
73:Self-supervised learning
68:Semi-supervised learning
3423:handwriting recognition
3359:, then the Gram matrix
3331:If the kernel function
3068:support-vector machines
2182:can be expressed as an
1856:handwriting recognition
1117:instance-based learners
1086:linear adaptive filters
261:Apprenticeship learning
3419:information extraction
3375:
3345:
3322:
3302:
3282:
3262:
3238:
3169:
3113:
3060:
3040:
3020:
2993:
2969:
2917:
2897:
2873:
2817:
2750:
2729:
2696:
2676:
2650:
2591:
2577:ensuring the function
2567:
2543:
2515:
2491:
2471:
2430:
2329:
2289:
2252:
2204:
2176:
2132:
2108:
2081:
2054:
2047:
1949:
1848:support-vector machine
1832:
1803:
1777:
1739:
1688:
1609:
1564:
1514:
1494:
1467:
1411:
1349:
1291:
1262:
1231:
1208:
1181:
1133:
977:support-vector machine
810:Bias–variance tradeoff
692:Reinforcement learning
668:Spiking neural network
78:Reinforcement learning
3466:Neural tangent kernel
3381:can also be called a
3376:
3346:
3323:
3303:
3283:
3263:
3239:
3170:
3114:
3061:
3041:
3021:
2994:
2979:), then the function
2970:
2918:
2898:
2874:
2818:
2730:
2709:
2697:
2677:
2651:
2592:
2568:
2544:
2516:
2492:
2472:
2431:
2330:
2290:
2253:
2205:
2177:
2133:
2109:
2082:
2048:
1950:
1869:
1833:
1804:
1778:
1740:
1689:
1610:
1565:
1515:
1495:
1468:
1412:
1329:
1292:
1263:
1232:
1209:
1207:{\displaystyle w_{i}}
1182:
1139:-th training example
1134:
1105:Rademacher complexity
646:Neural radiance field
468:Structured prediction
191:Structured prediction
63:Unsupervised learning
3363:
3335:
3312:
3308:is a Mercer kernel,
3292:
3272:
3252:
3179:
3123:
3080:
3050:
3030:
3006:
2983:
2927:
2907:
2883:
2827:
2706:
2686:
2660:
2615:
2581:
2553:
2533:
2501:
2481:
2440:
2339:
2299:
2279:
2214:
2190:
2142:
2138:, certain functions
2118:
2091:
2069:
1959:
1874:
1813:
1793:
1752:
1698:
1626:
1574:
1526:
1504:
1477:
1424:
1305:
1272:
1245:
1221:
1191:
1143:
1123:
1103:(for example, using
993:principal components
835:Statistical learning
733:Learning with humans
525:Local outlier factor
3864:Kernel-Machines Org
3754:2010MaGeo..42..487H
3548:Pattern Recognition
3499:Similarity learning
3494:Representer theorem
3353:covariance function
2903:and all choices of
2523:inner product space
2114:in the input space
2059:learning algorithms
1683:
1216:similarity function
1093:convex optimization
1082:spectral clustering
1025:Representer theorem
1017:similarity function
678:Electrochemical RAM
585:reservoir computing
316:Logistic regression
235:Supervised learning
221:Multimodal learning
196:Feature engineering
141:Generative modeling
103:Rule-based learning
98:Curriculum learning
58:Supervised learning
33:Part of a series on
3866:—community website
3371:
3357:Gaussian processes
3341:
3318:
3298:
3278:
3258:
3234:
3165:
3109:
3056:
3036:
3016:
2989:
2965:
2913:
2893:
2869:
2813:
2692:
2672:
2646:
2599:Mercer's condition
2587:
2563:
2539:
2511:
2487:
2467:
2426:
2325:
2285:
2248:
2200:
2172:
2128:
2104:
2077:
2055:
2043:
1945:
1828:
1799:
1773:
1735:
1684:
1663:
1605:
1560:
1510:
1490:
1463:
1407:
1287:
1258:
1227:
1204:
1177:
1129:
1066:Gaussian processes
246: •
161:Density estimation
3849:978-0-262-53657-8
3450:Polynomial kernel
3407:3D reconstruction
3383:covariance matrix
3344:{\displaystyle k}
3321:{\displaystyle k}
3301:{\displaystyle k}
3281:{\displaystyle k}
3261:{\displaystyle k}
3073:Theoretically, a
2992:{\displaystyle k}
2916:{\displaystyle n}
2695:{\displaystyle T}
2590:{\displaystyle k}
2288:{\displaystyle k}
2186:in another space
2063:decision boundary
1854:on tasks such as
1844:kernel perceptron
1825:
1622:labeled examples
1513:{\displaystyle y}
1436:
1317:
1299:binary classifier
1230:{\displaystyle k}
1132:{\displaystyle i}
1088:and many others.
1062:kernel perceptron
961:
960:
766:Model diagnostics
749:Human-in-the-loop
592:Boltzmann machine
505:Anomaly detection
301:Linear regression
216:Ontology learning
211:Grammar induction
186:Semantic analysis
181:Association rules
166:Anomaly detection
108:Neuro-symbolic AI
16:(Redirected from
3902:
3853:
3829:
3808:
3787:Shawe-Taylor, J.
3774:
3773:
3735:
3729:
3728:
3710:
3704:
3703:
3701:
3700:
3687:Sewell, Martin.
3684:
3678:
3677:
3656:
3650:
3649:
3639:
3629:
3605:
3599:
3598:
3596:
3580:
3568:
3562:
3561:
3543:
3537:
3536:
3534:
3533:
3519:
3380:
3378:
3377:
3372:
3370:
3350:
3348:
3347:
3342:
3327:
3325:
3324:
3319:
3307:
3305:
3304:
3299:
3287:
3285:
3284:
3279:
3267:
3265:
3264:
3259:
3243:
3241:
3240:
3235:
3230:
3229:
3224:
3215:
3214:
3209:
3194:
3193:
3174:
3172:
3171:
3166:
3161:
3160:
3155:
3140:
3139:
3134:
3119:with respect to
3118:
3116:
3115:
3110:
3108:
3107:
3096:
3087:
3065:
3063:
3062:
3057:
3045:
3043:
3042:
3037:
3025:
3023:
3022:
3017:
3015:
3014:
2998:
2996:
2995:
2990:
2974:
2972:
2971:
2966:
2961:
2960:
2942:
2941:
2922:
2920:
2919:
2914:
2902:
2900:
2899:
2894:
2892:
2891:
2878:
2876:
2875:
2870:
2865:
2864:
2859:
2844:
2843:
2838:
2822:
2820:
2819:
2814:
2806:
2805:
2796:
2795:
2783:
2782:
2777:
2768:
2767:
2762:
2749:
2744:
2728:
2723:
2701:
2699:
2698:
2693:
2681:
2679:
2678:
2673:
2655:
2653:
2652:
2647:
2645:
2637:
2610:counting measure
2596:
2594:
2593:
2588:
2572:
2570:
2569:
2564:
2562:
2561:
2548:
2546:
2545:
2540:
2527:Mercer's theorem
2520:
2518:
2517:
2512:
2510:
2509:
2496:
2494:
2493:
2488:
2476:
2474:
2473:
2468:
2466:
2465:
2464:
2435:
2433:
2432:
2427:
2422:
2421:
2420:
2407:
2406:
2385:
2365:
2364:
2352:
2335:which satisfies
2334:
2332:
2331:
2326:
2324:
2323:
2314:
2313:
2294:
2292:
2291:
2286:
2257:
2255:
2254:
2249:
2247:
2239:
2238:
2229:
2228:
2209:
2207:
2206:
2201:
2199:
2198:
2181:
2179:
2178:
2173:
2168:
2167:
2155:
2137:
2135:
2134:
2129:
2127:
2126:
2113:
2111:
2110:
2105:
2103:
2102:
2086:
2084:
2083:
2078:
2076:
2052:
2050:
2049:
2044:
2042:
2041:
2036:
2032:
2022:
2021:
2016:
2012:
1999:
1991:
1980:
1972:
1954:
1952:
1951:
1946:
1941:
1940:
1928:
1927:
1837:
1835:
1834:
1829:
1827:
1826:
1818:
1808:
1806:
1805:
1800:
1782:
1780:
1779:
1774:
1772:
1764:
1763:
1744:
1742:
1741:
1736:
1710:
1709:
1693:
1691:
1690:
1685:
1682:
1677:
1659:
1658:
1646:
1645:
1640:
1621:
1614:
1612:
1611:
1606:
1604:
1603:
1594:
1593:
1581:
1569:
1567:
1566:
1561:
1559:
1551:
1550:
1541:
1540:
1519:
1517:
1516:
1511:
1499:
1497:
1496:
1491:
1489:
1488:
1472:
1470:
1469:
1464:
1438:
1437:
1429:
1416:
1414:
1413:
1408:
1400:
1399:
1387:
1386:
1381:
1369:
1368:
1359:
1358:
1348:
1343:
1319:
1318:
1310:
1296:
1294:
1293:
1288:
1286:
1285:
1280:
1267:
1265:
1264:
1259:
1257:
1256:
1236:
1234:
1233:
1228:
1213:
1211:
1210:
1205:
1203:
1202:
1186:
1184:
1183:
1178:
1173:
1172:
1160:
1159:
1154:
1138:
1136:
1135:
1130:
1078:ridge regression
1032:kernel functions
981:pattern analysis
973:pattern analysis
965:machine learning
953:
946:
939:
900:Related articles
777:Confusion matrix
530:Isolation forest
475:Graphical models
254:
253:
206:Learning to rank
201:Feature learning
39:Machine learning
30:
29:
21:
3910:
3909:
3905:
3904:
3903:
3901:
3900:
3899:
3875:
3874:
3860:
3850:
3826:
3805:
3791:Cristianini, N.
3782:
3780:Further reading
3777:
3736:
3732:
3725:
3711:
3707:
3698:
3696:
3685:
3681:
3674:
3657:
3653:
3606:
3602:
3569:
3565:
3558:
3544:
3540:
3531:
3529:
3523:"Kernel method"
3521:
3520:
3516:
3512:
3504:Cover's theorem
3480:
3445:Kernel smoother
3431:
3429:Popular kernels
3415:cheminformatics
3391:
3366:
3364:
3361:
3360:
3336:
3333:
3332:
3313:
3310:
3309:
3293:
3290:
3289:
3273:
3270:
3269:
3253:
3250:
3249:
3225:
3220:
3219:
3210:
3205:
3204:
3186:
3182:
3180:
3177:
3176:
3156:
3151:
3150:
3135:
3130:
3129:
3124:
3121:
3120:
3097:
3092:
3091:
3083:
3081:
3078:
3077:
3051:
3048:
3047:
3031:
3028:
3027:
3010:
3009:
3007:
3004:
3003:
2984:
2981:
2980:
2956:
2952:
2937:
2933:
2928:
2925:
2924:
2908:
2905:
2904:
2887:
2886:
2884:
2881:
2880:
2860:
2855:
2854:
2839:
2834:
2833:
2828:
2825:
2824:
2801:
2797:
2791:
2787:
2778:
2773:
2772:
2763:
2758:
2757:
2745:
2734:
2724:
2713:
2707:
2704:
2703:
2687:
2684:
2683:
2661:
2658:
2657:
2641:
2633:
2616:
2613:
2612:
2582:
2579:
2578:
2557:
2556:
2554:
2551:
2550:
2534:
2531:
2530:
2505:
2504:
2502:
2499:
2498:
2482:
2479:
2478:
2460:
2459:
2455:
2441:
2438:
2437:
2416:
2415:
2411:
2399:
2398:
2381:
2357:
2356:
2348:
2340:
2337:
2336:
2319:
2318:
2309:
2308:
2300:
2297:
2296:
2280:
2277:
2276:
2265:kernel function
2243:
2234:
2233:
2224:
2223:
2215:
2212:
2211:
2210:. The function
2194:
2193:
2191:
2188:
2187:
2160:
2159:
2151:
2143:
2140:
2139:
2122:
2121:
2119:
2116:
2115:
2095:
2094:
2092:
2089:
2088:
2072:
2070:
2067:
2066:
2037:
2028:
2024:
2023:
2017:
2008:
2004:
2003:
1995:
1987:
1976:
1968:
1960:
1957:
1956:
1936:
1932:
1923:
1919:
1875:
1872:
1871:
1864:
1852:neural networks
1817:
1816:
1814:
1811:
1810:
1794:
1791:
1790:
1768:
1759:
1755:
1753:
1750:
1749:
1705:
1701:
1699:
1696:
1695:
1678:
1667:
1654:
1650:
1641:
1636:
1635:
1627:
1624:
1623:
1619:
1599:
1598:
1586:
1585:
1577:
1575:
1572:
1571:
1555:
1546:
1545:
1536:
1535:
1527:
1524:
1523:
1520:is of interest;
1505:
1502:
1501:
1481:
1480:
1478:
1475:
1474:
1428:
1427:
1425:
1422:
1421:
1392:
1391:
1382:
1377:
1376:
1364:
1360:
1354:
1350:
1344:
1333:
1309:
1308:
1306:
1303:
1302:
1281:
1276:
1275:
1273:
1270:
1269:
1249:
1248:
1246:
1243:
1242:
1222:
1219:
1218:
1198:
1194:
1192:
1189:
1188:
1168:
1164:
1155:
1150:
1149:
1144:
1141:
1140:
1124:
1121:
1120:
1113:
1001:classifications
969:kernel machines
957:
928:
927:
901:
893:
892:
853:
845:
844:
805:Kernel machines
800:
792:
791:
767:
759:
758:
739:Active learning
734:
726:
725:
694:
684:
683:
609:Diffusion model
545:
535:
534:
507:
497:
496:
470:
460:
459:
415:Factor analysis
410:
400:
399:
383:
346:
336:
335:
256:
255:
239:
238:
237:
226:
225:
131:
123:
122:
88:Online learning
53:
41:
28:
23:
22:
15:
12:
11:
5:
3908:
3898:
3897:
3892:
3887:
3873:
3872:
3867:
3859:
3858:External links
3856:
3855:
3854:
3848:
3830:
3824:
3809:
3803:
3781:
3778:
3776:
3775:
3748:(5): 487–517.
3730:
3723:
3705:
3679:
3672:
3660:Mohri, Mehryar
3651:
3600:
3594:10.1.1.17.7215
3563:
3556:
3538:
3513:
3511:
3508:
3507:
3506:
3501:
3496:
3491:
3486:
3479:
3476:
3475:
3474:
3468:
3463:
3461:String kernels
3458:
3452:
3447:
3442:
3437:
3430:
3427:
3411:bioinformatics
3390:
3387:
3369:
3340:
3317:
3297:
3277:
3257:
3233:
3228:
3223:
3218:
3213:
3208:
3203:
3200:
3197:
3192:
3189:
3185:
3164:
3159:
3154:
3149:
3146:
3143:
3138:
3133:
3128:
3106:
3103:
3100:
3095:
3090:
3086:
3055:
3035:
3013:
2988:
2964:
2959:
2955:
2951:
2948:
2945:
2940:
2936:
2932:
2912:
2890:
2868:
2863:
2858:
2853:
2850:
2847:
2842:
2837:
2832:
2812:
2809:
2804:
2800:
2794:
2790:
2786:
2781:
2776:
2771:
2766:
2761:
2756:
2753:
2748:
2743:
2740:
2737:
2733:
2727:
2722:
2719:
2716:
2712:
2691:
2671:
2668:
2665:
2644:
2640:
2636:
2632:
2629:
2626:
2623:
2620:
2586:
2560:
2538:
2508:
2486:
2463:
2458:
2454:
2451:
2448:
2445:
2425:
2419:
2414:
2410:
2405:
2402:
2397:
2394:
2391:
2388:
2384:
2380:
2377:
2374:
2371:
2368:
2363:
2360:
2355:
2351:
2347:
2344:
2322:
2317:
2312:
2307:
2304:
2284:
2246:
2242:
2237:
2232:
2227:
2222:
2219:
2197:
2171:
2166:
2163:
2158:
2154:
2150:
2147:
2125:
2101:
2098:
2075:
2040:
2035:
2031:
2027:
2020:
2015:
2011:
2007:
2002:
1998:
1994:
1990:
1986:
1983:
1979:
1975:
1971:
1967:
1964:
1944:
1939:
1935:
1931:
1926:
1922:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1863:
1860:
1840:
1839:
1824:
1821:
1798:
1784:
1771:
1767:
1762:
1758:
1746:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1708:
1704:
1681:
1676:
1673:
1670:
1666:
1662:
1657:
1653:
1649:
1644:
1639:
1634:
1631:
1616:
1602:
1597:
1592:
1589:
1584:
1580:
1558:
1554:
1549:
1544:
1539:
1534:
1531:
1521:
1509:
1487:
1484:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1435:
1432:
1406:
1403:
1398:
1395:
1390:
1385:
1380:
1375:
1372:
1367:
1363:
1357:
1353:
1347:
1342:
1339:
1336:
1332:
1328:
1325:
1322:
1316:
1313:
1284:
1279:
1255:
1252:
1226:
1201:
1197:
1176:
1171:
1167:
1163:
1158:
1153:
1148:
1128:
1112:
1109:
1043:inner products
1021:inner products
1005:feature vector
959:
958:
956:
955:
948:
941:
933:
930:
929:
926:
925:
920:
919:
918:
908:
902:
899:
898:
895:
894:
891:
890:
885:
880:
875:
870:
865:
860:
854:
851:
850:
847:
846:
843:
842:
837:
832:
827:
825:Occam learning
822:
817:
812:
807:
801:
798:
797:
794:
793:
790:
789:
784:
782:Learning curve
779:
774:
768:
765:
764:
761:
760:
757:
756:
751:
746:
741:
735:
732:
731:
728:
727:
724:
723:
722:
721:
711:
706:
701:
695:
690:
689:
686:
685:
682:
681:
675:
670:
665:
660:
659:
658:
648:
643:
642:
641:
636:
631:
626:
616:
611:
606:
601:
600:
599:
589:
588:
587:
582:
577:
572:
562:
557:
552:
546:
541:
540:
537:
536:
533:
532:
527:
522:
514:
508:
503:
502:
499:
498:
495:
494:
493:
492:
487:
482:
471:
466:
465:
462:
461:
458:
457:
452:
447:
442:
437:
432:
427:
422:
417:
411:
406:
405:
402:
401:
398:
397:
392:
387:
381:
376:
371:
363:
358:
353:
347:
342:
341:
338:
337:
334:
333:
328:
323:
318:
313:
308:
303:
298:
290:
289:
288:
283:
278:
268:
266:Decision trees
263:
257:
243:classification
233:
232:
231:
228:
227:
224:
223:
218:
213:
208:
203:
198:
193:
188:
183:
178:
173:
168:
163:
158:
153:
148:
143:
138:
136:Classification
132:
129:
128:
125:
124:
121:
120:
115:
110:
105:
100:
95:
93:Batch learning
90:
85:
80:
75:
70:
65:
60:
54:
51:
50:
47:
46:
35:
34:
26:
9:
6:
4:
3:
2:
3907:
3896:
3893:
3891:
3890:Geostatistics
3888:
3886:
3883:
3882:
3880:
3871:
3868:
3865:
3862:
3861:
3851:
3845:
3842:. MIT Press.
3841:
3840:
3835:
3834:Schölkopf, B.
3831:
3827:
3825:9781118211212
3821:
3817:
3816:
3810:
3806:
3804:9780511809682
3800:
3796:
3792:
3788:
3784:
3783:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3742:
3734:
3726:
3724:0-262-18253-X
3720:
3717:. MIT Press.
3716:
3709:
3695:on 2018-10-15
3694:
3690:
3683:
3675:
3673:9780262018258
3669:
3665:
3661:
3655:
3647:
3643:
3638:
3633:
3628:
3623:
3619:
3615:
3611:
3604:
3595:
3590:
3586:
3578:
3574:
3567:
3559:
3557:9780080949123
3553:
3549:
3542:
3528:
3524:
3518:
3514:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3481:
3473:(NNGP) kernel
3472:
3469:
3467:
3464:
3462:
3459:
3456:
3453:
3451:
3448:
3446:
3443:
3441:
3440:Graph kernels
3438:
3436:
3435:Fisher kernel
3433:
3432:
3426:
3424:
3420:
3416:
3412:
3408:
3404:
3400:
3396:
3395:geostatistics
3386:
3384:
3358:
3354:
3338:
3329:
3315:
3295:
3275:
3255:
3247:
3226:
3216:
3211:
3198:
3195:
3190:
3187:
3183:
3157:
3147:
3144:
3141:
3136:
3104:
3101:
3098:
3088:
3076:
3071:
3069:
3053:
3033:
3000:
2986:
2978:
2957:
2953:
2949:
2946:
2943:
2938:
2934:
2910:
2861:
2851:
2848:
2845:
2840:
2810:
2807:
2802:
2798:
2792:
2788:
2779:
2769:
2764:
2751:
2746:
2741:
2738:
2735:
2731:
2725:
2720:
2717:
2714:
2710:
2689:
2669:
2666:
2663:
2638:
2630:
2624:
2618:
2611:
2607:
2602:
2600:
2584:
2576:
2536:
2528:
2524:
2484:
2452:
2449:
2446:
2423:
2403:
2392:
2389:
2375:
2369:
2361:
2353:
2342:
2305:
2302:
2282:
2273:
2271:
2267:
2266:
2261:
2230:
2220:
2217:
2185:
2184:inner product
2164:
2156:
2145:
2099:
2064:
2060:
2038:
2018:
2000:
1992:
1984:
1973:
1962:
1937:
1933:
1929:
1924:
1920:
1916:
1913:
1910:
1907:
1901:
1892:
1889:
1886:
1877:
1868:
1859:
1857:
1853:
1849:
1845:
1819:
1796:
1789:
1788:sign function
1785:
1765:
1760:
1756:
1747:
1729:
1726:
1723:
1720:
1717:
1711:
1706:
1702:
1679:
1674:
1671:
1668:
1655:
1651:
1647:
1642:
1617:
1595:
1590:
1582:
1542:
1532:
1529:
1522:
1507:
1485:
1457:
1454:
1451:
1448:
1445:
1439:
1430:
1420:
1419:
1418:
1404:
1396:
1388:
1383:
1370:
1365:
1361:
1355:
1351:
1345:
1340:
1337:
1334:
1330:
1326:
1323:
1320:
1311:
1300:
1282:
1253:
1240:
1224:
1217:
1199:
1195:
1169:
1165:
1161:
1156:
1126:
1118:
1108:
1106:
1102:
1098:
1097:eigenproblems
1094:
1089:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1058:
1056:
1052:
1048:
1044:
1040:
1039:feature space
1037:
1033:
1028:
1026:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
974:
970:
966:
954:
949:
947:
942:
940:
935:
934:
932:
931:
924:
921:
917:
914:
913:
912:
909:
907:
904:
903:
897:
896:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
855:
849:
848:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
802:
796:
795:
788:
785:
783:
780:
778:
775:
773:
770:
769:
763:
762:
755:
752:
750:
747:
745:
744:Crowdsourcing
742:
740:
737:
736:
730:
729:
720:
717:
716:
715:
712:
710:
707:
705:
702:
700:
697:
696:
693:
688:
687:
679:
676:
674:
673:Memtransistor
671:
669:
666:
664:
661:
657:
654:
653:
652:
649:
647:
644:
640:
637:
635:
632:
630:
627:
625:
622:
621:
620:
617:
615:
612:
610:
607:
605:
602:
598:
595:
594:
593:
590:
586:
583:
581:
578:
576:
573:
571:
568:
567:
566:
563:
561:
558:
556:
555:Deep learning
553:
551:
548:
547:
544:
539:
538:
531:
528:
526:
523:
521:
519:
515:
513:
510:
509:
506:
501:
500:
491:
490:Hidden Markov
488:
486:
483:
481:
478:
477:
476:
473:
472:
469:
464:
463:
456:
453:
451:
448:
446:
443:
441:
438:
436:
433:
431:
428:
426:
423:
421:
418:
416:
413:
412:
409:
404:
403:
396:
393:
391:
388:
386:
382:
380:
377:
375:
372:
370:
368:
364:
362:
359:
357:
354:
352:
349:
348:
345:
340:
339:
332:
329:
327:
324:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
295:
291:
287:
286:Random forest
284:
282:
279:
277:
274:
273:
272:
269:
267:
264:
262:
259:
258:
251:
250:
245:
244:
236:
230:
229:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
171:Data cleaning
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
137:
134:
133:
127:
126:
119:
116:
114:
111:
109:
106:
104:
101:
99:
96:
94:
91:
89:
86:
84:
83:Meta-learning
81:
79:
76:
74:
71:
69:
66:
64:
61:
59:
56:
55:
49:
48:
45:
40:
37:
36:
32:
31:
19:
3838:
3814:
3794:
3745:
3739:
3733:
3714:
3708:
3697:. Retrieved
3693:the original
3682:
3663:
3654:
3627:math/0701907
3617:
3613:
3603:
3584:
3576:
3572:
3566:
3547:
3541:
3530:. Retrieved
3526:
3517:
3392:
3389:Applications
3330:
3072:
3001:
2603:
2274:
2263:
2259:
2056:
1841:
1238:
1114:
1090:
1059:
1051:kernel trick
1050:
1045:between the
1035:
1029:
1012:
1008:
997:correlations
968:
962:
830:PAC learning
517:
366:
361:Hierarchical
293:
247:
241:
18:Kernel trick
3355:as used in
3075:Gram matrix
2065:. For all
1237:, called a
1009:feature map
714:Multi-agent
651:Transformer
550:Autoencoder
306:Naive Bayes
44:data mining
3879:Categories
3699:2014-05-30
3579:: 821–837.
3532:2023-04-04
3510:References
3351:is also a
3244:, must be
2597:satisfies
1015:, i.e., a
699:Q-learning
597:Restricted
395:Mean shift
344:Clustering
321:Perceptron
249:regression
151:Clustering
146:Regression
3818:. Wiley.
3589:CiteSeerX
3581:Cited in
3145:…
3102:×
3089:∈
3054:φ
3034:φ
2947:…
2849:…
2808:≥
2732:∑
2711:∑
2667:⊂
2619:μ
2537:φ
2485:φ
2457:⟩
2453:⋅
2447:⋅
2444:⟨
2413:⟩
2393:φ
2376:φ
2373:⟨
2316:→
2306::
2303:φ
2241:→
2231:×
2221::
1993:⋅
1955:and thus
1878:φ
1823:^
1766:∈
1718:−
1712:∈
1596:∈
1553:→
1543:×
1533::
1446:−
1440:∈
1434:^
1331:∑
1327:
1315:^
858:ECML PKDD
840:VC theory
787:ROC curve
719:Self-play
639:DeepDream
480:Bayes net
271:Ensembles
52:Paradigms
3793:(2004).
3770:73657847
3646:88516979
3478:See also
2656:for all
2404:′
2362:′
2270:integral
2165:′
2100:′
2034:‖
2026:‖
2014:‖
2006:‖
1591:′
1486:′
1397:′
1254:′
1036:implicit
989:rankings
985:clusters
281:Boosting
130:Problems
3750:Bibcode
3399:kriging
2575:measure
1072:(PCA),
863:NeurIPS
680:(ECRAM)
634:AlexNet
276:Bagging
3846:
3822:
3801:
3768:
3721:
3670:
3644:
3591:
3554:
3527:Engati
2521:is an
2260:kernel
1417:where
1239:kernel
1055:graphs
1047:images
1013:kernel
656:Vision
512:RANSAC
390:OPTICS
385:DBSCAN
369:-means
176:AutoML
3766:S2CID
3642:S2CID
3622:arXiv
3620:(3).
3457:(RBF)
2975:(cf.
2262:or a
878:IJCAI
704:SARSA
663:Mamba
629:LeNet
624:U-Net
450:t-SNE
374:Fuzzy
351:BIRCH
3844:ISBN
3820:ISBN
3799:ISBN
3719:ISBN
3668:ISBN
3552:ISBN
3421:and
2087:and
1786:the
1748:the
888:JMLR
873:ICLR
868:ICML
754:RLHF
570:LSTM
356:CURE
42:and
3758:doi
3632:doi
2879:in
1797:sgn
1324:sgn
1107:).
1095:or
963:In
614:SOM
604:GAN
580:ESN
575:GRU
520:-NN
455:SDL
445:PGD
440:PCA
435:NMF
430:LDA
425:ICA
420:CCA
296:-NN
3881::
3789:;
3764:.
3756:.
3746:42
3744:.
3640:.
3630:.
3618:36
3616:.
3612:.
3577:25
3575:.
3525:.
3425:.
3417:,
3413:,
3409:,
3405:,
3401:,
3397:,
3385:.
2811:0.
2601:.
2272:.
1858:.
1084:,
1080:,
1076:,
1068:,
999:,
995:,
991:,
987:,
967:,
883:ML
3852:.
3828:.
3807:.
3772:.
3760::
3752::
3727:.
3702:.
3676:.
3648:.
3634::
3624::
3597:.
3560:.
3535:.
3368:K
3339:k
3316:k
3296:k
3276:k
3256:k
3232:)
3227:j
3222:x
3217:,
3212:i
3207:x
3202:(
3199:k
3196:=
3191:j
3188:i
3184:K
3163:}
3158:n
3153:x
3148:,
3142:,
3137:1
3132:x
3127:{
3105:n
3099:n
3094:R
3085:K
3012:X
2987:k
2963:)
2958:n
2954:c
2950:,
2944:,
2939:1
2935:c
2931:(
2911:n
2889:X
2867:)
2862:n
2857:x
2852:,
2846:,
2841:1
2836:x
2831:(
2803:j
2799:c
2793:i
2789:c
2785:)
2780:j
2775:x
2770:,
2765:i
2760:x
2755:(
2752:k
2747:n
2742:1
2739:=
2736:j
2726:n
2721:1
2718:=
2715:i
2690:T
2670:X
2664:T
2643:|
2639:T
2635:|
2631:=
2628:)
2625:T
2622:(
2585:k
2559:X
2507:V
2462:V
2450:,
2424:.
2418:V
2409:)
2401:x
2396:(
2390:,
2387:)
2383:x
2379:(
2370:=
2367:)
2359:x
2354:,
2350:x
2346:(
2343:k
2321:V
2311:X
2283:k
2245:R
2236:X
2226:X
2218:k
2196:V
2170:)
2162:x
2157:,
2153:x
2149:(
2146:k
2124:X
2097:x
2074:x
2039:2
2030:y
2019:2
2010:x
2001:+
1997:y
1989:x
1985:=
1982:)
1978:y
1974:,
1970:x
1966:(
1963:k
1943:)
1938:2
1934:b
1930:+
1925:2
1921:a
1917:,
1914:b
1911:,
1908:a
1905:(
1902:=
1899:)
1896:)
1893:b
1890:,
1887:a
1884:(
1881:(
1820:y
1770:R
1761:i
1757:w
1745:;
1733:}
1730:1
1727:+
1724:,
1721:1
1715:{
1707:i
1703:y
1680:n
1675:1
1672:=
1669:i
1665:}
1661:)
1656:i
1652:y
1648:,
1643:i
1638:x
1633:(
1630:{
1620:n
1615:;
1601:X
1588:x
1583:,
1579:x
1557:R
1548:X
1538:X
1530:k
1508:y
1483:x
1461:}
1458:1
1455:+
1452:,
1449:1
1443:{
1431:y
1405:,
1402:)
1394:x
1389:,
1384:i
1379:x
1374:(
1371:k
1366:i
1362:y
1356:i
1352:w
1346:n
1341:1
1338:=
1335:i
1321:=
1312:y
1283:i
1278:x
1251:x
1225:k
1200:i
1196:w
1175:)
1170:i
1166:y
1162:,
1157:i
1152:x
1147:(
1127:i
952:e
945:t
938:v
518:k
367:k
294:k
252:)
240:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.