Knowledge (XXG)

k-SVD

Source đź“ť

2245: 1957: 3277: 1768: 1639: 1199: 1320: 2240:{\displaystyle \|Y-DX\|_{F}^{2}=\left\|Y-\sum _{j=1}^{K}d_{j}x_{j}^{\text{T}}\right\|_{F}^{2}=\left\|\left(Y-\sum _{j\neq k}d_{j}x_{j}^{\text{T}}\right)-d_{k}x_{k}^{\text{T}}\right\|_{F}^{2}=\|E_{k}-d_{k}x_{k}^{\text{T}}\|_{F}^{2}} 2631: 986:
method, and it works by iteratively alternating between sparse coding the input data based on the current dictionary, and updating the atoms in the dictionary to better fit the data. It is structurally related to the
2931: 3112: 1650: 1518: 1085: 3101: 1210: 3015: 2862: 866: 3417: 904: 2809: 3458: 3349: 2685: 1073: 861: 3316: 2963: 2505: 2280: 851: 692: 2766: 2535: 1370: 3628: 2442: 3376: 3042: 2739: 2712: 2473: 2407: 1866: 1507: 1480: 2360: 899: 2314: 2380: 2334: 1949: 1929: 1909: 1889: 1835: 1815: 1795: 1453: 1418: 1394: 856: 707: 3472:-SVD operates by an iterative update which does not guarantee to find the global optimum. However, this is common to other algorithms for this purpose, and 438: 939: 742: 2543: 1841:
can be used for the calculation of the coefficients, as long as it can supply a solution with a fixed and predetermined number of nonzero entries
818: 988: 367: 2867: 3657: 3272:{\displaystyle \|E_{k}\Omega _{k}-d_{k}x_{k}^{\text{T}}\Omega _{k}\|_{F}^{2}=\|{\tilde {E}}_{k}-d_{k}{\tilde {x}}_{k}^{\text{T}}\|_{F}^{2}} 1763:{\displaystyle \quad \min \limits _{D,X}\sum _{i}\|x_{i}\|_{0}\qquad {\text{subject to }}\quad \forall i\;,\|Y-DX\|_{F}^{2}\leq \epsilon .} 876: 639: 174: 894: 727: 702: 651: 775: 770: 423: 1891:. However, finding the whole dictionary all at a time is impossible, so the process is to update only one column of the dictionary 1634:{\displaystyle \quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\quad \forall i\;,\|x_{i}\|_{0}\leq T_{0}.} 1194:{\displaystyle \quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\forall i,x_{i}=e_{k}{\text{ for some }}k.} 433: 71: 828: 932: 592: 413: 995:-SVD can be found widely in use in applications such as image processing, audio processing, biology, and document analysis. 803: 505: 281: 3647: 3490: 2445: 972: 760: 697: 607: 585: 428: 418: 3047: 1315:{\displaystyle \quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\quad \forall i,\|x_{i}\|_{0}=1} 911: 823: 808: 269: 91: 798: 2968: 2814: 871: 548: 443: 231: 164: 124: 3531: 3381: 925: 531: 299: 169: 3460:. After updating the whole dictionary, the process then turns to iteratively solve X, then iteratively solve D. 553: 473: 396: 314: 144: 106: 101: 61: 56: 3652: 1076: 500: 349: 249: 76: 2771: 3593: 3584:
Rubinstein, R., Bruckstein, A.M., and Elad, M. (2010), "Dictionaries for Sparse Representation Modeling",
3508: 3422: 3321: 680: 656: 558: 319: 294: 254: 66: 2639: 1027: 1021: 968: 634: 456: 408: 264: 179: 51: 3285: 2936: 2478: 2253: 1455:, the sparsity term of the constraint is relaxed so that the number of nonzero entries of each column 3546: 3485: 563: 513: 3598: 964: 953: 666: 602: 573: 478: 304: 237: 223: 209: 184: 134: 86: 46: 3611: 3562: 3500: 2744: 2513: 1335: 1014: 980: 644: 568: 354: 149: 2382:-th remains unknown. After this step, we can solve the minimization problem by approximate the 3622: 737: 580: 493: 289: 259: 204: 199: 154: 96: 3603: 3554: 2412: 1838: 765: 518: 468: 378: 362: 332: 194: 189: 139: 129: 27: 3354: 3020: 2717: 2690: 2451: 2385: 1844: 1837:
is hard, we use an approximation pursuit method. Any algorithm such as OMP, the orthogonal
1485: 1458: 793: 597: 463: 403: 2339: 3550: 2296: 3532:"K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation" 2365: 2319: 1934: 1914: 1894: 1874: 1820: 1800: 1780: 1438: 1403: 1379: 1329: 813: 344: 81: 3641: 3527: 732: 661: 543: 274: 159: 3615: 3566: 3495: 3468:
Choosing an appropriate "dictionary" for a dataset is a non-convex problem, and
538: 32: 3607: 2626:{\displaystyle \omega _{k}=\{i\mid 1\leq i\leq N,x_{k}^{\text{T}}(i)\neq 0\},} 1400:-SVD algorithm is to represent the signal as a linear combination of atoms in 687: 383: 309: 1024:. That is, finding the best possible codebook to represent the data samples 3558: 1871:
After the sparse coding task, the next is to search for a better dictionary
846: 627: 2926:{\displaystyle {\tilde {x}}_{k}^{\text{T}}=x_{k}^{\text{T}}\Omega _{k}} 622: 373: 617: 612: 339: 2965:
by discarding the zero entries. Similarly, the multiplication
1435:-means, in order to achieve a linear combination of atoms in 1376:-means algorithm to use only one atom (column) in dictionary 3106:
So the minimization problem as mentioned before becomes
3017:
is the subset of the examples that are current using the
905:
List of datasets in computer vision and image processing
3282:
and can be done by directly using SVD. SVD decomposes
3425: 3384: 3357: 3324: 3288: 3115: 3050: 3023: 2971: 2939: 2870: 2817: 2774: 2747: 2720: 2693: 2642: 2546: 2516: 2481: 2454: 2415: 2388: 2368: 2342: 2322: 2299: 2256: 1960: 1951:-th column is done by rewriting the penalty term as 1937: 1917: 1897: 1877: 1847: 1823: 1803: 1783: 1653: 1521: 1488: 1461: 1441: 1406: 1382: 1338: 1213: 1088: 1030: 1427:-SVD algorithm follows the construction flow of the 3452: 3411: 3378:is the first column of U, the coefficient vector 3370: 3343: 3310: 3271: 3095: 3036: 3009: 2957: 2925: 2856: 2803: 2760: 2733: 2706: 2679: 2625: 2529: 2499: 2475:with it. However, the new solution for the vector 2467: 2436: 2401: 2374: 2354: 2328: 2308: 2274: 2239: 1943: 1923: 1903: 1883: 1860: 1829: 1809: 1789: 1762: 1633: 1501: 1474: 1447: 1412: 1388: 1364: 1314: 1193: 1067: 3096:{\displaystyle {\tilde {E}}_{k}=E_{k}\Omega _{k}} 3579: 3577: 3575: 1797:is first fixed and the best coefficient matrix 2864:entries and zeros otherwise. When multiplying 1396:. To relax this constraint, the target of the 900:List of datasets for machine-learning research 3010:{\displaystyle {\tilde {Y}}_{k}=Y\Omega _{k}} 2857:{\displaystyle (i,\omega _{k}(i)){\text{th}}} 933: 8: 3627:: CS1 maint: multiple names: authors list ( 3255: 3195: 3178: 3116: 2657: 2643: 2617: 2560: 2223: 2181: 1977: 1961: 1737: 1721: 1695: 1681: 1606: 1592: 1572: 1558: 1542: 1539: 1297: 1283: 1264: 1250: 1234: 1231: 1139: 1125: 1109: 1106: 1045: 1031: 3412:{\displaystyle {\tilde {x}}_{k}^{\text{T}}} 1482:can be more than 1, but less than a number 3530:; Michael Elad; Alfred Bruckstein (2006), 1717: 1588: 1431:-means algorithm. However, in contrast to 940: 926: 18: 3597: 3424: 3403: 3398: 3387: 3386: 3383: 3362: 3356: 3335: 3323: 3302: 3291: 3290: 3287: 3263: 3258: 3248: 3243: 3232: 3231: 3224: 3211: 3200: 3199: 3186: 3181: 3171: 3161: 3156: 3146: 3133: 3123: 3114: 3087: 3077: 3064: 3053: 3052: 3049: 3028: 3022: 3001: 2985: 2974: 2973: 2970: 2949: 2944: 2938: 2917: 2907: 2902: 2889: 2884: 2873: 2872: 2869: 2849: 2831: 2816: 2796: 2790: 2781: 2773: 2752: 2746: 2725: 2719: 2698: 2692: 2671: 2660: 2650: 2641: 2596: 2591: 2551: 2545: 2521: 2515: 2491: 2486: 2480: 2459: 2453: 2414: 2393: 2387: 2367: 2341: 2336:rank 1 matrices, we can assume the other 2321: 2298: 2266: 2261: 2255: 2231: 2226: 2216: 2211: 2201: 2188: 2172: 2167: 2156: 2151: 2141: 2123: 2118: 2108: 2092: 2062: 2057: 2046: 2041: 2031: 2021: 2010: 1985: 1980: 1959: 1936: 1916: 1896: 1876: 1852: 1846: 1822: 1802: 1782: 1745: 1740: 1705: 1698: 1688: 1675: 1659: 1652: 1622: 1609: 1599: 1576: 1566: 1561: 1527: 1520: 1493: 1487: 1466: 1460: 1440: 1405: 1381: 1356: 1343: 1337: 1300: 1290: 1268: 1258: 1253: 1219: 1212: 1180: 1174: 1161: 1143: 1133: 1128: 1094: 1087: 1059: 1048: 1038: 1029: 1325:which is k-means that allows "weights". 967:algorithm for creating a dictionary for 3519: 1817:is found. As finding the truly optimal 989:expectation–maximization (EM) algorithm 26: 3620: 3539:IEEE Transactions on Signal Processing 3044:atom. The same effect can be seen on 2804:{\displaystyle N\times |\omega _{k}|} 7: 3476:-SVD works fairly well in practice. 3453:{\displaystyle V\times \Delta (1,1)} 3344:{\displaystyle U\Delta V^{\text{T}}} 1020:can be also regarded as a method of 1009:-SVD is a kind of generalization of 2680:{\displaystyle \{y_{i}\}_{i=1}^{N}} 1512:So, the objective function becomes 1068:{\displaystyle \{y_{i}\}_{i=1}^{M}} 895:Glossary of artificial intelligence 3432: 3328: 3168: 3130: 3084: 2998: 2914: 2749: 2293:By decomposing the multiplication 1711: 1582: 1274: 1148: 14: 2362:terms are assumed fixed, and the 1332:. The sparse representation term 3311:{\displaystyle {\tilde {E}}_{k}} 2958:{\displaystyle x_{k}^{\text{T}}} 2507:is not guaranteed to be sparse. 2500:{\displaystyle x_{k}^{\text{T}}} 2275:{\displaystyle x_{k}^{\text{T}}} 979:-SVD is a generalization of the 2741:that is nonzero). Then, define 1710: 1704: 1654: 1581: 1575: 1522: 1273: 1267: 1214: 1142: 1089: 3447: 3435: 3392: 3296: 3237: 3205: 3058: 2979: 2933:, this shrinks the row vector 2878: 2846: 2843: 2837: 2818: 2797: 2782: 2608: 2602: 2163: 2073: 2053: 1996: 1204:which is nearly equivalent to 315:Relevance vector machine (RVM) 1: 2510:To cure this problem, define 1644:or in another objective form 804:Computational learning theory 368:Expectation–maximization (EM) 16:Dictionary learning algorithm 3491:Singular value decomposition 2446:singular value decomposition 973:singular value decomposition 761:Coefficient of determination 608:Convolutional neural network 320:Support vector machine (SVM) 3658:Cluster analysis algorithms 2761:{\displaystyle \Omega _{k}} 2530:{\displaystyle \omega _{k}} 1656: 1524: 1365:{\displaystyle x_{i}=e_{k}} 1216: 1091: 912:Outline of machine learning 809:Empirical risk minimization 3674: 3608:10.1109/JPROC.2010.2040551 549:Feedforward neural network 300:Artificial neural networks 2636:which points to examples 1328:The letter F denotes the 532:Artificial neural network 1911:each time, while fixing 1013:-means, as follows. The 841:Journals and conferences 788:Mathematical foundations 698:Temporal difference (TD) 554:Recurrent neural network 474:Conditional random field 397:Dimensionality reduction 145:Dimensionality reduction 107:Quantum machine learning 102:Neuromorphic engineering 62:Self-supervised learning 57:Semi-supervised learning 3586:Proceedings of the IEEE 3559:10.1109/TSP.2006.881199 3419:as the first column of 250:Apprenticeship learning 3509:Low-rank approximation 3454: 3413: 3372: 3345: 3312: 3273: 3097: 3038: 3011: 2959: 2927: 2858: 2805: 2762: 2735: 2708: 2681: 2627: 2531: 2501: 2469: 2438: 2437:{\displaystyle rank-1} 2403: 2376: 2356: 2330: 2310: 2276: 2241: 2026: 1945: 1925: 1905: 1885: 1862: 1831: 1811: 1791: 1764: 1635: 1503: 1476: 1449: 1414: 1390: 1366: 1316: 1195: 1069: 969:sparse representations 799:Bias–variance tradeoff 681:Reinforcement learning 657:Spiking neural network 67:Reinforcement learning 3455: 3414: 3373: 3371:{\displaystyle d_{k}} 3346: 3313: 3274: 3098: 3039: 3037:{\displaystyle d_{k}} 3012: 2960: 2928: 2859: 2806: 2768:as a matrix of size 2763: 2736: 2734:{\displaystyle x_{i}} 2714:(also the entries of 2709: 2707:{\displaystyle d_{k}} 2682: 2628: 2532: 2502: 2470: 2468:{\displaystyle d_{k}} 2439: 2404: 2402:{\displaystyle E_{k}} 2377: 2357: 2331: 2311: 2277: 2242: 2006: 1946: 1926: 1906: 1886: 1863: 1861:{\displaystyle T_{0}} 1832: 1812: 1792: 1765: 1636: 1504: 1502:{\displaystyle T_{0}} 1477: 1475:{\displaystyle x_{i}} 1450: 1415: 1391: 1367: 1317: 1196: 1070: 1022:sparse representation 635:Neural radiance field 457:Structured prediction 180:Structured prediction 52:Unsupervised learning 3486:Sparse approximation 3423: 3382: 3355: 3322: 3286: 3113: 3048: 3021: 2969: 2937: 2868: 2815: 2772: 2745: 2718: 2691: 2640: 2544: 2514: 2479: 2452: 2413: 2386: 2366: 2340: 2320: 2297: 2254: 1958: 1935: 1931:. The update of the 1915: 1895: 1875: 1845: 1821: 1801: 1781: 1777:-SVD algorithm, the 1651: 1519: 1486: 1459: 1439: 1404: 1380: 1336: 1211: 1182: for some  1086: 1028: 824:Statistical learning 722:Learning with humans 514:Local outlier factor 3648:Norms (mathematics) 3551:2006ITSP...54.4311A 3408: 3351:. The solution for 3268: 3253: 3191: 3166: 2954: 2912: 2894: 2811:, with ones on the 2676: 2601: 2496: 2355:{\displaystyle K-1} 2271: 2236: 2221: 2177: 2161: 2128: 2067: 2051: 1990: 1750: 1571: 1263: 1138: 1064: 965:dictionary learning 954:applied mathematics 667:Electrochemical RAM 574:reservoir computing 305:Logistic regression 224:Supervised learning 210:Multimodal learning 185:Feature engineering 130:Generative modeling 92:Rule-based learning 87:Curriculum learning 47:Supervised learning 22:Part of a series on 3450: 3409: 3385: 3368: 3341: 3308: 3269: 3254: 3230: 3177: 3152: 3093: 3034: 3007: 2955: 2940: 2923: 2898: 2871: 2854: 2801: 2758: 2731: 2704: 2677: 2656: 2623: 2587: 2527: 2497: 2482: 2465: 2434: 2399: 2372: 2352: 2326: 2309:{\displaystyle DX} 2306: 2272: 2257: 2237: 2222: 2207: 2147: 2114: 2103: 2071: 2037: 1994: 1976: 1941: 1921: 1901: 1881: 1858: 1827: 1807: 1787: 1760: 1736: 1680: 1670: 1631: 1557: 1538: 1499: 1472: 1445: 1410: 1386: 1362: 1312: 1249: 1230: 1191: 1124: 1105: 1065: 1044: 235: • 150:Density estimation 3545:(11): 4311–4322, 3504:-means clustering 3406: 3395: 3338: 3299: 3251: 3240: 3208: 3164: 3061: 2982: 2952: 2910: 2892: 2881: 2852: 2599: 2494: 2375:{\displaystyle k} 2329:{\displaystyle K} 2269: 2219: 2159: 2126: 2088: 2049: 1944:{\displaystyle k} 1924:{\displaystyle X} 1904:{\displaystyle D} 1884:{\displaystyle D} 1830:{\displaystyle X} 1810:{\displaystyle X} 1790:{\displaystyle D} 1708: 1671: 1655: 1579: 1523: 1448:{\displaystyle D} 1413:{\displaystyle D} 1389:{\displaystyle D} 1271: 1215: 1183: 1146: 1090: 1018:-means clustering 984:-means clustering 950: 949: 755:Model diagnostics 738:Human-in-the-loop 581:Boltzmann machine 494:Anomaly detection 290:Linear regression 205:Ontology learning 200:Grammar induction 175:Semantic analysis 170:Association rules 155:Anomaly detection 97:Neuro-symbolic AI 3665: 3633: 3632: 3626: 3618: 3601: 3592:(6): 1045–1057, 3581: 3570: 3569: 3536: 3524: 3459: 3457: 3456: 3451: 3418: 3416: 3415: 3410: 3407: 3404: 3402: 3397: 3396: 3388: 3377: 3375: 3374: 3369: 3367: 3366: 3350: 3348: 3347: 3342: 3340: 3339: 3336: 3317: 3315: 3314: 3309: 3307: 3306: 3301: 3300: 3292: 3278: 3276: 3275: 3270: 3267: 3262: 3252: 3249: 3247: 3242: 3241: 3233: 3229: 3228: 3216: 3215: 3210: 3209: 3201: 3190: 3185: 3176: 3175: 3165: 3162: 3160: 3151: 3150: 3138: 3137: 3128: 3127: 3102: 3100: 3099: 3094: 3092: 3091: 3082: 3081: 3069: 3068: 3063: 3062: 3054: 3043: 3041: 3040: 3035: 3033: 3032: 3016: 3014: 3013: 3008: 3006: 3005: 2990: 2989: 2984: 2983: 2975: 2964: 2962: 2961: 2956: 2953: 2950: 2948: 2932: 2930: 2929: 2924: 2922: 2921: 2911: 2908: 2906: 2893: 2890: 2888: 2883: 2882: 2874: 2863: 2861: 2860: 2855: 2853: 2850: 2836: 2835: 2810: 2808: 2807: 2802: 2800: 2795: 2794: 2785: 2767: 2765: 2764: 2759: 2757: 2756: 2740: 2738: 2737: 2732: 2730: 2729: 2713: 2711: 2710: 2705: 2703: 2702: 2686: 2684: 2683: 2678: 2675: 2670: 2655: 2654: 2632: 2630: 2629: 2624: 2600: 2597: 2595: 2556: 2555: 2536: 2534: 2533: 2528: 2526: 2525: 2506: 2504: 2503: 2498: 2495: 2492: 2490: 2474: 2472: 2471: 2466: 2464: 2463: 2443: 2441: 2440: 2435: 2408: 2406: 2405: 2400: 2398: 2397: 2381: 2379: 2378: 2373: 2361: 2359: 2358: 2353: 2335: 2333: 2332: 2327: 2315: 2313: 2312: 2307: 2281: 2279: 2278: 2273: 2270: 2267: 2265: 2246: 2244: 2243: 2238: 2235: 2230: 2220: 2217: 2215: 2206: 2205: 2193: 2192: 2176: 2171: 2166: 2162: 2160: 2157: 2155: 2146: 2145: 2133: 2129: 2127: 2124: 2122: 2113: 2112: 2102: 2066: 2061: 2056: 2052: 2050: 2047: 2045: 2036: 2035: 2025: 2020: 1989: 1984: 1950: 1948: 1947: 1942: 1930: 1928: 1927: 1922: 1910: 1908: 1907: 1902: 1890: 1888: 1887: 1882: 1867: 1865: 1864: 1859: 1857: 1856: 1839:matching pursuit 1836: 1834: 1833: 1828: 1816: 1814: 1813: 1808: 1796: 1794: 1793: 1788: 1769: 1767: 1766: 1761: 1749: 1744: 1709: 1707:subject to  1706: 1703: 1702: 1693: 1692: 1679: 1669: 1640: 1638: 1637: 1632: 1627: 1626: 1614: 1613: 1604: 1603: 1580: 1578:subject to  1577: 1570: 1565: 1537: 1508: 1506: 1505: 1500: 1498: 1497: 1481: 1479: 1478: 1473: 1471: 1470: 1454: 1452: 1451: 1446: 1419: 1417: 1416: 1411: 1395: 1393: 1392: 1387: 1371: 1369: 1368: 1363: 1361: 1360: 1348: 1347: 1321: 1319: 1318: 1313: 1305: 1304: 1295: 1294: 1272: 1270:subject to  1269: 1262: 1257: 1229: 1200: 1198: 1197: 1192: 1184: 1181: 1179: 1178: 1166: 1165: 1147: 1145:subject to  1144: 1137: 1132: 1104: 1077:nearest neighbor 1074: 1072: 1071: 1066: 1063: 1058: 1043: 1042: 942: 935: 928: 889:Related articles 766:Confusion matrix 519:Isolation forest 464:Graphical models 243: 242: 195:Learning to rank 190:Feature learning 28:Machine learning 19: 3673: 3672: 3668: 3667: 3666: 3664: 3663: 3662: 3638: 3637: 3636: 3619: 3583: 3582: 3573: 3534: 3526: 3525: 3521: 3517: 3482: 3466: 3421: 3420: 3380: 3379: 3358: 3353: 3352: 3331: 3320: 3319: 3289: 3284: 3283: 3220: 3198: 3167: 3142: 3129: 3119: 3111: 3110: 3083: 3073: 3051: 3046: 3045: 3024: 3019: 3018: 2997: 2972: 2967: 2966: 2935: 2934: 2913: 2866: 2865: 2827: 2813: 2812: 2786: 2770: 2769: 2748: 2743: 2742: 2721: 2716: 2715: 2694: 2689: 2688: 2646: 2638: 2637: 2547: 2542: 2541: 2517: 2512: 2511: 2477: 2476: 2455: 2450: 2449: 2411: 2410: 2389: 2384: 2383: 2364: 2363: 2338: 2337: 2318: 2317: 2295: 2294: 2252: 2251: 2197: 2184: 2137: 2104: 2081: 2077: 2076: 2072: 2027: 1999: 1995: 1956: 1955: 1933: 1932: 1913: 1912: 1893: 1892: 1873: 1872: 1848: 1843: 1842: 1819: 1818: 1799: 1798: 1779: 1778: 1694: 1684: 1649: 1648: 1618: 1605: 1595: 1517: 1516: 1489: 1484: 1483: 1462: 1457: 1456: 1437: 1436: 1402: 1401: 1378: 1377: 1352: 1339: 1334: 1333: 1296: 1286: 1209: 1208: 1170: 1157: 1084: 1083: 1034: 1026: 1025: 1004: 946: 917: 916: 890: 882: 881: 842: 834: 833: 794:Kernel machines 789: 781: 780: 756: 748: 747: 728:Active learning 723: 715: 714: 683: 673: 672: 598:Diffusion model 534: 524: 523: 496: 486: 485: 459: 449: 448: 404:Factor analysis 399: 389: 388: 372: 335: 325: 324: 245: 244: 228: 227: 226: 215: 214: 120: 112: 111: 77:Online learning 42: 30: 17: 12: 11: 5: 3671: 3669: 3661: 3660: 3655: 3653:Linear algebra 3650: 3640: 3639: 3635: 3634: 3599:10.1.1.160.527 3571: 3518: 3516: 3513: 3512: 3511: 3506: 3498: 3493: 3488: 3481: 3478: 3465: 3462: 3449: 3446: 3443: 3440: 3437: 3434: 3431: 3428: 3401: 3394: 3391: 3365: 3361: 3334: 3330: 3327: 3305: 3298: 3295: 3280: 3279: 3266: 3261: 3257: 3246: 3239: 3236: 3227: 3223: 3219: 3214: 3207: 3204: 3197: 3194: 3189: 3184: 3180: 3174: 3170: 3159: 3155: 3149: 3145: 3141: 3136: 3132: 3126: 3122: 3118: 3090: 3086: 3080: 3076: 3072: 3067: 3060: 3057: 3031: 3027: 3004: 3000: 2996: 2993: 2988: 2981: 2978: 2947: 2943: 2920: 2916: 2905: 2901: 2897: 2887: 2880: 2877: 2848: 2845: 2842: 2839: 2834: 2830: 2826: 2823: 2820: 2799: 2793: 2789: 2784: 2780: 2777: 2755: 2751: 2728: 2724: 2701: 2697: 2687:that use atom 2674: 2669: 2666: 2663: 2659: 2653: 2649: 2645: 2634: 2633: 2622: 2619: 2616: 2613: 2610: 2607: 2604: 2594: 2590: 2586: 2583: 2580: 2577: 2574: 2571: 2568: 2565: 2562: 2559: 2554: 2550: 2524: 2520: 2489: 2485: 2462: 2458: 2448:, then update 2433: 2430: 2427: 2424: 2421: 2418: 2396: 2392: 2371: 2351: 2348: 2345: 2325: 2305: 2302: 2264: 2260: 2248: 2247: 2234: 2229: 2225: 2214: 2210: 2204: 2200: 2196: 2191: 2187: 2183: 2180: 2175: 2170: 2165: 2154: 2150: 2144: 2140: 2136: 2132: 2121: 2117: 2111: 2107: 2101: 2098: 2095: 2091: 2087: 2084: 2080: 2075: 2070: 2065: 2060: 2055: 2044: 2040: 2034: 2030: 2024: 2019: 2016: 2013: 2009: 2005: 2002: 1998: 1993: 1988: 1983: 1979: 1975: 1972: 1969: 1966: 1963: 1940: 1920: 1900: 1880: 1855: 1851: 1826: 1806: 1786: 1771: 1770: 1759: 1756: 1753: 1748: 1743: 1739: 1735: 1732: 1729: 1726: 1723: 1720: 1716: 1713: 1701: 1697: 1691: 1687: 1683: 1678: 1674: 1668: 1665: 1662: 1658: 1642: 1641: 1630: 1625: 1621: 1617: 1612: 1608: 1602: 1598: 1594: 1591: 1587: 1584: 1574: 1569: 1564: 1560: 1556: 1553: 1550: 1547: 1544: 1541: 1536: 1533: 1530: 1526: 1496: 1492: 1469: 1465: 1444: 1409: 1385: 1359: 1355: 1351: 1346: 1342: 1330:Frobenius norm 1323: 1322: 1311: 1308: 1303: 1299: 1293: 1289: 1285: 1282: 1279: 1276: 1266: 1261: 1256: 1252: 1248: 1245: 1242: 1239: 1236: 1233: 1228: 1225: 1222: 1218: 1202: 1201: 1190: 1187: 1177: 1173: 1169: 1164: 1160: 1156: 1153: 1150: 1141: 1136: 1131: 1127: 1123: 1120: 1117: 1114: 1111: 1108: 1103: 1100: 1097: 1093: 1062: 1057: 1054: 1051: 1047: 1041: 1037: 1033: 1003: 1002:-SVD algorithm 997: 948: 947: 945: 944: 937: 930: 922: 919: 918: 915: 914: 909: 908: 907: 897: 891: 888: 887: 884: 883: 880: 879: 874: 869: 864: 859: 854: 849: 843: 840: 839: 836: 835: 832: 831: 826: 821: 816: 814:Occam learning 811: 806: 801: 796: 790: 787: 786: 783: 782: 779: 778: 773: 771:Learning curve 768: 763: 757: 754: 753: 750: 749: 746: 745: 740: 735: 730: 724: 721: 720: 717: 716: 713: 712: 711: 710: 700: 695: 690: 684: 679: 678: 675: 674: 671: 670: 664: 659: 654: 649: 648: 647: 637: 632: 631: 630: 625: 620: 615: 605: 600: 595: 590: 589: 588: 578: 577: 576: 571: 566: 561: 551: 546: 541: 535: 530: 529: 526: 525: 522: 521: 516: 511: 503: 497: 492: 491: 488: 487: 484: 483: 482: 481: 476: 471: 460: 455: 454: 451: 450: 447: 446: 441: 436: 431: 426: 421: 416: 411: 406: 400: 395: 394: 391: 390: 387: 386: 381: 376: 370: 365: 360: 352: 347: 342: 336: 331: 330: 327: 326: 323: 322: 317: 312: 307: 302: 297: 292: 287: 279: 278: 277: 272: 267: 257: 255:Decision trees 252: 246: 232:classification 222: 221: 220: 217: 216: 213: 212: 207: 202: 197: 192: 187: 182: 177: 172: 167: 162: 157: 152: 147: 142: 137: 132: 127: 125:Classification 121: 118: 117: 114: 113: 110: 109: 104: 99: 94: 89: 84: 82:Batch learning 79: 74: 69: 64: 59: 54: 49: 43: 40: 39: 36: 35: 24: 23: 15: 13: 10: 9: 6: 4: 3: 2: 3670: 3659: 3656: 3654: 3651: 3649: 3646: 3645: 3643: 3630: 3624: 3617: 3613: 3609: 3605: 3600: 3595: 3591: 3587: 3580: 3578: 3576: 3572: 3568: 3564: 3560: 3556: 3552: 3548: 3544: 3540: 3533: 3529: 3528:Michal Aharon 3523: 3520: 3514: 3510: 3507: 3505: 3503: 3499: 3497: 3494: 3492: 3489: 3487: 3484: 3483: 3479: 3477: 3475: 3471: 3463: 3461: 3444: 3441: 3438: 3429: 3426: 3399: 3389: 3363: 3359: 3332: 3325: 3303: 3293: 3264: 3259: 3244: 3234: 3225: 3221: 3217: 3212: 3202: 3192: 3187: 3182: 3172: 3157: 3153: 3147: 3143: 3139: 3134: 3124: 3120: 3109: 3108: 3107: 3104: 3088: 3078: 3074: 3070: 3065: 3055: 3029: 3025: 3002: 2994: 2991: 2986: 2976: 2945: 2941: 2918: 2903: 2899: 2895: 2885: 2875: 2840: 2832: 2828: 2824: 2821: 2791: 2787: 2778: 2775: 2753: 2726: 2722: 2699: 2695: 2672: 2667: 2664: 2661: 2651: 2647: 2620: 2614: 2611: 2605: 2592: 2588: 2584: 2581: 2578: 2575: 2572: 2569: 2566: 2563: 2557: 2552: 2548: 2540: 2539: 2538: 2522: 2518: 2508: 2487: 2483: 2460: 2456: 2447: 2444:matrix using 2431: 2428: 2425: 2422: 2419: 2416: 2409:term with a 2394: 2390: 2369: 2349: 2346: 2343: 2323: 2303: 2300: 2291: 2289: 2285: 2262: 2258: 2232: 2227: 2212: 2208: 2202: 2198: 2194: 2189: 2185: 2178: 2173: 2168: 2152: 2148: 2142: 2138: 2134: 2130: 2119: 2115: 2109: 2105: 2099: 2096: 2093: 2089: 2085: 2082: 2078: 2068: 2063: 2058: 2042: 2038: 2032: 2028: 2022: 2017: 2014: 2011: 2007: 2003: 2000: 1991: 1986: 1981: 1973: 1970: 1967: 1964: 1954: 1953: 1952: 1938: 1918: 1898: 1878: 1869: 1853: 1849: 1840: 1824: 1804: 1784: 1776: 1757: 1754: 1751: 1746: 1741: 1733: 1730: 1727: 1724: 1718: 1714: 1699: 1689: 1685: 1676: 1672: 1666: 1663: 1660: 1647: 1646: 1645: 1628: 1623: 1619: 1615: 1610: 1600: 1596: 1589: 1585: 1567: 1562: 1554: 1551: 1548: 1545: 1534: 1531: 1528: 1515: 1514: 1513: 1510: 1494: 1490: 1467: 1463: 1442: 1434: 1430: 1426: 1421: 1407: 1399: 1383: 1375: 1357: 1353: 1349: 1344: 1340: 1331: 1326: 1309: 1306: 1301: 1291: 1287: 1280: 1277: 1259: 1254: 1246: 1243: 1240: 1237: 1226: 1223: 1220: 1207: 1206: 1205: 1188: 1185: 1175: 1171: 1167: 1162: 1158: 1154: 1151: 1134: 1129: 1121: 1118: 1115: 1112: 1101: 1098: 1095: 1082: 1081: 1080: 1079:, by solving 1078: 1060: 1055: 1052: 1049: 1039: 1035: 1023: 1019: 1017: 1012: 1008: 1001: 998: 996: 994: 990: 985: 983: 978: 974: 970: 966: 962: 960: 955: 943: 938: 936: 931: 929: 924: 923: 921: 920: 913: 910: 906: 903: 902: 901: 898: 896: 893: 892: 886: 885: 878: 875: 873: 870: 868: 865: 863: 860: 858: 855: 853: 850: 848: 845: 844: 838: 837: 830: 827: 825: 822: 820: 817: 815: 812: 810: 807: 805: 802: 800: 797: 795: 792: 791: 785: 784: 777: 774: 772: 769: 767: 764: 762: 759: 758: 752: 751: 744: 741: 739: 736: 734: 733:Crowdsourcing 731: 729: 726: 725: 719: 718: 709: 706: 705: 704: 701: 699: 696: 694: 691: 689: 686: 685: 682: 677: 676: 668: 665: 663: 662:Memtransistor 660: 658: 655: 653: 650: 646: 643: 642: 641: 638: 636: 633: 629: 626: 624: 621: 619: 616: 614: 611: 610: 609: 606: 604: 601: 599: 596: 594: 591: 587: 584: 583: 582: 579: 575: 572: 570: 567: 565: 562: 560: 557: 556: 555: 552: 550: 547: 545: 544:Deep learning 542: 540: 537: 536: 533: 528: 527: 520: 517: 515: 512: 510: 508: 504: 502: 499: 498: 495: 490: 489: 480: 479:Hidden Markov 477: 475: 472: 470: 467: 466: 465: 462: 461: 458: 453: 452: 445: 442: 440: 437: 435: 432: 430: 427: 425: 422: 420: 417: 415: 412: 410: 407: 405: 402: 401: 398: 393: 392: 385: 382: 380: 377: 375: 371: 369: 366: 364: 361: 359: 357: 353: 351: 348: 346: 343: 341: 338: 337: 334: 329: 328: 321: 318: 316: 313: 311: 308: 306: 303: 301: 298: 296: 293: 291: 288: 286: 284: 280: 276: 275:Random forest 273: 271: 268: 266: 263: 262: 261: 258: 256: 253: 251: 248: 247: 240: 239: 234: 233: 225: 219: 218: 211: 208: 206: 203: 201: 198: 196: 193: 191: 188: 186: 183: 181: 178: 176: 173: 171: 168: 166: 163: 161: 160:Data cleaning 158: 156: 153: 151: 148: 146: 143: 141: 138: 136: 133: 131: 128: 126: 123: 122: 116: 115: 108: 105: 103: 100: 98: 95: 93: 90: 88: 85: 83: 80: 78: 75: 73: 72:Meta-learning 70: 68: 65: 63: 60: 58: 55: 53: 50: 48: 45: 44: 38: 37: 34: 29: 25: 21: 20: 3589: 3585: 3542: 3538: 3522: 3501: 3473: 3469: 3467: 3281: 3105: 2635: 2509: 2316:into sum of 2292: 2287: 2283: 2282:denotes the 2249: 1870: 1774: 1772: 1643: 1511: 1432: 1428: 1424: 1422: 1397: 1373: 1327: 1324: 1203: 1015: 1010: 1006: 1005: 999: 992: 981: 976: 958: 957: 951: 819:PAC learning 506: 355: 350:Hierarchical 282: 236: 230: 3496:Matrix norm 3464:Limitations 2286:-th row of 703:Multi-agent 640:Transformer 539:Autoencoder 295:Naive Bayes 33:data mining 3642:Categories 3515:References 975:approach. 688:Q-learning 586:Restricted 384:Mean shift 333:Clustering 310:Perceptron 238:regression 140:Clustering 135:Regression 3594:CiteSeerX 3433:Δ 3430:× 3393:~ 3329:Δ 3297:~ 3256:‖ 3238:~ 3218:− 3206:~ 3196:‖ 3179:‖ 3169:Ω 3140:− 3131:Ω 3117:‖ 3085:Ω 3059:~ 2999:Ω 2980:~ 2915:Ω 2879:~ 2829:ω 2788:ω 2779:× 2750:Ω 2612:≠ 2579:≤ 2573:≤ 2567:∣ 2549:ω 2519:ω 2429:− 2347:− 2224:‖ 2195:− 2182:‖ 2135:− 2097:≠ 2090:∑ 2086:− 2008:∑ 2004:− 1978:‖ 1968:− 1962:‖ 1755:ϵ 1752:≤ 1738:‖ 1728:− 1722:‖ 1712:∀ 1696:‖ 1682:‖ 1673:∑ 1616:≤ 1607:‖ 1593:‖ 1583:∀ 1559:‖ 1549:− 1543:‖ 1372:enforces 1298:‖ 1284:‖ 1275:∀ 1251:‖ 1241:− 1235:‖ 1149:∀ 1126:‖ 1116:− 1110:‖ 847:ECML PKDD 829:VC theory 776:ROC curve 708:Self-play 628:DeepDream 469:Bayes net 260:Ensembles 41:Paradigms 3623:citation 3480:See also 2164:‖ 2074:‖ 2054:‖ 1997:‖ 971:, via a 270:Boosting 119:Problems 3616:2176046 3567:7477309 3547:Bibcode 1773:In the 852:NeurIPS 669:(ECRAM) 623:AlexNet 265:Bagging 3614:  3596:  3565:  2250:where 645:Vision 501:RANSAC 379:OPTICS 374:DBSCAN 358:-means 165:AutoML 3612:S2CID 3563:S2CID 3535:(PDF) 3318:into 963:is a 867:IJCAI 693:SARSA 652:Mamba 618:LeNet 613:U-Net 439:t-SNE 363:Fuzzy 340:BIRCH 3629:link 1423:The 961:-SVD 877:JMLR 862:ICLR 857:ICML 743:RLHF 559:LSTM 345:CURE 31:and 3604:doi 3555:doi 2537:as 1657:min 1525:min 1217:min 1092:min 1075:by 952:In 603:SOM 593:GAN 569:ESN 564:GRU 509:-NN 444:SDL 434:PGD 429:PCA 424:NMF 419:LDA 414:ICA 409:CCA 285:-NN 3644:: 3625:}} 3621:{{ 3610:, 3602:, 3590:98 3588:, 3574:^ 3561:, 3553:, 3543:54 3541:, 3537:, 3103:. 2851:th 2290:. 1868:. 1509:. 1420:. 991:. 956:, 872:ML 3631:) 3606:: 3557:: 3549:: 3502:k 3474:k 3470:k 3448:) 3445:1 3442:, 3439:1 3436:( 3427:V 3405:T 3400:k 3390:x 3364:k 3360:d 3337:T 3333:V 3326:U 3304:k 3294:E 3265:2 3260:F 3250:T 3245:k 3235:x 3226:k 3222:d 3213:k 3203:E 3193:= 3188:2 3183:F 3173:k 3163:T 3158:k 3154:x 3148:k 3144:d 3135:k 3125:k 3121:E 3089:k 3079:k 3075:E 3071:= 3066:k 3056:E 3030:k 3026:d 3003:k 2995:Y 2992:= 2987:k 2977:Y 2951:T 2946:k 2942:x 2919:k 2909:T 2904:k 2900:x 2896:= 2891:T 2886:k 2876:x 2847:) 2844:) 2841:i 2838:( 2833:k 2825:, 2822:i 2819:( 2798:| 2792:k 2783:| 2776:N 2754:k 2727:i 2723:x 2700:k 2696:d 2673:N 2668:1 2665:= 2662:i 2658:} 2652:i 2648:y 2644:{ 2621:, 2618:} 2615:0 2609:) 2606:i 2603:( 2598:T 2593:k 2589:x 2585:, 2582:N 2576:i 2570:1 2564:i 2561:{ 2558:= 2553:k 2523:k 2493:T 2488:k 2484:x 2461:k 2457:d 2432:1 2426:k 2423:n 2420:a 2417:r 2395:k 2391:E 2370:k 2350:1 2344:K 2324:K 2304:X 2301:D 2288:X 2284:k 2268:T 2263:k 2259:x 2233:2 2228:F 2218:T 2213:k 2209:x 2203:k 2199:d 2190:k 2186:E 2179:= 2174:2 2169:F 2158:T 2153:k 2149:x 2143:k 2139:d 2131:) 2125:T 2120:j 2116:x 2110:j 2106:d 2100:k 2094:j 2083:Y 2079:( 2069:= 2064:2 2059:F 2048:T 2043:j 2039:x 2033:j 2029:d 2023:K 2018:1 2015:= 2012:j 2001:Y 1992:= 1987:2 1982:F 1974:X 1971:D 1965:Y 1939:k 1919:X 1899:D 1879:D 1854:0 1850:T 1825:X 1805:X 1785:D 1775:k 1758:. 1747:2 1742:F 1734:X 1731:D 1725:Y 1719:, 1715:i 1700:0 1690:i 1686:x 1677:i 1667:X 1664:, 1661:D 1629:. 1624:0 1620:T 1611:0 1601:i 1597:x 1590:, 1586:i 1573:} 1568:2 1563:F 1555:X 1552:D 1546:Y 1540:{ 1535:X 1532:, 1529:D 1495:0 1491:T 1468:i 1464:x 1443:D 1433:k 1429:k 1425:k 1408:D 1398:k 1384:D 1374:k 1358:k 1354:e 1350:= 1345:i 1341:x 1310:1 1307:= 1302:0 1292:i 1288:x 1281:, 1278:i 1265:} 1260:2 1255:F 1247:X 1244:D 1238:Y 1232:{ 1227:X 1224:, 1221:D 1189:. 1186:k 1176:k 1172:e 1168:= 1163:i 1159:x 1155:, 1152:i 1140:} 1135:2 1130:F 1122:X 1119:D 1113:Y 1107:{ 1102:X 1099:, 1096:D 1061:M 1056:1 1053:= 1050:i 1046:} 1040:i 1036:y 1032:{ 1016:k 1011:k 1007:k 1000:k 993:k 982:k 977:k 959:k 941:e 934:t 927:v 507:k 356:k 283:k 241:) 229:(

Index

Machine learning
data mining
Supervised learning
Unsupervised learning
Semi-supervised learning
Self-supervised learning
Reinforcement learning
Meta-learning
Online learning
Batch learning
Curriculum learning
Rule-based learning
Neuro-symbolic AI
Neuromorphic engineering
Quantum machine learning
Classification
Generative modeling
Regression
Clustering
Dimensionality reduction
Density estimation
Anomaly detection
Data cleaning
AutoML
Association rules
Semantic analysis
Structured prediction
Feature engineering
Feature learning
Learning to rank

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

↑