Improved Coupled Tensor Factorization with Its Applications in Health Data Analysis

<div>An illustration of a joint decomposition with a third-order tensor and a matrix factorization of <svg height="8.8423pt" id="M70" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 8.28119 8.8423" width="8.28119pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"></path></g></svg> components. <svg height="9.49473pt" id="M71" style="vertical-align:-0.3238297pt" version="1.1" viewbox="-0.0498162 -9.1709 13.579 9.49473" width="13.579pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M664 567C637 648 562 699 450 699C391 699 338 686 281 647C223 607 163 539 163 467C163 371 228 323 310 323C393 323 433 371 467 439L423 457C396 406 376 378 325 378C280 378 254 412 254 461C254 553 348 644 445 644C520 644 544 594 544 503C544 455 511 342 452 228C388 102 323 34 213 34C188 34 160 43 155 54V55C199 55 221 94 221 117C221 153 197 188 150 188C99 188 64 141 64 98C64 22 145 -21 215 -21C304 -21 383 8 443 87H445C477 20 514 -18 603 -18C686 -18 766 49 842 139L804 169C720 73 673 41 634 41C607 41 584 64 584 112C584 175 633 326 684 451C737 582 788 644 871 644C896 644 914 635 919 624V623C875 623 853 584 853 561C853 525 879 492 926 492C977 492 1010 538 1010 580C1010 656 939 699 869 699C769 699 706 626 666 567H664Z"></path></g></svg> and <svg height="8.79011pt" id="M72" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -8.7403 8.64656 8.79011" width="8.64656pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M646 650H414V615C488 606 489 592 473 552C449 491 398 397 369 340C330 414 289 488 254 558C237 594 236 606 301 615V650H16V615C71 607 88 597 125 531C185 425 222 358 249 304C258 284 262 274 262 249V138C262 47 253 41 168 34V0H493V34C403 41 396 47 396 138V266C396 297 399 308 414 335C462 429 489 472 526 531S583 608 646 615V650Z"></path></g></svg> share one dimension; that is, <svg height="12.5794pt" id="M73" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 98.6068 12.5794" width="98.6068pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M687 0V35C623 43 611 56 577 144C512 316 450 490 386 665L335 652L130 138C97 54 84 43 18 35V0H242V35C169 44 166 58 181 106C193 146 206 182 217 213H409C424 173 439 130 449 100C465 56 460 44 401 35V0H687ZM390 266H234C260 337 285 410 313 483H316L390 266Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.81,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.073,0)"><path d="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.558,0)"><path d="M465 45L455 77C444 72 433 69 421 69C404 69 389 81 389 128V293C389 349 380 392 337 425C310 445 280 454 239 454C182 443 111 409 73 383C51 366 38 353 38 335C38 309 74 285 101 285C121 285 136 293 143 319C151 350 160 374 171 388C180 400 191 406 205 406C241 406 265 371 265 308V278C242 258 165 234 106 212C54 192 30 168 30 116C30 54 81 -12 165 -12C185 -12 232 21 270 46C274 34 281 21 295 9C308 -3 329 -12 356 -12L465 45ZM265 79C249 67 238 63 223 63C200 63 163 84 163 139C163 174 179 192 210 208C231 219 247 229 265 239V79Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,34.616,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g><g transform="matrix(.013,0,0,-0.013,39.562,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.705,0)"><path d="M465 45L455 77C444 72 433 69 421 69C404 69 389 81 389 128V293C389 349 380 392 337 425C310 445 280 454 239 454C182 443 111 409 73 383C51 366 38 353 38 335C38 309 74 285 101 285C121 285 136 293 143 319C151 350 160 374 171 388C180 400 191 406 205 406C241 406 265 371 265 308V278C242 258 165 234 106 212C54 192 30 168 30 116C30 54 81 -12 165 -12C185 -12 232 21 270 46C274 34 281 21 295 9C308 -3 329 -12 356 -12L465 45ZM265 79C249 67 238 63 223 63C200 63 163 84 163 139C163 174 179 192 210 208C231 219 247 229 265 239V79Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,50.763,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g><g transform="matrix(.013,0,0,-0.013,55.71,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,60.853,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,65.996,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,71.139,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,76.316,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,81.459,0)"><path d="M465 45L455 77C444 72 433 69 421 69C404 69 389 81 389 128V293C389 349 380 392 337 425C310 445 280 454 239 454C182 443 111 409 73 383C51 366 38 353 38 335C38 309 74 285 101 285C121 285 136 293 143 319C151 350 160 374 171 388C180 400 191 406 205 406C241 406 265 371 265 308V278C242 258 165 234 106 212C54 192 30 168 30 116C30 54 81 -12 165 -12C185 -12 232 21 270 46C274 34 281 21 295 9C308 -3 329 -12 356 -12L465 45ZM265 79C249 67 238 63 223 63C200 63 163 84 163 139C163 174 179 192 210 208C231 219 247 229 265 239V79Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,87.517,3.132)"><path d="M636 22C611 29 589 38 560 74C536 104 518 134 495 173L430 300C518 328 592 392 592 492C592 555 563 594 528 618C491 643 435 650 377 650H140L131 618C222 610 223 601 208 518L137 136C121 43 116 40 27 32L18 0H295L300 32C212 40 206 42 223 138L248 279H296C334 279 350 272 369 230C393 174 413 140 441 94C482 24 523 1 626 -8L636 22ZM497 480C497 422 468 374 417 343C387 324 353 315 304 315H255L298 556C303 583 309 598 319 603C331 610 348 613 367 613C448 613 497 573 497 480Z"></path></g><g transform="matrix(.013,0,0,-0.013,93.824,0)"><path d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z"></path></g></svg> as the common factor matrix. Figure adapted from [<a href="/journals/complexity/2019/1574240/#B32">32</a>].</div>

Complexity

fig2

Figure 2

Figure 2: Improved Coupled Tensor Factorization with Its Applications in Health Data Analysis 