Outlier-Resistant Localization for Smart Cities: Low-Rank Approximation-Based Approaches

<div>An illustration of the scenario where the triangular inequality is broken; in this scenario, <span class="nowrap"><svg height="16.1809pt" id="M39" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -12.8898 87.9816 16.1809" width="87.9816pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,.895,-3.549)"><path d="M1410 559C1171 602 927 646 716 700H694C485 646 237 602 0 559L5 532L705 625L1405 532L1410 559Z"></path></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.15,3.132)"><path d="M722 32C646 41 636 48 622 116L544 496L509 667L472 658L153 123C106 45 93 42 27 32L18 0H259L268 32C178 40 172 48 204 107L272 233H509C523 180 533 137 540 96C548 50 546 39 466 32L457 0H714L722 32ZM502 283H299L448 547H450L502 283Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.453,3.132)"><path d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.82,0)"><path d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"></path></g><g transform="matrix(.013,0,0,-0.013,35.083,0)"><path d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,42.233,3.132)"><path d="M722 32C646 41 636 48 622 116L544 496L509 667L472 658L153 123C106 45 93 42 27 32L18 0H259L268 32C178 40 172 48 204 107L272 233H509C523 180 533 137 540 96C548 50 546 39 466 32L457 0H714L722 32ZM502 283H299L448 547H450L502 283Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,48.7,3.132)"><path d="M598 511C598 627 497 650 401 650H141L131 618C219 611 222 606 210 538L130 119C117 43 108 39 25 32L18 0H236C324 0 396 5 456 30C531 59 585 119 585 193C585 285 520 331 428 351V353C529 373 598 423 598 511ZM502 508C502 418 431 368 330 368H263L298 559C303 584 308 595 317 600C329 608 345 612 369 612C428 612 502 595 502 508ZM483 197C483 95 404 39 309 39C225 39 206 58 219 139L256 330H317C395 330 483 301 483 197Z"></path></g><g transform="matrix(.013,0,0,-0.013,57.685,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,68.221,0)"><path d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,75.371,3.132)"><path d="M598 511C598 627 497 650 401 650H141L131 618C219 611 222 606 210 538L130 119C117 43 108 39 25 32L18 0H236C324 0 396 5 456 30C531 59 585 119 585 193C585 285 520 331 428 351V353C529 373 598 423 598 511ZM502 508C502 418 431 368 330 368H263L298 559C303 584 308 595 317 600C329 608 345 612 369 612C428 612 502 595 502 508ZM483 197C483 95 404 39 309 39C225 39 206 58 219 139L256 330H317C395 330 483 301 483 197Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,81.069,3.132)"><path d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"></path></g></svg>,</span> a result that defies common sense and implies that outliers are introduced, at least on the edge AC.</div>

Scientific Programming

fig3

Figure 3

Figure 3: Outlier-Resistant Localization for Smart Cities: Low-Rank Approximation-Based Approaches 