Intrinsic Dimension Estimation: Relevant Techniques and a Benchmark Framework

<div>(a) Samples from <svg height="8.38558pt" id="M909" style="vertical-align:-0.1933403pt" version="1.1" viewbox="-0.0498162 -8.19224 41.2032 8.38558" width="41.2032pt" 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="M77 28C79 11 88 2 105 0H420C436 1 447 13 448 28V37C447 53 436 62 420 65H299V546H420C436 548 445 557 448 574V583C447 599 436 608 420 611H105C88 608 79 599 77 583V574C79 557 88 548 105 546H226V65H105C88 63 79 54 77 37V28Z" id="g204-42"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.851,0)"><path d="M52 16C55 1 64 -8 79 -11H88C108 -8 107 -2 122 33C163 4 219 -11 288 -11C388 -11 472 68 472 167C472 247 414 320 335 338L204 369C159 380 121 417 121 461C121 490 133 513 156 530C180 548 207 557 236 557C315 557 361 515 373 432C376 418 384 410 399 408H418C434 410 443 419 445 435V595C443 611 434 620 418 622H410C386 616 387 608 375 577C338 607 292 622 236 622C143 625 50 550 52 457C52 380 113 316 186 298L317 268C366 256 403 213 403 162C403 101 348 54 288 54C196 54 125 89 125 176C123 192 114 201 98 203H79C64 201 55 192 52 176V16Z" id="g204-52"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,13.701,0)"><path d="M89 39C118 -5 178 -11 262 -11C336 -11 398 -8 435 39C455 71 465 136 466 196C467 226 468 263 468 306C468 391 468 445 457 504C452 533 445 556 435 572C406 615 346 622 262 622S118 615 89 572C69 540 60 475 57 415C56 385 56 349 56 306C56 242 57 202 61 151C65 102 73 64 89 39ZM150 96C131 133 129 207 129 318V374C129 387 130 405 131 427C135 472 141 502 150 519C170 554 207 557 262 557C318 557 354 552 374 519C395 478 395 408 395 318C395 207 391 131 374 96S317 54 262 54C209 54 167 61 150 96Z" id="g204-48"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,20.552,0)"><path d="M15 28C17 11 26 2 43 0H153C169 1 180 12 181 28V37C180 54 169 62 153 65H128V538C129 525 160 423 222 234C229 215 246 204 262 204S295 215 302 234C364 423 395 525 396 538V65H371C355 63 346 54 343 37V28C346 11 355 2 371 0H481C498 2 507 11 509 28V37C507 54 498 63 481 65H456V546H481C498 548 507 557 509 574V583C507 600 498 609 481 611H398C378 611 361 600 356 581C319 469 263 293 262 277C262 291 205 466 168 581C163 600 146 611 126 611H43C26 609 17 600 15 583V574C17 557 26 548 43 546H68V65H43C26 63 17 54 15 37V28Z" id="g204-46"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,27.402,0)"><path d="M28 28C30 11 39 2 56 0H191C207 1 218 13 219 28V37C218 53 207 62 191 65H162L184 159H340C349 118 355 94 362 65H333C316 63 307 54 305 37V28C307 11 316 2 333 0H468C484 1 495 12 496 28V37C495 54 484 62 468 65H437L316 593C312 610 295 623 278 623H246C229 623 212 610 208 593L87 65H56C39 63 30 54 28 37V28ZM198 224C213 287 225 351 240 416C255 483 262 526 262 538H263C263 527 269 492 281 433S308 305 326 224H198Z" id="g204-34"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,34.253,0)"><path d="M24 28C26 11 35 2 52 0H211C228 2 237 11 239 28V37C237 54 228 63 211 65H168V241H295C396 241 480 327 480 426C480 527 396 611 295 611H52C35 608 26 599 24 583V574C25 558 37 547 52 546H95V65H52C35 63 26 54 24 37V28ZM168 546H277C312 546 342 535 368 512S407 461 407 426C407 358 343 306 277 306H168V546Z" id="g204-49"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g></svg> face database. (b) Samples from digit “0” to digit “9” in <svg height="8.38558pt" id="M910" style="vertical-align:-0.1933403pt" version="1.1" viewbox="-0.0498162 -8.19224 34.3526 8.38558" width="34.3526pt" 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="M15 28C17 11 26 2 43 0H153C169 1 180 12 181 28V37C180 54 169 62 153 65H128V538C129 525 160 423 222 234C229 215 246 204 262 204S295 215 302 234C364 423 395 525 396 538V65H371C355 63 346 54 343 37V28C346 11 355 2 371 0H481C498 2 507 11 509 28V37C507 54 498 63 481 65H456V546H481C498 548 507 557 509 574V583C507 600 498 609 481 611H398C378 611 361 600 356 581C319 469 263 293 262 277C262 291 205 466 168 581C163 600 146 611 126 611H43C26 609 17 600 15 583V574C17 557 26 548 43 546H68V65H43C26 63 17 54 15 37V28Z" id="g204-46"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.851,0)"><path d="M188 0C205 2 214 11 216 28V37C214 54 205 63 188 65H151V538C152 531 183 445 224 326C267 209 303 113 332 30C339 13 355 0 375 0H405C421 1 432 12 433 28V546H470C487 548 496 557 498 574V583C496 600 487 609 470 611H336C319 609 310 600 308 583V574C310 557 319 548 336 546H373V73C373 75 364 100 347 148L296 291L192 581C185 600 169 611 149 611H54C37 609 28 600 26 583V574C28 557 37 548 54 546H91V65H54C37 63 28 54 26 37V28C28 11 37 2 54 0H188Z" id="g204-47"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,13.701,0)"><path d="M77 28C79 11 88 2 105 0H420C436 1 447 13 448 28V37C447 53 436 62 420 65H299V546H420C436 548 445 557 448 574V583C447 599 436 608 420 611H105C88 608 79 599 77 583V574C79 557 88 548 105 546H226V65H105C88 63 79 54 77 37V28Z" id="g204-42"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,20.552,0)"><path d="M52 16C55 1 64 -8 79 -11H88C108 -8 107 -2 122 33C163 4 219 -11 288 -11C388 -11 472 68 472 167C472 247 414 320 335 338L204 369C159 380 121 417 121 461C121 490 133 513 156 530C180 548 207 557 236 557C315 557 361 515 373 432C376 418 384 410 399 408H418C434 410 443 419 445 435V595C443 611 434 620 418 622H410C386 616 387 608 375 577C338 607 292 622 236 622C143 625 50 550 52 457C52 380 113 316 186 298L317 268C366 256 403 213 403 162C403 101 348 54 288 54C196 54 125 89 125 176C123 192 114 201 98 203H79C64 201 55 192 52 176V16Z" id="g204-52"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,27.402,0)"><path d="M134 28C137 11 146 2 162 0H363C379 1 390 12 391 28V37C390 54 379 62 363 65H299V546H425V480C427 463 436 454 453 452H470C487 454 496 463 498 480V583C496 600 487 609 470 611H54C37 609 28 600 26 583V480C28 463 37 454 54 452H71C87 455 98 463 99 480V546H226V65H162C146 63 137 54 134 37V28Z" id="g204-53"></path><glyph.data ascent="698" descent="-233" horiz-adv-x="525" vert-adv-y="525"></glyph.data></g></svg> database.</div>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Intrinsic Dimension Estimation: Relevant Techniques and a Benchmark Framework 