An Efficient Algorithm for Learning Dictionary under Coherence Constraint

<div>The MSE performance. (a) Results of <svg height="11.5695pt" id="M324" style="vertical-align:-3.18152pt" version="1.1" viewbox="-0.0498162 -8.38798 33.9451 11.5695" width="33.9451pt" 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="M43 650V622C120 616 128 612 128 526V124C128 39 120 33 34 27V0H270C392 0 492 25 567 83C643 141 690 230 690 350C690 444 655 517 605 565C543 625 450 650 323 650H43ZM213 547C213 587 217 598 226 604C236 612 262 617 304 617C371 617 429 604 474 576C554 529 592 439 592 336C592 176 505 36 319 36C246 36 213 55 213 131V547Z" id="g121-66"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="735" vert-adv-y="735"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.555,0)"><path d="M135 536C164 536 186 560 186 587C186 617 164 639 136 639C109 639 85 617 85 587C85 560 109 536 135 536ZM252 0V26C188 32 181 38 181 106V451C138 433 90 420 39 412V388C99 379 102 374 102 312V106C102 38 95 32 32 26V0H252Z" id="g121-103"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="268" vert-adv-y="268"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,13.039,0)"><path d="M390 111C344 68 312 56 269 56C212 56 118 102 118 241C118 346 175 401 241 401C277 401 312 388 342 360C350 352 355 349 361 349C372 349 394 371 394 392C394 403 391 411 378 422C362 436 329 449 288 449H287C250 449 190 432 138 392C71 341 37 274 37 197C37 90 112 -12 238 -12C297 -12 363 32 407 90L390 111Z" id="g121-97"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="423" vert-adv-y="423"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,18.629,0)"><path d="M298 36L289 62C276 55 253 45 228 45C202 45 169 60 169 141V397H276C289 405 292 426 282 437H169V574L155 576L90 509V437H45L17 408L21 397H90V107C90 28 125 -12 188 -12C198 -12 213 -8 230 1L298 36Z" id="g121-114"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="305" vert-adv-y="305"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,22.679,3.132)"><path d="M384 354C435 370 515 408 515 502C515 549 494 593 459 616C426 639 384 650 308 650H45V618C127 613 130 606 130 524V127C130 45 125 38 37 33V0H262C342 0 409 12 461 40C521 73 558 124 558 192C558 295 481 336 384 352V354ZM217 365V556C217 587 221 597 229 604C238 611 261 615 288 615C382 615 421 558 421 490C421 406 374 365 266 365H217ZM217 327H264C384 327 457 284 457 183C457 80 383 38 301 38C235 38 217 55 217 127V327Z" id="g58-64"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="598" vert-adv-y="598"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,28.093,3.132)"><path d="M46 650V618C122 613 130 611 130 522V128C130 46 123 40 40 32V0H319V32C227 40 217 42 217 128V279L292 260C303 259 322 258 338 261C436 272 534 337 534 467C534 534 508 580 469 609C428 640 371 650 299 650H46ZM217 558C217 585 220 596 228 603C235 610 256 615 284 615C367 615 435 571 435 463C435 339 358 304 291 304C256 304 229 310 217 317V558Z" id="g58-78"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="573" vert-adv-y="573"></glyph.data></g></svg> versus <svg height="9.39034pt" id="M325" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -5.96091 6.54569 9.39034" width="6.54569pt" 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="M471 401C471 442 455 448 440 448C421 448 389 439 366 424C314 390 233 331 165 242H163L187 345C193 372 197 393 197 409C197 435 189 448 174 448C146 448 80 408 23 349L40 327C64 351 97 373 107 373C115 373 118 366 111 337L29 -4L36 -12C56 -4 83 3 110 9C123 69 136 126 150 172C227 283 336 381 377 381C392 381 396 370 379 298L326 74C290 -77 273 -172 273 -197C273 -219 316 -261 346 -261L353 -246C348 -227 352 -174 374 -66L457 310C467 352 471 381 471 401Z" id="g113-229"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g></svg>. (b) Results of <svg height="11.6836pt" id="M326" style="vertical-align:-3.29562pt" version="1.1" viewbox="-0.0498162 -8.38798 39.7909 11.6836" width="39.7909pt" 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="M43 650V622C120 616 128 612 128 526V124C128 39 120 33 34 27V0H270C392 0 492 25 567 83C643 141 690 230 690 350C690 444 655 517 605 565C543 625 450 650 323 650H43ZM213 547C213 587 217 598 226 604C236 612 262 617 304 617C371 617 429 604 474 576C554 529 592 439 592 336C592 176 505 36 319 36C246 36 213 55 213 131V547Z" id="g121-66"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="735" vert-adv-y="735"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.555,0)"><path d="M135 536C164 536 186 560 186 587C186 617 164 639 136 639C109 639 85 617 85 587C85 560 109 536 135 536ZM252 0V26C188 32 181 38 181 106V451C138 433 90 420 39 412V388C99 379 102 374 102 312V106C102 38 95 32 32 26V0H252Z" id="g121-103"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="268" vert-adv-y="268"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,13.039,0)"><path d="M390 111C344 68 312 56 269 56C212 56 118 102 118 241C118 346 175 401 241 401C277 401 312 388 342 360C350 352 355 349 361 349C372 349 394 371 394 392C394 403 391 411 378 422C362 436 329 449 288 449H287C250 449 190 432 138 392C71 341 37 274 37 197C37 90 112 -12 238 -12C297 -12 363 32 407 90L390 111Z" id="g121-97"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="423" vert-adv-y="423"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,18.629,0)"><path d="M298 36L289 62C276 55 253 45 228 45C202 45 169 60 169 141V397H276C289 405 292 426 282 437H169V574L155 576L90 509V437H45L17 408L21 397H90V107C90 28 125 -12 188 -12C198 -12 213 -8 230 1L298 36Z" id="g121-114"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="305" vert-adv-y="305"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,22.679,3.132)"><path d="M416 501L399 642C356 655 314 665 271 665C140 665 63 582 63 486C63 372 163 325 229 290C304 250 359 217 359 142C359 70 316 25 240 25C138 25 94 125 78 186L44 178C52 123 64 45 72 21C81 16 96 8 121 0S179 -15 220 -15C353 -15 445 68 445 176C445 289 350 335 270 375C191 414 143 449 143 520C143 572 177 627 253 627C340 627 366 556 382 495L416 501Z" id="g58-81"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="484" vert-adv-y="484"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,26.983,3.132)"><path d="M43 650V617C122 612 130 606 130 522V128C130 44 122 37 34 32V0H275C400 0 497 25 573 82C652 142 698 229 698 350C698 445 662 518 613 565C549 625 456 650 327 650H43ZM217 544C217 580 220 591 229 599C240 607 264 613 311 613C380 613 437 599 482 570C561 523 598 436 598 336C598 177 511 40 324 40C248 40 217 61 217 135V544Z" id="g58-66"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="744" vert-adv-y="744"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,33.689,3.132)"><path d="M384 354C435 370 515 408 515 502C515 549 494 593 459 616C426 639 384 650 308 650H45V618C127 613 130 606 130 524V127C130 45 125 38 37 33V0H262C342 0 409 12 461 40C521 73 558 124 558 192C558 295 481 336 384 352V354ZM217 365V556C217 587 221 597 229 604C238 611 261 615 288 615C382 615 421 558 421 490C421 406 374 365 266 365H217ZM217 327H264C384 327 457 284 457 183C457 80 383 38 301 38C235 38 217 55 217 127V327Z" id="g58-64"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="598" vert-adv-y="598"></glyph.data></g></svg> versus <svg height="13.9704pt" id="M327" style="vertical-align:-3.429399pt" version="1.1" viewbox="-0.0498162 -10.541 33.7749 13.9704" width="33.7749pt" 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="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z" id="g113-225"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="501" vert-adv-y="501"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.148,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,21.411,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,27.651,-5.741)"><path d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z" id="g50-113"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="597" vert-adv-y="597"></glyph.data></g></svg>.</div>

Mathematical Problems in Engineering

fig6

Figure 6

Figure 6: An Efficient Algorithm for Learning Dictionary under Coherence Constraint 