A Simple Gaussian Measurement Bound for Exact Recovery of Block-Sparse Signals

<div>Plots of the exact recovery frequency versus the number of measurements, for the use of the mixed <svg height="15.3808pt" id="M287" style="vertical-align:-5.7029pt" version="1.1" viewbox="-0.0498162 -9.6779 22.7209 15.3808" width="22.7209pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z" id="g113-109"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="269" vert-adv-y="269"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,3.306,3.264)"><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" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,8.44,0)"><path d="M368 703H309L44 -163H104L368 703Z" id="g113-48"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="412" vert-adv-y="412"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,14.027,0)"><path d="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z" id="g113-109"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="269" vert-adv-y="269"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,17.333,3.264)"><path d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z" id="g50-114"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="499" vert-adv-y="499"></glyph.data></g></svg> and the standard <svg height="15.3808pt" id="M288" style="vertical-align:-5.7029pt" version="1.1" viewbox="-0.0498162 -9.6779 8.66602 15.3808" width="8.66602pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z" id="g113-109"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="269" vert-adv-y="269"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,3.306,3.264)"><path d="M481 430L460 436C440 442 393 451 366 451C348 451 325 448 313 446C267 437 195 409 147 367C79 308 24 211 24 105C24 37 56 -12 93 -12C121 -12 156 1 199 32C246 66 313 122 349 168H351L286 -147C274 -205 261 -215 210 -224L199 -226L193 -255L438 -243L442 -214L417 -212C364 -208 359 -200 370 -140L433 204C455 324 469 389 481 430ZM391 385C382 336 366 260 358 238C321 178 209 59 143 59C129 59 115 83 115 131C115 207 152 317 220 373C243 392 282 403 314 403C348 403 374 395 391 385Z" id="g50-114"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="499" vert-adv-y="499"></glyph.data></g></svg> minimization methods with different values of <svg height="9.7821pt" id="M289" style="vertical-align:-3.57194pt" version="1.1" viewbox="-0.0498162 -6.21016 6.77673 9.7821" width="6.77673pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z" id="g113-114"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="491" vert-adv-y="491"></glyph.data></g></svg>.</div>

Discrete Dynamics in Nature and Society

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Figure 1

Figure 1: A Simple Gaussian Measurement Bound for Exact Recovery of Block-Sparse Signals 