Failure structures of message-passing algorithms in erasure decoding and compressed sensing
Kuupäev
2019-02-12
Autorid
Ajakirja pealkiri
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Kirjastaja
Abstrakt
Esitatud tulemused on näiliselt kahest erinevast valdkonnast, nimelt käsitleme sõnumivahetuskanali dekodeerimise ja hõreda signaalihõive (ingl k. compressed sensing) meetodeid.
Kanali dekodeerimine aitab edastada informatsiooni veakindlalt. Antud juhul uurisime me kahendkustuskanalit (ingl k. binary erasure channel, BEC). Sellise kanali puhul infoühik kas jõuab veatult kohale või kustub, kusjuures info kustumine on vastuvõtjale tuvastatav. Shannoni fundamentaalne järeldus oli, et ükskõik kui halva kanali korral on alati võimalik informatsiooni edastada veakindlalt, kodeerides andmeid piisavalt suurel hulgal. Üks praegu populaarne dekodeerimise viis on kasutada sõnumivahetusalgoritmi, mis on kiire kuid mitte optimaalne, kuna mõnikord see annab tõrke, kuigi taastamine on siiski võimalik keerulisema algoritmiga. Käesolevas dissertatsioonis me uurime, kuidas ühendada neid kahte meetodit.
Teine eelmainitud uurimisvaldkondadest, hõre signaalihõive, põhineb järgneval tähelepanekul. Mitmeid olulisi signaale saab esitada hõredate vektoritena, st. vektoritena kus on palju nulle. Pakuti välja vastuvõetud signaale jooksvalt hõrendada, korrutades neid kaudselt läbi mõõtemaatriksiga. Me uurisime üht suboptimaalset algoritmi, intervallivahetusalgoritmi, ja millistel juhtudel antud algoritm annab tõrke. Me kirjeldasime täieliku graafiteoreetilise kriteeriumi, mille korral tõrked esinevad.
Me uurisime sõnumivahetusalgoritme kustutuste dekodeerimises ja hõredas signaalihõives. See tõi nende algoritmide vahel esile mitmed sarnasused ja võimaldab ühtlustada uurimisvahendeid nende analüüsiks.
The presented results are from two different- on the face of it- fields, message-passing channel decoding and compressed sensing. Channel decoding deals with reliable transmission of information. In our case, we consider the binary erasure channel (BEC). This means that a bit one sends is received either unchanged, or erased completely. A fundamental result by Shannon was as follows: whatever bad channel you have, there is always a way to send information reliably (i.e. with vanishing probability of error) if you encode large enough blocks of information together. One of the most popular algorithms used for decoding nowadays is message-passing, which is fast but not optimal, i.e. sometimes it fails while the reconstruction is still possible with a more sophisticated method. In this thesis, we try to build a bridge between these two methods. Another aforementioned problem, compressed sensing, is based on the following observation. Many important signals can be represented as sparse vectors, i.e. vectors with many zeroes. It was suggested to compress such signals on-the-fly, by implicitly multiplying them by a measurement matrix. We consider one of the suboptimal algorithms, the interval-passing algorithm. More precisely, we ask a question on what are the conditions for the algorithm to fail. We were able to give a complete graph-theoretic criterion of reconstruction success. We considered failures of both message-passing decoding and the interval-passing algorithm for compressed sensing. This allowed us to unveil many similarities and how one can translate the research tools between the algorithms.
The presented results are from two different- on the face of it- fields, message-passing channel decoding and compressed sensing. Channel decoding deals with reliable transmission of information. In our case, we consider the binary erasure channel (BEC). This means that a bit one sends is received either unchanged, or erased completely. A fundamental result by Shannon was as follows: whatever bad channel you have, there is always a way to send information reliably (i.e. with vanishing probability of error) if you encode large enough blocks of information together. One of the most popular algorithms used for decoding nowadays is message-passing, which is fast but not optimal, i.e. sometimes it fails while the reconstruction is still possible with a more sophisticated method. In this thesis, we try to build a bridge between these two methods. Another aforementioned problem, compressed sensing, is based on the following observation. Many important signals can be represented as sparse vectors, i.e. vectors with many zeroes. It was suggested to compress such signals on-the-fly, by implicitly multiplying them by a measurement matrix. We consider one of the suboptimal algorithms, the interval-passing algorithm. More precisely, we ask a question on what are the conditions for the algorithm to fail. We were able to give a complete graph-theoretic criterion of reconstruction success. We considered failures of both message-passing decoding and the interval-passing algorithm for compressed sensing. This allowed us to unveil many similarities and how one can translate the research tools between the algorithms.
Kirjeldus
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Märksõnad
message handling systems, algorithms, decoding, signal processing