{
  "id": "1402.5324",
  "version": "v2",
  "published": "2014-02-21T15:36:15.000Z",
  "updated": "2015-07-07T22:12:01.000Z",
  "title": "On Asymptotic Incoherence and its Implications for Compressed Sensing of Inverse Problems",
  "authors": [
    "Alex D. Jones",
    "Ben Adcock",
    "Anders C. Hansen"
  ],
  "categories": [
    "cs.IT",
    "math.IT",
    "math.NA"
  ],
  "abstract": "Recently, it has been shown that incoherence is an unrealistic assumption for compressed sensing when applied to many inverse problems. Instead, the key property that permits efficient recovery in such problems is so-called local incoherence. Similarly, the standard notion of sparsity is also inadequate for many real world problems. In particular, in many applications, the optimal sampling strategy depends on asymptotic incoherence and the signal sparsity structure. The purpose of this paper is to study asymptotic incoherence and its implications towards the design of optimal sampling strategies and efficient sparsity bases. It is determined how fast asymptotic incoherence can decay in general for isometries. Furthermore it is shown that Fourier sampling and wavelet sparsity, whilst globally coherent, yield optimal asymptotic incoherence as a power law up to a constant factor. Sharp bounds on the asymptotic incoherence for Fourier sampling with polynomial bases are also provided. A numerical experiment is also presented to demonstrate the role of asymptotic incoherence in finding good subsampling strategies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-21T15:36:15.000Z",
      "abstract": "Recently, it has been shown that incoherence is an unrealistic assumption for compressed sensing when applied to infinite-dimensional inverse problems. Instead, the key property that permits efficient recovery in such problems is so-called asymptotic incoherence. The purpose of this paper is to study this new concept, and its implications towards the design of optimal sampling strategies. It is determined how fast asymptotic incoherence can decay in general for isometries. Furthermore it is shown that Fourier sampling and wavelet sparsity, whilst globally coherent, yield optimal asymptotic incoherence as a power law up to a constant factor. Sharp bounds on the asymptotic incoherence for Fourier sampling with polynomial bases are also provided. A numerical experiment is also presented to demonstrate the role of asymptotic incoherence in finding good subsampling strategies.",
      "comment": null,
      "journal": null,
      "doi": null,
      "authors": [
        "Alex Jones",
        "Ben Adcock",
        "Anders C. Hansen"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-07-07T22:12:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compressed sensing",
      "implications",
      "yield optimal asymptotic incoherence",
      "fast asymptotic incoherence",
      "permits efficient recovery"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.5324J"
    }
  }
}