About consistency in an inverse problem formulation












1














I'm a beginner with inverse problems and I was reading about regularization techniques.



Consider the problem:
$$d=Kf_{text{true}}$$
$d$ is a data vector, $K$ is an linear operator, $d=hat{d}+eta$ where $eta$ is the noise and $delta=||eta||$ (noise level).



In regularization by filtering (linear problems), Vogel (Computational Methods for Inverse Problems) defines the error as the error due to truncation, plus the error due to noise:



$$e_{alpha} = e^{text{truc}}_{alpha} +e^{text{noise}}_{alpha}$$



and in a model for an inverse problem, is important to choose $alpha$ such that:
$$e^{text{truc}}_{alpha} to 0, e^{text{noise}}_{alpha} to 0 (star)$$
when $deltato 0$. This is a kind of consistency of the method.



A real problem does not satisfy the $(star)$ condition, but the model of the problem needs to satisfy it, at least in agreement with my interpretation.




Now, here is my question: why is important that a classical
regularization model of an inverse problem satisfies $(star)$?




For example, I've heard that in statistics crooss validation does not satisfy $(star)$, and some statisticians said that is not really important because a real problem does not satisfy $(star)$.



My intention is not to create a discussion, but rather have a proper justification of why consistency is necessary or not in the modeling of an inverse problem.



Thank you very much.






statistics regularization inverse-problems







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    1














    I'm a beginner with inverse problems and I was reading about regularization techniques.



    Consider the problem:
    $$d=Kf_{text{true}}$$
    $d$ is a data vector, $K$ is an linear operator, $d=hat{d}+eta$ where $eta$ is the noise and $delta=||eta||$ (noise level).



    In regularization by filtering (linear problems), Vogel (Computational Methods for Inverse Problems) defines the error as the error due to truncation, plus the error due to noise:



    $$e_{alpha} = e^{text{truc}}_{alpha} +e^{text{noise}}_{alpha}$$



    and in a model for an inverse problem, is important to choose $alpha$ such that:
    $$e^{text{truc}}_{alpha} to 0, e^{text{noise}}_{alpha} to 0 (star)$$
    when $deltato 0$. This is a kind of consistency of the method.



    A real problem does not satisfy the $(star)$ condition, but the model of the problem needs to satisfy it, at least in agreement with my interpretation.




    Now, here is my question: why is important that a classical
    regularization model of an inverse problem satisfies $(star)$?




    For example, I've heard that in statistics crooss validation does not satisfy $(star)$, and some statisticians said that is not really important because a real problem does not satisfy $(star)$.



    My intention is not to create a discussion, but rather have a proper justification of why consistency is necessary or not in the modeling of an inverse problem.



    Thank you very much.






    statistics regularization inverse-problems







    share|cite|improve this question



























      1












      1








      1







      I'm a beginner with inverse problems and I was reading about regularization techniques.



      Consider the problem:
      $$d=Kf_{text{true}}$$
      $d$ is a data vector, $K$ is an linear operator, $d=hat{d}+eta$ where $eta$ is the noise and $delta=||eta||$ (noise level).



      In regularization by filtering (linear problems), Vogel (Computational Methods for Inverse Problems) defines the error as the error due to truncation, plus the error due to noise:



      $$e_{alpha} = e^{text{truc}}_{alpha} +e^{text{noise}}_{alpha}$$



      and in a model for an inverse problem, is important to choose $alpha$ such that:
      $$e^{text{truc}}_{alpha} to 0, e^{text{noise}}_{alpha} to 0 (star)$$
      when $deltato 0$. This is a kind of consistency of the method.



      A real problem does not satisfy the $(star)$ condition, but the model of the problem needs to satisfy it, at least in agreement with my interpretation.




      Now, here is my question: why is important that a classical
      regularization model of an inverse problem satisfies $(star)$?




      For example, I've heard that in statistics crooss validation does not satisfy $(star)$, and some statisticians said that is not really important because a real problem does not satisfy $(star)$.



      My intention is not to create a discussion, but rather have a proper justification of why consistency is necessary or not in the modeling of an inverse problem.



      Thank you very much.






      statistics regularization inverse-problems







      share|cite|improve this question















      I'm a beginner with inverse problems and I was reading about regularization techniques.



      Consider the problem:
      $$d=Kf_{text{true}}$$
      $d$ is a data vector, $K$ is an linear operator, $d=hat{d}+eta$ where $eta$ is the noise and $delta=||eta||$ (noise level).



      In regularization by filtering (linear problems), Vogel (Computational Methods for Inverse Problems) defines the error as the error due to truncation, plus the error due to noise:



      $$e_{alpha} = e^{text{truc}}_{alpha} +e^{text{noise}}_{alpha}$$



      and in a model for an inverse problem, is important to choose $alpha$ such that:
      $$e^{text{truc}}_{alpha} to 0, e^{text{noise}}_{alpha} to 0 (star)$$
      when $deltato 0$. This is a kind of consistency of the method.



      A real problem does not satisfy the $(star)$ condition, but the model of the problem needs to satisfy it, at least in agreement with my interpretation.




      Now, here is my question: why is important that a classical
      regularization model of an inverse problem satisfies $(star)$?




      For example, I've heard that in statistics crooss validation does not satisfy $(star)$, and some statisticians said that is not really important because a real problem does not satisfy $(star)$.



      My intention is not to create a discussion, but rather have a proper justification of why consistency is necessary or not in the modeling of an inverse problem.



      Thank you very much.






      statistics regularization inverse-problems




      statistics regularization inverse-problems






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      edited Jan 4 at 13:41









      amWhy

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      asked Feb 11 '15 at 18:35









      HiperionHiperion

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          One proves as much as is possible. The ideal thing to prove is a theorem which gives an explicit upper bound on the error in terms of the noise level in some norm. The condition you mention is a weaker version of this; it states that the error vanishes as the noise does, but does not given an explicit bound.



          The condition gives some faith in the method being reasonable. If one has other mathematical or practical justifications for the method being reasonable, maybe one can ignore this one.






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            One proves as much as is possible. The ideal thing to prove is a theorem which gives an explicit upper bound on the error in terms of the noise level in some norm. The condition you mention is a weaker version of this; it states that the error vanishes as the noise does, but does not given an explicit bound.



            The condition gives some faith in the method being reasonable. If one has other mathematical or practical justifications for the method being reasonable, maybe one can ignore this one.






            share|cite|improve this answer


























              0














              One proves as much as is possible. The ideal thing to prove is a theorem which gives an explicit upper bound on the error in terms of the noise level in some norm. The condition you mention is a weaker version of this; it states that the error vanishes as the noise does, but does not given an explicit bound.



              The condition gives some faith in the method being reasonable. If one has other mathematical or practical justifications for the method being reasonable, maybe one can ignore this one.






              share|cite|improve this answer
























                0












                0








                0






                One proves as much as is possible. The ideal thing to prove is a theorem which gives an explicit upper bound on the error in terms of the noise level in some norm. The condition you mention is a weaker version of this; it states that the error vanishes as the noise does, but does not given an explicit bound.



                The condition gives some faith in the method being reasonable. If one has other mathematical or practical justifications for the method being reasonable, maybe one can ignore this one.






                share|cite|improve this answer












                One proves as much as is possible. The ideal thing to prove is a theorem which gives an explicit upper bound on the error in terms of the noise level in some norm. The condition you mention is a weaker version of this; it states that the error vanishes as the noise does, but does not given an explicit bound.



                The condition gives some faith in the method being reasonable. If one has other mathematical or practical justifications for the method being reasonable, maybe one can ignore this one.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 4 at 12:32









                Tommi BranderTommi Brander

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