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Convex set for similarity constraint

  1. Jul 18, 2011 #1
    I am trying to ultimately find the projector onto a convex set defined in a non-explicit way, for a seismic processing application.

    The signals in question are members of some Hilbert Space H and the set membership requires that they must correlate with each other above some scalar [itex]\rho[/itex], given that the known signal [itex]\textbf{w}[/itex] is in the set. Symbolically, I want to find a projector [itex]\textit{P}[/itex] onto the convex set [itex]\textit{C}[/itex]:

    \begin{equation}

    C = \left\{\mathbf{u}(t) : \left\langle \hat{\mathbf{u}}(t),\hat{\mathbf{v}}(t) \right\rangle \geq\rho_{0}, \forall \mathbf{v}(t) \in C, \quad where \quad \mathbf{w}(t) \in C \right\},

    \end{equation}

    Any intermediate help is appreciated, i.e., is there an equivalent way to formulate this set, that make finding the projector easier?
     
    Last edited: Jul 18, 2011
  2. jcsd
  3. Jul 18, 2011 #2

    Office_Shredder

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    I'm a bit confused by how C references itself in its definition.
     
  4. Jul 18, 2011 #3
    I understand the confusion--that is what makes the defining characteristic a little awkward. The basic idea is this: each element in the convex set [itex]\textit{C}[/itex] must correlate with every other element above [itex]\rho[/itex]. But we also know that a given (known) element [itex]\textbf{w}(t)[/itex] is contained in [itex]\textit{C}[/itex]. Is that less confusing of a statement?
     
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