Maryam Fazel
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Maryam Fazel

Assistant Professor

Electrical Engineering Department,
University of Washington

Address:
University of Washington
Department of Electrical Engineering
Paul Allen Center - Room AE100R
Campus Box 352500
Seattle, WA 98195-2500
Phone: (206) 616-4781
Fax: (206) 543-3842
E-mail:
mfazel (at) ee(dot)washington(dot)edu
URL:
http://faculty.washington.edu/mfazel

I joined UW EE in December 2007. Prior to that, I was a Research Scientist at the Control and Dynamical Systems Department at Caltech, working with Prof. John Doyle. I received my PhD in Electrical Engineering from Stanford University under the supervision of Prof. Stephen Boyd.

Research Interests

  • Optimization, in particular convex optimization; convex relaxations and approximations; algebraic methods such as Sum of Squares
  • Systems and Control; Linear Matrix Inequalities
  • Finding ``simple" models (or Parsimonious Modeling) and Matrix Rank Minimization; convex relaxations for rank minimization
  • Compressed sensing and sparse signal recovery
  • Network resource allocation; distributed optimization
  • Applications in: Networking, Systems Biology, Physiological Modeling

Teaching

EE/AA/ME/ChemE 591: Topics in Optimization and its Engineering Applications
Spring 2008, University of Washington
Seminar Webpage

EE/AA/ME 578: Optimization in System Sciences (a.k.a. Convex optimization and applications)
Winter 2008, University of Washington
Course Webpage
Course Announcement

Education

  • Ph.D. Electrical Engineering, Stanford University, CA, March 2002
  • M.S. Electrical Engineering, Stanford University, CA, June 1997
  • B.Sc. Electrical Engineering, Sharif University of Technology, Tehran, Iran, 1995

Publications

Ph.D. Thesis

  • Matrix Rank Minimization with Applications. Elec. Eng. Dept, Stanford University, March 2002.
    Thesis: (ps file), (pdf file), (gzipped ps file)


    My PhD work concerns the problem of minimizing the rank of a matrix over a convex set. In many engineering applications, notions such as order, complexity, or dimension of a model or design can be expressed as the rank of an appropriate matrix. If the set of feasible models or designs is convex, then choosing the simplest model can be cast as a Rank Minimization Problem (RMP). For example, a low-rank matrix could correspond to a low-order controller for a plant, a low-degree statistical model fit for a random process, a shape that can be embedded in a low-dimensional space, or a design with a small number of components. It is thus not surprising that rank minimization has diverse applications: following Occam's razor, we often seek simple explanations and designs.

    The RMP is known to be NP-hard. If the variable in an RMP is a positive semidefinite matrix, a simple and effective relaxation commonly used is trace minimization. We prove that any general RMP involving non-positive semidefinite or even non-square matrices, can be embedded in a larger, positive semidefinite RMP. This semidefinite embedding lemma is then used to extend the trace relaxation, as well as other methods using semidefiniteness, to general matrices. The generalized trace relaxation minimizes the dual of the matrix 2-norm, referred to as the nuclear norm. Furthermore, we show that this relaxation is equivalent to minimizing the convex envelope of the rank function over the unit ball, thus providing theoretical support for its use. We also propose another heuristic based on (locally) minimizing the logarithm of the determinant (log-det) of the matrix, to refine the result of the trace relaxation and reduce the rank further. This heuristic is in fact equivalent to iteratively minimizing a weighted trace objective, and thus also admits an interpretation in terms of convex envelopes. Finally, applying these methods to application examples shows they work well in practice.

    Related papers:

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Last modified: Dec 2007