Frame theory intersects geometry
July 29 to August 2, 2013
American Institute of Mathematics,
San Jose, California
and Tim Roemer
This workshop will be devoted to outstanding problems that are in the intersection of frame
theory and geometry. Frames are families of vectors in Hilbert spaces which provide stable expansions. These families
are more general than orthonormal bases because they can incorporate linear dependencies. Applications in
engineering, quantum theory and in pure mathematics have lead to several design problems for which optimal frames
satisfy spectral as well as geometric properties. The simplest frame design problem, the construction of equal-norm
Parseval frames, can be rephrased as a minimization problem on a Stiefel manifold. Similar reformulations exist
for the construction of equiangular tight frames, including the special case of symmetric informationally complete
positive operator-valued measures in quantum information theory. Another design problem with geometric character
is that of frames for phase retrieval; these frames allow the recovery of a point in complex projective space based on the
magnitudes of its frame coefficients.
Recent advances on these types of problems have incorporated more and more geometric techniques in their analysis.
A strong interaction between researchers in frame theory with those in real and complex geometry, algebraic geometry
and algebraic topology is expected to boost progress on these outstanding problems.
Particular topics envisioned for the workshop are the following:
Finding minimal frames with this property is an open problem.
- Existence of complex equiangular line sets and equiangular Parseval frames as affine algebraic varieties.
The construction of equiangular Parseval frames amounts to solving a number of polynomial equations
defining a possibly non-trivial algebraic variety. The existence of such varieties could be shown by constructing
Gr�bner bases for the associated ideal space.
- Connectedness of equal-norm Parseval frames and convergence of gradient flows towards global minimizers.
The connectedness is a necessary condition for the feasibility of constructing Grassmannian frames by the minimization
of suitable frame potentials. Convergence results are expected from local convexity.
- Characterization of frames which allow phase retrieval from magnitudes of frame coefficients. The construction
of such frames is equivalent to having uniqueness of solutions for certain quadratic equations in projective space.
Material from the workshop
A list of participants.
The workshop schedule.
A report on the workshop activities.
A list of open problems.
Papers arising from the workshop:
Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames
by Jameson Cahill, Dustin G. Mixon and Nate Strawn, SIAM J. Appl. Algebra Geom. 1 (2017), no. 1, 38–72 MR3633768
Complex Two-Graphs via Equiangular Tight Frames
by Thomas Hoffman and James Solazzo
An algebraic characterization of injectivity in phase retrieval
by Aldo Conca, Dan Edidin, Milena Hering and Cynthia Vinzant, Appl. Comput. Harmon. Anal. 38 (2015), no. 2, 346–356 MR3303679
Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns
by David Gross, Felix Krahmer and Richard Kueng, Appl. Comput. Harmon. Anal. 42 (2017), no. 1, 37-64 MR3574560
A strong restricted isometry property, with an application to phaseless compressed sensing
by Vladislav Voroninski and Zhiqiang Xu, Appl. Comput. Harmon. Anal. 40 (2016), no. 2, 386-395 MR3440178
All complex equiangular tight frames in dimension 3
by Ferenc Szöllősi
Frame potentials and the geometry of frames
by Bernhard G. Bodmann and John Haas, J. Fourier Anal. Appl. 21 (2015), no. 6, 1344-1383 MR3421919