The Kadison-Singer and Paulsen Problems in Finite Frame Theory

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24 Peter G. Casazzathe frame operator is unchanged for any value of θ. Now choose the θ yielding‖ψ i ‖ 2 = λ. Repeating this process at most M − 1 times yields an equal normframe with the same frame operator as {ϕ i } M i=1 .Using a Parseval frame in Proposition 6, we do get to an equal normParseval frame. The problem, again, is that we do not have any quantitativemeasure of how close these two Parseval frames are.There is an obvious approach towards solving the Paulsen Problem. Givenan ɛ-nearly equal norm ɛ-nearly Parseval frame {ϕ i } M i=1 for HN with frameoperator S, we can switch to the closest Parseval frame {S −1/2 ϕ i } M i=1 . Thenswitch to the closest equal norm frame to {S −1/2 ϕ i } M i=1 , call it {ψ i} M i=1 withframe operator S 1 . Now switch to {S −1/21 ψ i } M i=1 and again switch to the closestequal norm frame and continue. Unfortunately, even if we could show thatthis process converges and we could check the distance traveled through thisprocess, we would still not have an answer to the Paulsen Problem becausethis process does not have to converge to an equal norm Parseval frame. Inparticular, there is a fixed point of this process which is not an equal normParseval frame.Example 1. Let {e i } N i=1 be an orthonormal basis for lN 2 and let {ϕ i } N+1i=1 bea equiangular unit norm tight frame for l N 2 . Then {e i ⊕ 0} N i=1 ∪ {0 ⊕ ϕ i} N+1i=1in l N 2 ⊕ l N 2 is a ɛ = 1 N -nearly equal norm and 1 N- nearly Parseval frame withframe operator, say S. A direct calculation shows that taking S −1/2 of theframe vectors and switching to the closest equal norm frame leaves the frameunchanged.The Paulsen Problem has proven to be intractable for over 12 years. Recently,two partial solutions to the problem were given in [10, 20] each havingsome advantages. Since each of these papers is technical, we will only outlinethe ideas here.In [10], a new technique is introduced. This is a system of vector valuedODE’s which starts with a given Parseval frame and has the property that allframes in the flow are still Parseval while approaching an equal norm Parsevalframe. They then bound the arc length of the system of ODE’s by the frameenergy. Finally, giving an exponential bound on the frame energy, they havea quantitative estimate for the distance between the initial, ɛ-nearly equalnormand ɛ-nearly Parseval frame F and the equal-norm Parseval frame G.For the method to work, they must assume that the dimension N of theHilbert space and the number M of frame vectors are relatively prime. Theauthors show that in practice, this is not a serious restriction. The main resultof [10] isTheorem 13. Let N, M ∈ N be relatively prime, let 0 < ɛ < 1 2, and assumeΦ = {ϕ i } M i=1 is an ɛ-nearly equal-norm and ɛ-nearly Parseval frame for areal or complex Hilbert space of dimension N. Then there is an equal-normParseval frame Ψ = {ψ i } M i=1 such that
1 The Kadison-Singer and Paulsen Problems in Finite Frame Theory 25‖Φ − Ψ‖ ≤ 298 N 2 M(M − 1) 8 ɛ.In [20], the authors present a new iterative algorithm—gradient descent ofthe frame potential—for increasing the degree of tightness of any finite unitnorm frame. The algorithm itself is trivial to implement, and it preservescertain group structures present in the initial frame. In the special case wherethe number of frame elements is relatively prime to the dimension of theunderlying space, they show that this algorithm converges to a unit normtight frame at a linear rate, provided the initial unit norm frame is alreadysufficiently close to being tight. The main difference between this approachand the approach in [10] is that in [10], the authors start with a nearly equalnorm Parseval frame and improve its closeness to an equal norm frame whilemaintaining Parseval, and in [20] the authors start with an equal norm nearlyParseval frame and give an algorithm for improving its algebraic propertieswhile changing its transform as little as possible. The main result from [20]is:1Theorem 14. Suppose M and N are relatively prime. Pick t ∈ (0,2M), andlet Φ 0 = {ϕ i } M i=1 be a unit norm frame with analysis operator T 0 satisfying‖T0 ∗ T 0 − M N I‖2 HS ≤ 2N. Now, iterate the gradient descent of the frame potential3method to obtain Φ k . Then, Φ ∞ := lim k Φ k exists and is a unit norm tightframe satisfying‖Φ ∞ − Φ 0 ‖ HS ≤ 4N 20 M 8.51−2Mt∥ T∗0 T 0 − M N I∥ ∥HS.In [10], the authors showed there is a connection between the PaulsenProblem and a fundamental open problem in operator theory.Problem 3 (Projection Problem). Let H N be an N-dimensional Hilbertspace with orthonormal basis {e i } N i=1 . Find the function g(ɛ, N, M) satisfyingthe following. If P is a projection of rank M on H N satisfying(1 − ɛ) M N ≤ ‖P e i‖ 2 ≤ (1 + ɛ) M , for all i = 1, 2, . . . , N,Nthen there is a projection Q with ‖Qe i ‖ 2 = M Nfor all i = 1, 2, . . . , N satisfyingN∑‖P e i − Qe i ‖ 2 ≤ g(ɛ, N, M).i=1In [14], it was shown that the Paulsen problem is equivalent to the ProjectionProblem and that their closeness functions are within a factor of 2 ofone another. The proof of this result gives several exact connections betweenthe distance between frames and the distance between the ranges of theiranalysis operators.
Page 2 and 3: 2 Peter G. CasazzaThe Kadison-Singe
Page 4 and 5: 4 Peter G. Casazzaimplies PC follow
Page 6 and 7: 6 Peter G. Casazza(2) The square su
Page 8 and 9: 8 Peter G. CasazzaSimilarly, ‖
Page 10: 10 Peter G. Casazza‖N∑a i T e i
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The Kadison-Singer and Paulsen Problems in Finite Frame Theory

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