arXiv subjects

Dustin G. Mixon

Publications and source records attributed to Dustin G. Mixon.

115 records · Page 7Linked to original sources

arXiv2011

Frame completions for optimally robust reconstruction

In information fusion, one is often confronted with the following problem: given a preexisting set of measurements about an unknown quantity, what new measurements should one collect in order to accomplish a given fusion task with optimal accuracy and efficiency. We illustrate just how difficult this problem can become by considering one of its more simple forms: when the unknown quantity is a vector in a Hilbert space, the task itself is vector reconstruction, and the measurements are linear functionals, that is, inner products of the unknown vector with given measurement vectors. Such reconstruction problems are the subject of frame theory. Here, we can measure the quality of a given frame by the average reconstruction error induced by noisy measurements; the mean square error is known to be the trace of the inverse of the frame operator. We discuss preliminary results which help indicate how to add new vectors to a given frame in order to reduce this mean square error as much as possible.

math.FA↗

arXiv2011

Constructing finite frames of a given spectrum and set of lengths

When constructing finite frames for a given application, the most important consideration is the spectrum of the frame operator. Indeed, the minimum and maximum eigenvalues of the frame operator are the optimal frame bounds, and the frame is tight precisely when this spectrum is constant. Often, the second-most important design consideration is the lengths of frame vectors: Gabor, wavelet, equiangular and Grassmannian frames are all special cases of equal norm frames, and unit norm tight frame-based encoding is known to be optimally robust against additive noise and erasures. We consider the problem of constructing frames whose frame operator has a given spectrum and whose vectors have prescribed lengths. For a given spectrum and set of lengths, the existence of such frames is characterized by the Schur-Horn Theorem---they exist if and only if the spectrum majorizes the squared lengths---the classical proof of which is nonconstructive. Certain construction methods, such as harmonic frames and spectral tetris, are known in the special case of unit norm tight frames, but even these provide but a few examples from the manifold of all such frames, the dimension of which is known and nontrivial. In this paper, we provide a new method for explicitly constructing any and all frames whose frame operator has a prescribed spectrum and whose vectors have prescribed lengths. The method itself has two parts. In the first part, one chooses eigensteps---a sequence of interlacing spectra---that transform the trivial spectrum into the desired one. The second part is to explicitly compute the frame vectors in terms of these eigensteps; though nontrivial, this process is nevertheless straightforward enough to be implemented by hand, involving only arithmetic, square roots and matrix multiplication.

math.FA↗

arXiv2011

Frame Coherence and Sparse Signal Processing

The sparse signal processing literature often uses random sensing matrices to obtain performance guarantees. Unfortunately, in the real world, sensing matrices do not always come from random processes. It is therefore desirable to evaluate whether an arbitrary matrix, or frame, is suitable for sensing sparse signals. To this end, the present paper investigates two parameters that measure the coherence of a frame: worst-case and average coherence. We first provide several examples of frames that have small spectral norm, worst-case coherence, and average coherence. Next, we present a new lower bound on worst-case coherence and compare it to the Welch bound. Later, we propose an algorithm that decreases the average coherence of a frame without changing its spectral norm or worst-case coherence. Finally, we use worst-case and average coherence, as opposed to the Restricted Isometry Property, to garner near-optimal probabilistic guarantees on both sparse signal detection and reconstruction in the presence of noise. This contrasts with recent results that only guarantee noiseless signal recovery from arbitrary frames, and which further assume independence across the nonzero entries of the signal---in a sense, requiring small average coherence replaces the need for such an assumption.

math.FA↗

arXiv2010

Steiner equiangular tight frames

We provide a new method for constructing equiangular tight frames (ETFs). The construction is valid in both the real and complex settings, and shows that many of the few previously-known examples of ETFs are but the first representatives of infinite families of such frames. It provides great freedom in terms of the frame's size and redundancy. This method also explicitly constructs the frame vectors in their native domain, as opposed to implicitly defining them via their Gram matrix. Moreover, in this domain, the frame vectors are very sparse. The construction is extremely simple: a tensor-like combination of a Steiner system and a regular simplex. This simplicity permits us to resolve an open question regarding ETFs and the restricted isometry property (RIP): we show that the RIP behavior of some ETFs is unfortunately no better than their coherence indicates.

math.FA↗

arXiv2010

Auto-tuning unit norm frames

Finite unit norm tight frames provide Parseval-like decompositions of vectors in terms of redundant components of equal weight. They are known to be exceptionally robust against additive noise and erasures, and as such, have great potential as encoding schemes. Unfortunately, up to this point, these frames have proven notoriously difficult to construct. Indeed, though the set of all unit norm tight frames, modulo rotations, is known to contain manifolds of nontrivial dimension, we have but a small finite number of known constructions of such frames. In this paper, we present a new iterative algorithm---gradient descent of the frame potential---for increasing the degree of tightness of any finite unit norm frame. The algorithm itself is trivial to implement, and it preserves certain group structures present in the initial frame. In the special case where the number of frame elements is relatively prime to the dimension of the underlying space, we show that this algorithm converges to a unit norm tight frame at a linear rate, provided the initial unit norm frame is already sufficiently close to being tight. By slightly modifying this approach, we get a similar, but weaker, result in the non-relatively-prime case, providing an explicit answer to the Paulsen problem: "How close is a frame which is almost tight and almost unit norm to some unit norm tight frame?"

math.FA↗

arXiv2010

The Bourgain-Tzafriri conjecture and concrete constructions of non-pavable projections

It is known that the Kadison-Singer Problem (KS) and the Paving Conjecture (PC) are equivalent to the Bourgain-Tzafriri Conjecture (BT). Also, it is known that (PC) fails for $2$-paving projections with constant diagonal $1/2$. But the proofs of this fact are existence proofs. We will use variations of the discrete Fourier Transform matrices to construct concrete examples of these projections and projections with constant diagonal $1/r$ which are not $r$-pavable in a very strong sense. In 1989, Bourgain and Tzafriri showed that the class of zero diagonal matrices with small entries (on the order of $\le 1/log^{1+ε}n$, for an $n$-dimensional Hilbert space) are {\em pavable}. It has always been assumed that this result also holds for the BT-Conjecture - although no one formally checked it. We will show that this is not the case. We will show that if the BT-Conjecture is true for vectors with small coefficients (on the order of $\le C/\sqrt{n}$) then the BT-Conjecture is true and hence KS and PC are true.

math.FA↗

arXiv2010

Filter Bank Fusion Frames

In this paper we characterize and construct novel oversampled filter banks implementing fusion frames. A fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way. When properly designed, fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures. However, up to this point, few implementable constructions of such frames were known; we show how to construct them using oversampled filter banks. In this work, we first provide polyphase domain characterizations of filter bank fusion frames. We then use these characterizations to construct filter bank fusion frame versions of discrete wavelet and Gabor transforms, emphasizing those specific finite impulse response filters whose frequency responses are well-behaved.

cs.IT↗