arXiv subjects

Peter G. Casazza

Publications and source records attributed to Peter G. Casazza.

At least 55 records · Page 3Linked to original sources

arXiv2014

Phase retrieval and norm retrieval

Phase retrieval has become a very active area of research. We will classify when phase retrieval by Parseval frames passes to the Naimark complement and when phase retrieval by projections passes to the orthogonal complements. We introduce a new concept we call norm retrieval and show that this is what is necessary for passing phase retrieval to complements. This leads to a detailed study of norm retrieval and its relationship to phase retrieval. One fundamental result: a frame $\{φ_i\}_{i=1}^M$ yields phase retrieval if and only if $\{Tφ_i\}_{i=1}^M$ yields norm retrieval for every invertible operator $T$.

math.FA↗

arXiv2014

The fundamentals of Spectral Tetris frame constructions

The landmark paper "Constructing tight fusion frames" by Casazza, Fickus, Mixon, Wang and Zhou introduced a fundamental method for constructing unit norm tight frames, which they called Spectral Tetris. This was a significant advancement for finite frame theory - especially constructions of finite frames. This paper then generated a vast amount of literature as Spectral Tetris was steadily developed, refined, and generalized until today we have a complete picture of what are the broad applications as well as the limitations of Spectral Tetris. In this paper, we will put this vast body of literature into a coherent theory.

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arXiv2013

Tight and random nonorthogonal fusion frames

First we show that tight nonorthogonal fusion frames a relatively easy to com by. In order to do this we need to establish a classification of how to to wire a self adjoint operator as a product of (nonorthogonal) projection operators. We also discuss the link between nonorthogonal fusion frames and positive operator valued measures, we define and study a nonorthogonal fusion frame potential, and we introduce the idea of random nonorthogonal fusion frames.

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arXiv2013

Phase Retrieval By Projections

The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists in engineering: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. There exist semidefinite programming algorithms to solve this problem, but much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps toward these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with $2M-1$ projections of arbitrary rank in $\HH_M$. Finally we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.

math.FA↗

arXiv2013

Every Hilbert space frame has a Naimark complement

Naimark complements for Hilbert space Parseval frames are one of the most fundamental and useful results in the field of frame theory. We will show that actually all Hilbert space frames have Naimark complements which possess all the usual properties for Naimark complements with one notable exception. So these complements can be used for equiangular frames, RIP property, fusion frames etc. Along the way, we will correct a mistake in a recent fusion frame paper where chordal distances for Naimark complements are computed incorrectly.

math.FA↗

arXiv2012

The simplified version of the Spielman and Srivastava algorithm for proving the Bourgain-Tzafriri restricted invertiblity theorem

By giving up the best constants, we will see that the original argument of Spielman and Srivastava for proving the Bourgain-Tzafriri Restricted Invertibility Theorem \cite{SS} still works - and is much simplier than the final version. We do not intend on publishing this since it is their argument with just a trivial modification, but we want to make it available to the mathematics community since several people have requested it already.

math.FA↗

arXiv2012

Weighted Fusion Frame Construction via Spectral Tetris

Fusion frames consist of a sequence of subspaces from a Hilbert space and corresponding positive weights so that the sum of weighted orthogonal projections onto these subspaces is an invertible operator on the space. Given a spectrum for a desired fusion frame operator and dimensions for subspaces, one existing method for creating unit-weight fusion frames with these properties is the flexible and elementary procedure known as spectral tetris. Despite the extensive literature on fusion frames, until now there has been no construction of fusion frames with prescribed weights. In this paper we use spectral tetris to construct more general, arbitrarily weighted fusion frames. Moreover, we provide necessary and sufficient conditions for when a desired fusion frame can be constructed via spectral tetris.

math.NA↗

arXiv2012

Spectral Tetris Fusion Frame Constructions

Spectral tetris is a fexible and elementary method to construct unit norm frames with a given frame operator, having all of its eigenvalues greater than or equal to two. One important application of spectral tetris is the construction of fusion frames. We first show how the assumption on the spectrum of the frame operator can be dropped and extend the spectral tetris algorithm to construct unit norm frames with any given spectrum of the frame operator. We then provide a suffcient condition for using this generalization of spectral tetris to construct fusion frames with prescribed spectrum for the fusion frame operator and with prescribed dimensions for the subspaces. This condition is shown to be necessary in the tight case of redundancy greater than two.

math.NA↗

arXiv2011

Fusion Frames and the Restricted Isometry Property

We will show that tight frames satisfying the restricted isometry property give rise to nearly tight fusion frames which are nearly orthogonal and hence are nearly equi-isoclinic. We will also show how to replace parts of the RIP frame with orthonormal sets while maintaining the RIP property.

math.FA↗

arXiv2011

An elementary, illustrative proof of the Rado-Horn Theorem

The Rado-Horn theorem provides necessary and sufficient conditions for when a collection of vectors can be partitioned into a fixed number of linearly independent sets. Such partitions exist if and only if every subset of the vectors satisfies the so-called Rado-Horn inequality. Today there are at least six proofs of the Rado-Horn theorem, but these tend to be extremely delicate or require intimate knowledge of matroid theory. In this paper we provide an elementary proof of the Rado-Horn theorem as well as elementary proofs for several generalizations including results for the redundant case when the hypotheses of the Rado-Horn theorem fail. Another problem with the existing proofs of the Rado-Horn Theorem is that they give no information about how to actually partition the vectors. We start by considering a specific partition of the vectors, and the proof consists of showing that this is an optimal partition. We further show how certain structures we construct in the proof are at the heart of the Rado-Horn theorem by characterizing subsets of vectors which maximize the Rado-Horn inequality. Lastly, we demonsrate how these results may be used to select an optimal partition with respect to spanning properties of the vectors.

math.FA↗

arXiv2011

Optimally Sparse Frames

Frames have established themselves as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. However, when the signal dimension is large, the computation of the frame measurements of a signal typically requires a large number of additions and multiplications, and this makes a frame decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we focus on frames in finite-dimensional Hilbert spaces and introduce sparsity for such frames as a new paradigm. In our terminology, a sparse frame is a frame whose elements have a sparse representation in an orthonormal basis, thereby enabling low-complexity frame decompositions. To introduce a precise meaning of optimality, we take the sum of the numbers of vectors needed of this orthonormal basis when expanding each frame vector as sparsity measure. We then analyze the recently introduced algorithm Spectral Tetris for construction of unit norm tight frames and prove that the tight frames generated by this algorithm are in fact optimally sparse with respect to the standard unit vector basis. Finally, we show that even the generalization of Spectral Tetris for the construction of unit norm frames associated with a given frame operator produces optimally sparse frames.

math.NA↗

arXiv2011

The Paulsen Problem in Operator Theory

The Paulsen Problem in Hilbert space frame theory has proved to be one of the most intractable problems in the field. We will help explain why by showing that this problem is equivalent to a fundamental, deep problem in operator theory. Along the way we will give a new exact computation for chordal distances, we will give a generalization of these problems and we will spell out exactly the complementary versions of the problem.

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.

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arXiv2010

A quantitative notion of redundancy for infinite frames

Bodmann, Casazza and Kutyniok introduced a quantitative notion of redundancy for finite frames - which they called {\em upper and lower redundancies} - that match better with an intuitive understanding of redundancy for finite frames in a Hilbert space. The objective of this paper is to see how much of this theory generalizes to infinite frames.

math.FA↗

arXiv2010

Concrete constructions of non-pavable projections

It is known that the paving conjecture fails for 2-paving projections with constant diagonal 1/2. But the proofs of this fact are existence proofs. We will give concrete examples of these projections and projections with constant diagonal $1/r$ which are not $r$-pavable in a very strong sense.

math.FA↗

arXiv2010

Spanning and independence properties of frame partitions

We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in finite dimensional Hilbert spaces, Parseval frames with norms bounded away from 1 can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. We also prove, assuming the Kadison-Singer conjecture is true, that this holds for infinite dimensional Hilbert spaces. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent, and the complement of each set to contain a number of disjoint, spanning sets.

math.FA↗

arXiv2009

A quantitative notion of redundancy for finite frames

The objective of this paper is to improve the customary definition of redundancy by providing quantitative measures in its place, which we coin upper and lower redundancies, that match better with an intuitive understanding of redundancy for finite frames in a Hilbert space. This motivates a carefully chosen list of desired properties for upper and lower redundancies. The means to achieve these properties is to consider the maximum and minimum of a redundancy function, which is interesting in itself. The redundancy function is defined on the sphere of the Hilbert space and measures the concentration of frame vectors around each point. A complete characterization of functions on the sphere which coincide with a redundancy function for some frame is given. The upper and lower redundancies obtained from this function are shown to satisfy all of the intuitively desirable properties. In addition, the range of values they assume is characterized.

math.FA↗