arXiv subjects

Pete Casazza

Publications and source records attributed to Pete Casazza.

4 recordsLinked to original sources

arXiv2023

A Survey of Fusion Frames in Hilbert Spaces

Fusion frames are a very active area of research today because of their myriad of applications in pure mathematics, applied mathematics, engineering, medicine, signal and image processing and much more. They provide a great flexibility for designing sets of vectors for applications and are therefore prominent in all these areas, including e.g. mitigating the effects of noise in a signal or giving robustness to erasures. In this chapter, we present the fundamentals of fusion frame theory with an emphasis on their delicate relation to frame theory. The goal here is to provide researchers and students with an easy entry into this topic. Proofs for fusion frames will be self-contained and differences between frames and fusion frames are analyzed. In particular, we focus on the subtleties of fusion frame duality. We also provide a reproducible research implementation.

math.FA↗

arXiv2020

Preserving Injectivity under Subgaussian Mappings and Its Application to Compressed Sensing

The field of compressed sensing has become a major tool in high-dimensional analysis, with the realization that vectors can be recovered from relatively very few linear measurements as long as the vectors lie in a low-dimensional structure, typically the vectors that are zero in most coordinates with respect to a basis. However, there are many applications where we instead want to recover vectors that are sparse with respect to a dictionary rather than a basis. That is, we assume the vectors are linear combinations of at most $s$ columns of a $d \times n$ matrix $\mathbf{D}$, where $s$ is very small relative to $n$ and the columns of $\mathbf{D}$ form a (typically overcomplete) spanning set. In this direction, we show that as a matrix $\mathbf{D}$ stays bounded away from zero in norm on a set $S$ and a provided map ${\boldsymbol Φ}$ comprised of i.i.d. subgaussian rows has number of measurements at least proportional to the square of $w(\mathbf{D}S)$, the Gaussian width of the related set $\mathbf{D}S$, then with high probability the composition ${\boldsymbol Φ} \mathbf{D}$ also stays bounded away from zero. As a specific application, we obtain that the null space property of order $s$ is preserved under such subgaussian maps with high probability. Consequently, we obtain stable recovery guarantees for dictionary-sparse signals via the $\ell_1$-synthesis method with only $O(s\log(n/s))$ random measurements and a minimal condition on $\mathbf{D}$, which complements the compressed sensing literature.

cs.IT↗

arXiv2009

Redundancy for localized and Gabor frames

Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist, the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one. We will answer this question in the affirmative for $\ell^1$-localized frames. We then specialize our results to Gabor multi-frames with generators in $M^1(\R^d)$, and Gabor molecules with envelopes in $W(C,l^1)$. As a main tool in this work, we show there is a universal function $g(x)$ so that for every $ε>0$, every Parseval frame $\{f_i\}_{i=1}^M$ for an $N$-dimensional Hilbert space $H_N$ has a subset of fewer than $(1+ε)N$ elements which is a frame for $H_N$ with lower frame bound $g(ε/(2\frac{M}{N}-1))$. This work provides the first meaningful quantative notion of redundancy for a large class of infinite frames. In addition, the results give compelling new evidence in support of a general definition of reudndancy given in [7].

math.FA↗

arXiv2004

On signal reconstruction without noisy phase

We construct new classes of Parseval frames for a Hilbert space which allow signal reconstruction from the absolute value of the frame coefficients. As a consequence, signal reconstruction can be done without using noisy phase or its estimation. This verifies a longstanding conjecture of the speech processing community.

math.FA↗