Related papers: Sequences that do frame reconstruction
The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and…
Recovering a signal up to a unimodular constant from the magnitudes of linear measurements has been popular and well studied in recent years. However, numerous unsolved problems regarding phase retrieval still exist. Given a phase retrieval…
A novel framework for designing image reconstruction algorithms for linear forward problems is proposed. The framework is based on the novel concept of conserving the information in the data during image reconstruction rather than…
In the paper, we introduce an unconstrained analysis model based on the $\ell_{1}-\alpha \ell_{2}$ $(0< \alpha \leq1)$ minimization for the signal and image reconstruction. We develop some new technology lemmas for tight frame, and the…
We consider sequences in a Hilbert space $\mathcal H$ of the form $(T^nf_0)_{n\in I},$ with a linear operator $T$, the index set being either $I = \mathbb N$ or $I = \mathbb Z$, a vector $f_0\in \mathcal H$, and answer the following two…
Transformer architectures are typically described in algorithmic and statistical terms, leaving their internal mechanics without a familiar structural language for researchers trained in physical theories. To bridge this gap, we develop a…
Motivated by the dynamical sampling problem, we study frames in an infinite dimensional Hilbert space generated by the iterates of a bounded operator T, also known as dynamical frames. We first characterize the operators that generate…
We consider sampling strategies for a class of multivariate bandlimited functions $f$ that have a spectrum consisting of disjoint frequency bands. Taking advantage of the special spectral structure, we provide formulas relating $f$ to the…
Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this subsequence has length (1 - \epsilon) n. On the other hand, there is a frame which does not contain bases with…
We introduce the notions of kernel map and kernel set of a bounded linear operator on a Hilbert space relative to a subspace lattice. The characterization of the kernel maps and kernel sets of finite rank operators leads to showing that…
Fusion frames are extensively studied due to their effectiveness in recovering signals from large-scale data. They are applicable in distributed processing, wireless sensor networks, and packet encoding systems due to their robustness and…
We consider a one-parameter family of piecewise isometries of a rhombus. The rotational component is fixed, and its coefficients belong to the quadratic number field $K=\mathbb{Q}(\sqrt{2})$. The translations depend on a parameter $s$ which…
In this paper it is investigated how to find a matrix representation of operators on a Hilbert space with Bessel sequences, frames and Riesz bases. In many applications these sequences are often preferable to orthonormal bases (ONBs).…
Let $A\colon H\rightarrow H$ be a normal operator on an infinite-dimensional separable Hilbert space $H$ and let $S\subseteq H$ be a finite subset such that $\{A^nx\}_{n\geq 0,\,x\in S}$ can be rescaled to form a frame for $H$. That is,…
Phase retrieval is a nonlinear inverse problem that arises in a wide range of imaging modalities, from electron microscopy to Fourier ptychography. In particular, the reconstruction is facilitated when the sensing matrix is i.i.d. random,…
Despite their impressive performance, contemporary neural networks often lack structural safeguards that promote stable learning and interpretable behavior. In this work, we introduce a reformulation of layer-level transformations that…
We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays…
Identifying quantum phase transitions poses a significant challenge in condensed matter physics, as this requires methods that both provide accurate results and scale well with system size. In this work, we demonstrate how relaxation…
We propose a sparse reconstruction framework for solving inverse problems. Opposed to existing sparse regularization techniques that are based on frame representations, we train an encoder-decoder network by including an $\ell^1$-penalty.…
Recovering a signal from its Fourier magnitude is referred to as phase retrieval, which occurs in different fields of engineering and applied physics. This paper gives a new characterization of the phase retrieval problem. Particularly…