Compressed sensing has demonstrated that a general signal $\boldsymbol{x} \in \mathbb{F}^n$ ($\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$) can be estimated from few linear measurements with an error {proportional to} the best $k$-term approximation error, a property known as instance optimality. In this paper, we investigate instance optimality in the context of phaseless measurements using the $\ell_p$-minimization decoder, where $p \in (0, 1]$, for both real and complex cases. More specifically, we prove that $(2,1)$ and $(1,1)$-instance optimality of order $k$ can be achieved with $m =O(k \log(n/k))$ phaseless measurements, paralleling results from linear measurements. These results imply that one can stably recover approximately $k$-sparse signals from $m = O(k \log(n/k))$ phaseless measurements. Our approach leverages the phaseless bi-Lipschitz condition. Additionally, we present a non-uniform version of $(2,2)$-instance optimality result in probability applicable to any fixed vector $\boldsymbol{x} \in \mathbb{F}^n$. These findings reveal striking parallels between compressive phase retrieval and classical compressed sensing, enhancing our understanding of both phase retrieval and instance optimality.

This work addresses the problem of uniquely determining a rotational motion from continuous time-dependent measurements within the frameworks of parallel-beam and diffraction tomography. The motivation stems from the challenge of imaging trapped biological samples manipulated and rotated using optical or acoustic tweezers. We analyze the conditions under which the rotation of the unknown sample can be uniquely recovered using the infinitesimal common line and circle method, respectively. We provide explicit criteria for the sample's structure and the induced motion that guarantee unique reconstruction of all rotation parameters. Moreover, we demonstrate that the set of objects for which uniqueness fails is nowhere dense.

The Effros-Shen algebra corresponding to an irrational number $\theta$ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of $\theta$ encodes the dimensions of the summands, and how the matrix algebras at the $n$th level fit into the summands at the $(n+1)$th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the $C^*$-algebra of a category of paths -- a generalization of a directed graph -- determined by the continued fraction expansion of $\theta$. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. In the present work, we define a spectral triple in terms of the category of paths presentation of an Effros-Shen algebra, drawing on a construction by Christensen and Ivan. This article describes categories of paths, the example of Mitscher and Spielberg, and the spectral triple construction.

The paper investigates the existence of a limit in the operator norm for a family of operators $T_z(H)= F(H-z)^{-1}F^*$ for $z$ tending to the real axis. The conditions for the $H$ operator and the rigging operator $F$ are established, under which the limit exists. Special attention is paid to the separation of cases when the limit point belongs and does not belong to the point spectrum $H$.

The purpose of this work is an attempt to expand the results obtained by A. K. Lakmon and Y. Mensah on embeddings of quantum Sobolev spaces $\mathfrak{H}_\gamma^{s,p}(G,H)$ consisting of Hilbert-Schmidt operators, with $p\neq 2$.

We extend the theory of gradient flows beyond metric spaces by studying evolution variational inequalities (EVIs) driven by general cost functions $c$, including Bregman and entropic transport divergences. We establish several properties of the resulting flows, including stability and energy identities. Using novel notions of convexity related to costs $c$, we prove that EVI flows are the limit of splitting schemes, providing assumptions for both implicit and explicit iterations.

For $G$ a connected, non-compact, semi-simple Lie group with finite center and $K$ a maximal compact subgroup of $G$, we study the $K$-biinvariant, real-valued solutions $f \in L^1(G)$ of the iterated convolution inequality \begin{equation*} f \ge \displaystyle\sum_{n=2}^N a_n \left(*^n f\right) \:, \end{equation*} where $N \ge 2$ is an integer, for $2 \le n \le N$, $a_n$ are non-negative integers with at least one of them positive and \begin{equation*} *^n f = \underbrace{f * \cdots *f }_{n-times}\:. \end{equation*} We prove that $f$ must be non-negative and satisfy the non-trivial bound $\|f\|_{L^1(G)}\le t_{\mathcal{Q}}$ where \begin{equation*} \mathcal{Q}(t):=t-\displaystyle\sum_{n=2}^N a_n\:t^n\:, \end{equation*} and $t_{\mathcal{Q}}$ is the unique zero of $\mathcal{Q}'$ in $(0,\infty)$. Our arguments involve Spherical Fourier Analysis on Riemannian Symmetric Spaces of non-compact type and Complex analysis. We then apply our result to obtain an a priori estimate for solutions of an integro-differential equation which is related to the physical problem of the ground state energy of the Bose gas in the classical Euclidean setting. Our results are new even for $\mathbb{R}^n$ and generalize recent works by Carlen-Jauslin-Lieb-Loss (Int Math Res Not 24:18604-18612, 2021) and Nakamura-Sawano (J Geom Anal 35:68, 2025) for two-fold and $m$-fold monomial convolution inequalities respectively, on $\mathbb{R}^n$.

This paper devotes to combine the chirp basis function transformation and symplectic coordinates transformation to yield a novel Wigner distribution (WD) associated with the linear canonical transform (LCT), named as the symplectic WD in the LCT domain (SWDL). It incorporates the merits of the symplectic WD (SWD) and the WD in the LCT domain (WDL), achieving stronger capability in the linear frequency-modulated (LFM) signal frequency rate feature extraction while maintaining the same level of computational complexity. Some essential properties of the SWDL are derived, including marginal distributions, energy conservations, unique reconstruction, Moyal formula, complex conjugate symmetry, time reversal symmetry, scaling property, time translation property, frequency modulation property, and time translation and frequency modulation property. Heisenberg's uncertainty principles of the SWDL are formulated, giving rise to three kinds of lower bounds attainable respectively by Gaussian enveloped complex exponential signal, Gaussian signal and Gaussian enveloped chirp signal. The optimal symplectic matrices corresponding to the highest time-frequency resolution are generated by solving the lower bound optimization (minimization) problem. The time-frequency resolution of the SWDL is compared with those of the SWD and WDL to demonstrate its superiority in LFM signals time-frequency energy concentration. A synthesis example is also carried out to verify the feasibility and reliability of the theoretical analysis.

The dynamics of probability density functions has been extensively studied in science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics that can be formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. The goal of this paper is to parallel the success of techniques using functional inequalities, for dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences as energy functionals. Such dynamics take the form of a nonlocal differential equation, for which existing analysis critically relies on using the explicit solution formula in special cases. We provide a comprehensive study on functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of the obtained functional inequalities is that they do not depend on the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posed) is uniform across general target distributions, making them potentially desirable dynamics for posterior sampling applications in Bayesian inference.

This paper introduces a hypothesis space for deep learning based on deep neural networks (DNNs). By treating a DNN as a function of two variables - the input variable and the parameter variable - we consider the set of DNNs where the parameter variable belongs to a space of weight matrices and biases determined by a prescribed depth and layer widths. To construct a Banach space of functions of the input variable, we take the weak* closure of the linear span of this DNN set. We prove that the resulting Banach space is a reproducing kernel Banach space (RKBS) and explicitly construct its reproducing kernel. Furthermore, we investigate two learning models - regularized learning and the minimum norm interpolation (MNI) problem - within the RKBS framework by establishing representer theorems. These theorems reveal that the solutions to these learning problems can be expressed as a finite sum of kernel expansions based on training data.

The purpose of this paper is to explore more properties and representations of the W-weighted m-weak group (in short, W-m-WG) inverse. We first explore an interesting relation between two projectors with respect to the W-m-WG inverse. Then, the W-m-WG inverse is represented by various generalized inverses including W-weighted Drazin inverse, W-weighted weak group inverse, W-weighted core inverse, etc. We also give three concise explicit expressions for the W-m-WG inverse. Moreover, a canonical form of the W-m-WG inverse is presented in terms of the singular value decomposition. Finally, several numerical examples are designed to illustrate some results given in the paper.

The Hardy-Littlewood-Polya inequality of majorization is extended for the {\omega}-m-star-convex functions to the framework of ordered Banach spaces. Several open problems which seem of interest for further extensions of the Hardy-Littlewood-Polya inequality are also included.

We establish a sharp comparison inequality between the negative moments and the second moment of the magnitude of sums of independent random vectors uniform on three-dimensional Euclidean spheres. This provides a probabilistic extension of the Oleszkiewicz-Pelczy\'nski polydisc slicing result. The Haagerup-type phase transition occurs exactly when the p-norm recovers volume, in contrast to the real case. We also obtain partial results in higher dimensions.

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in $L^p(\mathbb{R}^n)$.

We study projection-free methods for constrained Riemannian optimization. In particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex problems, and to a critical point for nonconvex objectives. We also present a practical setting under which RFW can attain a linear convergence rate. As a concrete example, we specialize RFW to the manifold of positive definite matrices and apply it to two tasks: (i) computing the matrix geometric mean (Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter. Both tasks involve geodesically convex interval constraints, for which we show that the Riemannian "linear" oracle required by RFW admits a closed-form solution; this result may be of independent interest. We further specialize RFW to the special orthogonal group and show that here too, the Riemannian "linear" oracle can be solved in closed form. Here, we describe an application to the synchronization of data matrices (Procrustes problem). We complement our theoretical results with an empirical comparison of RFW against state-of-the-art Riemannian optimization methods and observe that RFW performs competitively on the task of computing Riemannian centroids.

How should we place $n$ great circles on a sphere to minimize the furthest distance between a point on the sphere and its nearest great circle? Fejes Tóth conjectured that the optimum is attained by placing $n$ circles evenly spaced all passing through the north and south poles. This conjecture was recently proved by Jiang and Polyanskii. We present a short simplification of Ortega-Moreno's alternate proof of this conjecture.

In this article we prove martingale type pointwise convergence theorems pertaining to tensor product splines defined on $d$-dimensional Euclidean space ($d$ is a positive integer), where conditional expectations are replaced by their corresponding tensor spline orthoprojectors. Versions of Doob's maximal inequality, the martingale convergence theorem and the characterization of the Radon-Nikodým property of Banach spaces $X$ in terms of pointwise $X$-valued martingale convergence are obtained in this setting. Those assertions are in full analogy to their martingale counterparts and hold independently of filtration, spline degree, and dimension $d$.