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1. A Duality Principle for Groups II: Multi-frames Meet Super-Frames

   https://doi.org/10.1007/s00041-020-09792-0  

   Balan, R.; Dutkay, D.; Han, D.; Larson, D.; Luef, F. (December 2020, Journal of Fourier Analysis and Applications)

   null (Ed.)

   Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d . 

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2. Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation

   https://doi.org/10.1353/ajm.2023.a913295  

   Farah, Luiz Gustavo; Holmer, Justin; Roudenko, Svetlana; Yang, Kai (December 2023, American Journal of Mathematics)

   Abstract: We consider the quadratic Zakharov-Kuznetsov equation $$\partial_t u + \partial_x \Delta u + \partial_x u^2=0$$ on $$\Bbb{R}^3$$. A solitary wave solution is given by $Q(x-t,y,z)$, where $$Q$$ is the ground state solution to $$-Q+\Delta Q+Q^2=0$$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $$Q$$ in the energy space, evolves to a solution that, as $$t\to\infty$$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $$x>\delta t-\tan\theta\sqrt{y^2+z^2}$$ for $$0\leq\theta\leq{\pi\over 3}-\delta$$. 

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3. Inference in high-dimensional linear regression via lattice basis reduction and integer relation detection

   https://doi.org/10.1109/TIT.2021.3113921  

   Gamarnik, D.; Kizildag, E.C.; Ilias Zadik, I. (January 2021, IEEE transactions on information theory)

   We consider the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector $$\beta^*\in\mathbb{R}^p$$ from its linear measurements, using a small number $$n$$ of samples. Unlike most of the literature, we make no sparsity assumption on $$\beta^*$$, but instead adopt a different regularization: In the noiseless setting, we assume $$\beta^*$$ consists of entries, which are either rational numbers with a common denominator $$Q\in\mathbb{Z}^+$$ (referred to as $Q-$$rationality); or irrational numbers taking values in a rationally independent set of bounded cardinality, known to learner; collectively called as the mixed-range assumption. Using a novel combination of the Partial Sum of Least Squares (PSLQ) integer relation detection, and the Lenstra-Lenstra-Lov\'asz (LLL) lattice basis reduction algorithms, we propose a polynomial-time algorithm which provably recovers a $$\beta^*\in\mathbb{R}^p$ enjoying the mixed-range assumption, from its linear measurements $$Y=X\beta^*\in\mathbb{R}^n$$ for a large class of distributions for the random entries of $$X$$, even with one measurement ($n=1$). In the noisy setting, we propose a polynomial-time, lattice-based algorithm, which recovers a $$\beta^*\in\mathbb{R}^p$$ enjoying the $Q-$rationality property, from its noisy measurements $$Y=X\beta^*+W\in\mathbb{R}^n$$, even from a single sample ($n=1$). We further establish that for large $$Q$$, and normal noise, this algorithm tolerates information-theoretically optimal level of noise. We then apply these ideas to develop a polynomial-time, single-sample algorithm for the phase retrieval problem. Our methods address the single-sample ($n=1$) regime, where the sparsity-based methods such as the Least Absolute Shrinkage and Selection Operator (LASSO) and the Basis Pursuit are known to fail. Furthermore, our results also reveal algorithmic connections between the high-dimensional linear regression problem, and the integer relation detection, randomized subset-sum, and shortest vector problems. 

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4. An inhomogeneous Dirichlet theorem via shrinking targets

   https://doi.org/10.1112/S0010437X1900719X  

   Kleinbock, Dmitry; Wadleigh, Nick (July 2019, Compositio Mathematica)

   We give an integrability criterion on a real-valued non-increasing function $$\unicode[STIX]{x1D713}$$ guaranteeing that for almost all (or almost no) pairs $$(A,\mathbf{b})$$ , where $$A$$ is a real $$m\times n$$ matrix and $$\mathbf{b}\in \mathbb{R}^{m}$$ , the system $$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n} 

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5. Optimal orthogonal group synchronization and rotation group synchronization

   https://doi.org/10.1093/imaiai/iaac022  

   Gao, Chao; Zhang, Anderson Y. (August 2022, Information and Inference: A Journal of the IMA)

   Abstract We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model is $$Y_{ij} = Z_i^* Z_j^{*T} + \sigma W_{ij}\in{\mathbb{R}}^{d\times d}$$ where $$W_{ij}$$ is a Gaussian random matrix and $$Z_i^*$$ is either an orthogonal matrix or a rotation matrix, and each $$Y_{ij}$$ is observed independently with probability $$p$$. We analyze an iterative polar decomposition algorithm for the estimation of $Z^*$ and show it has an error of $$(1+o(1))\frac{\sigma ^2 d(d-1)}{2np}$$ when initialized by spectral methods. A matching minimax lower bound is further established that leads to the optimality of the proposed algorithm as it achieves the exact minimax risk. 

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