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1. Uniqueness theorems for meromorphic inner functions and canonical systems

   https://doi.org/10.4153/S0008439524000614  

   Hatinoğlu, Burak (March 2025, Canadian Mathematical Bulletin)

   Abstract We consider uniqueness problems for meromorphic inner functions on the upper half-plane. In these problems, we consider spectral data depending partially or fully on the spectrum, derivative values at the spectrum, Clark measure, or the spectrum of the negative of a meromorphic inner function. Moreover, we consider applications of these uniqueness results to inverse spectral theory of canonical Hamiltonian systems and obtain generalizations of the Borg-Levinson two-spectra theorem for canonical Hamiltonian systems and unique determination of a Hamiltonian from its spectral measure under some conditions. 

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2. Influence Diagnostics under Self-concordance

   Fisher,  Jillian; Liu,  Lang; Pillutla,  Krishna; Choi,  Yejin; Harchaoui, Zaid (April 2023, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics)

   Ruiz, Francisco; Dy, Jennifer; van de Meent, Jan-Willem (Ed.)

   Influence diagnostics such as influence functions and approximate maximum influence perturbations are popular in machine learning and in AI domain applications. Influence diagnostics are powerful statistical tools to identify influential datapoints or subsets of datapoints. We establish finite-sample statistical bounds, as well as computational complexity bounds, for influence functions and approximate maximum influence perturbations using efficient inverse-Hessian-vector product implementations. We illustrate our results with generalized linear models and large attention based models on synthetic and real data. 

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3. Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems

   https://doi.org/10.1007/s11071-023-09135-w  

   Colbrook, Matthew_J; Li, Qin; Raut, Ryan_V; Townsend, Alex (December 2023, Nonlinear Dynamics)

   Abstract Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and dynamic mode decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models. 

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4. Adaptive photonic RF spectral shaper

   https://doi.org/10.1364/OE.398833  

   Liu, Qidi; Fok, Mable_P (August 2020, Optics Express)

   The radio frequency spectral shaper is an essential component in emerging multi-service mobile communications, multiband satellite and radar systems, and future 5G/6G radio frequency systems for equalizing spectral unevenness, removing out-of-band noise and interference, and manipulating multi-band signal simultaneously. While it is easy to achieve simple spectral functions using either conventional microwave photonic filters or the optical spectrum to microwave spectra mapping techniques, it is challenging to enable complex spectral shaping functions over tens of GHz bandwidth as well as to achieve point-by-point shaping capability to fulfill the needs in dynamic wireless communications. In this paper, we proposed and demonstrated a novel spectral shaping system, which utilizes a two-section algorithm to automatically decompose the target RF response into a series of Gaussian functions and to reconstruct the desired RF response by microwave photonic techniques. The devised spectral shaping system is capable of manipulating the spectral function in various bands (S, C, and X) simultaneously with step resolution of as fine as tens of MHz. The resolution limitation in optical spectral processing is mitigated using the discrete convolution technique. Over 10 dynamic and independently adjustable spectral control points are experimentally achieved based on the proposed spectral shaper. 

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5. Efficient approximation of deep ReLU networks for functions on low dimensional manifolds

   Chen, Minshuo; Jiang, Haomin; Liao, Wenjing; Zhao, Tuo (December 2019, Conference on Neural Information Processing Systems)

   Deep neural networks have revolutionized many real world applications, due to their flexibility in data fitting and accurate predictions for unseen data. A line of research reveals that neural networks can approximate certain classes of functions with an arbitrary accuracy, while the size of the network scales exponentially with respect to the data dimension. Empirical results, however, suggest that networks of moderate size already yield appealing performance. To explain such a gap, a common belief is that many data sets exhibit low dimensional structures, and can be modeled as samples near a low dimensional manifold. In this paper, we prove that neural networks can efficiently approximate functions supported on low dimensional manifolds. The network size scales exponentially in the approximation error, with an exponent depending on the intrinsic dimension of the data and the smoothness of the function. Our result shows that exploiting low dimensional data structures can greatly enhance the efficiency in function approximation by neural networks. We also implement a sub-network that assigns input data to their corresponding local neighborhoods, which may be of independent interest. 

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