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1. Learning neural contracting dynamics: Extended linearization and global guarantees

   Jaffe, Sean; Davydov, Alexander; Lapsekili, Deniz; Singh, Ambuj K; Bullo, Francesco (December 2024, 38th Conference on Neural Information Processing Systems (NeurIPS 2024))

   Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty. We present Extended Linearized Contracting Dynamics (ELCD), the first neural network-based dynamical system with global contractivity guarantees in arbitrary metrics. The key feature of ELCD is a parametrization of the extended linearization of the nonlinear vector field. In its most basic form, ELCD is guaranteed to be (i) globally exponentially stable,(ii) equilibrium contracting, and (iii) globally contracting with respect to some metric. To allow for contraction with respect to more general metrics in the data space, we train diffeomorphisms between the data space and a latent space and enforce contractivity in the latent space, which ensures global contractivity in the data space. We demonstrate the performance of ELCD on the high dimensional LASA, multi-link pendulum, and Rosenbrock datasets. 

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2. Maximally mixing active nematics

   https://doi.org/10.1103/PhysRevE.109.014606  

   Mitchell, Kevin A.; Sabbir, Md Mainul; Geumhan, Kevin; Smith, Spencer A.; Klein, Brandon; Beller, Daniel A. (January 2024, Physical Review E)

   Active nematics are an important new paradigm in soft condensed matter systems. They consist of rodlike components with an internal driving force pushing them out of equilibrium. The resulting fluid motion exhibits chaotic advection, in which a small patch of fluid is stretched exponentially in length. Using simulation, this paper shows that this system can exhibit stable periodic motion when confined to a sufficiently small square with periodic boundary conditions. Moreover, employing tools from braid theory, we show that this motion is maximally mixing, in that it optimizes the (dimensionless) “topological entropy”—the exponential stretching rate of a material line advected by the fluid. That is, this periodic motion of the defects, counterintuitively, produces more chaotic mixing than chaotic motion of the defects. We also explore the stability of the periodic state. Importantly, we show how to stabilize this orbit into a larger periodic tiling, a critical necessity for it to be seen in future experiments. 

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3. Quantum control and noise protection of a Floquet $0\text{−}\pi$ qubit

   https://doi.org/10.1103/PhysRevA.109.042607  

   Wang, Zhaoyou; Safavi-Naeini, Amir H. (April 2024, Physical Review A)

   Time-periodic systems allow engineering new effective Hamiltonians from limited physical interactions. For example, the inverted position of the Kapitza pendulum emerges as a stable equilibrium with rapid drive of its pivot point. In this work we propose the Kapitzonium: a Floquet qubit that is the superconducting circuit analog of a mechanical Kapitza pendulum. Under periodic driving, the bit- and phase-flip rates of the emerging qubit states are exponentially suppressed with respect to the ratio of the effective Josephson energy to charging energy. However, we find that dissipation causes leakage out of the Floquet qubit subspace. We engineer a passive cooling scheme to stabilize the qubit subspace, which is crucial for high-fidelity quantum control under dissipation. Furthermore, we introduce a hardware-efficient fluorescence-based method for qubit measurement and discuss the experimental implementation of the Floquet qubit. Our work provides the fundamental steps to develop more complex Floquet quantum systems from the ground up to realize large-scale protected engineered dynamics. 

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4. Building Rapport through Dynamic Models of Acoustic-Prosodic Entrainment

   https://doi.org/10.1145/3027063.3027132  

   Lubold, Nichola (January 2017, Proceedings of the 2017 CHI Conference Extended Abstracts on Human Factors in Computing Systems)

   As dialogue systems become more prevalent in the form of personalized assistants, there is an increasingly important role for systems which can socially engage the user by influencing social factors like rapport. For example, learning companions enhance learning through socio-motivational support and are more successful when users feel rapport. In this work, I explore social engagement in dialogue systems in terms of acoustic-prosodic entrainment; entrainment is a phenomenon where over the course of a conversation, speakers adapt their acoustic-prosodic features, becoming more similar in their pitch, intensity, or speaking rate. Correlated with rapport and task success, entrainment plays a significant role in how individuals connect; a system which can entrain has potential to improve social engagement by enhancing these factors. As a result of this work, I introduce a dialogue system which can entrain and investigate its effects on social factors like rapport. 

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5. A Converse to Banach's Fixed Point Theorem and its CLS Completeness

   Daskalakis, Constantinos; Tzamos, Christos; Zampetakis, Manolis (June 2018, Proceedings of the annual ACM Symposium on Theory of Computing)

   Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for establishing global convergence of iterative methods to unique fixed points, and for bounding their convergence rate. In other words, we show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and which can be used to bound the number of iterations until convergence. We illustrate our approach in the widely used power method, providing a new way of bounding its convergence rate through contraction arguments. We next consider the computational complexity of Banach's fixed point theorem. Making the proof of our converse theorem constructive, we show that computing a fixed point whose existence is guaranteed by Banach's fixed point theorem is CLS-complete. We thus provide the first natural complete problem for the class CLS, which was defined in [DP11] to capture the complexity of problems such as P-matrix LCP, computing KKT-points, and finding mixed Nash equilibria in congestion and network coordination games. 

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