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1. Bi-Lipschitz embeddings of quasiconformal trees

   https://doi.org/10.1090/proc/16252  

   David, Guy; Eriksson-Bique, Sylvester; Vellis, Vyron (May 2023, Proceedings of the American Mathematical Society)

   A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 of David and Vellis [Illinois J. Math. 66 (2022), pp. 189–244]. 

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2. Quasiconformal model with CNN features for large deformation image registration

   https://doi.org/10.3934/ipi.2022010  

   Law, Ho; Choi, Gary P.; Lam, Ka Chun; Lui, Lok Ming (January 2022, Inverse Problems and Imaging)

   Image registration has been widely studied over the past several decades, with numerous applications in science, engineering and medicine. Most of the conventional mathematical models for large deformation image registration rely on prescribed landmarks, which usually require tedious manual labeling. In recent years, there has been a surge of interest in the use of machine learning for image registration. In this paper, we develop a novel method for large deformation image registration by a fusion of quasiconformal theory and convolutional neural network (CNN). More specifically, we propose a quasiconformal energy model with a novel fidelity term that incorporates the features extracted using a pre-trained CNN, thereby allowing us to obtain meaningful registration results without any guidance of prescribed landmarks. Moreover, unlike many prior image registration methods, the bijectivity of our method is guaranteed by quasiconformal theory. Experimental results are presented to demonstrate the effectiveness of the proposed method. More broadly, our work sheds light on how rigorous mathematical theories and practical machine learning approaches can be integrated for developing computational methods with improved performance. 

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3. Uniformly branching trees

   https://doi.org/10.1090/tran/8404  

   Bonk, Mario; Meyer, Daniel (April 2022, Transactions of the American Mathematical Society)

   A quasiconformal tree T is a (compact) metric tree that is doubling and of bounded turning. We call T trivalent if every branch point of T has exactly three branches. If the set of branch points is uniformly relatively separated and uniformly relatively dense, we say that T is uniformly branching. We prove that a metric space T is quasisymmetrically equivalent to the continuum self-similar tree if and only if it is a trivalent quasiconformal tree that is uniformly branching. In particular, any two trees of this type are quasisymmetrically equivalent. 

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4. Spatial Metaphor and the Development of Cross-Domain Mappings in Early Childhood

   Starr, Ariel; Srinivasan, Mahesh (April 2018, Developmental psychology)

   Spatial language is often used metaphorically to describe other domains, including time (long sound) and pitch (high sound). How does experience with these metaphors shape the ability to associate space with other domains? Here, we tested 3- to 6-year-old English-speaking children and adults with a cross-domain matching task. We probed cross-domain relations that are expressed in English metaphors for time and pitch (length-time and height-pitch), as well as relations that are unconventional in English but expressed in other languages (size-time and thickness-pitch). Participants were tested with a perceptual matching task, in which they matched between spatial stimuli and sounds of different durations or pitches, and a linguistic matching task, in which they matched between a label denoting a spatial attribute, duration, or pitch, and a picture or sound representing another dimension. Contrary to previous claims that experience with linguistic metaphors is necessary for children to make cross-domain mappings, children performed above chance for both familiar and unfamiliar relations in both tasks, as did adults. Children’s performance was also better when a label was provided for one of the dimensions, but only when making length-time, size-time, and height-pitch mappings (not thickness-pitch mappings). These findings suggest that, although experience with metaphorical language is not necessary to make cross-domain mappings, labels can promote these mappings, both when they have familiar metaphorical uses (e.g., English ‘long’ denotes both length and duration), and when they describe dimensions that share a common ordinal reference frame (e.g., size and duration, but not thickness and pitch). 

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5. Hybrid Control for Learning Motor Skills

   Ian Abraham, Alexander Broad (June 2020, Workshop on the Algorithmic Foundations of Robotics (WAFR))

   We develop a hybrid control approach for robot learning based on combining learned predictive models with experience-based state-action policy mappings to improve the learning capabilities of robotic systems. Predictive models provide an understanding of the task and the physics (which improves sample-efficiency), while experience-based pol-icy mappings are treated as “muscle memory” that encode favorable actions as experiences that override planned actions. Hybrid control tools are used to create an algorithmic approach for combining learned predictive models with experience-based learning. Hybrid learning is presented as a method for efficiently learning motor skills by systematically combining and improving the performance of predictive models and experience-based policies. A deterministic variation of hybrid learning is derived and extended into a stochastic implementation that relaxes some of the key assumptions in the original derivation. Each variation is tested on experience-based learning methods (where the robot interacts with the environment to gain experience) as well as imitation learning methods(where experience is provided through demonstrations and tested in the environment). The results show that our method is capable of improving the performance and sample-efficiency of learning motor skills in a variety of experimental domains. 

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