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1. Metric Dimension

   https://doi.org/10.4249/scholarpedia.53881  

   Tillquist, Richard; Frongillo, Rafael; Lladser, Manuel (January 2019, Scholarpedia)
2. Nonparametric Statistical Inference via Metric Distribution Function in Metric Spaces

   https://doi.org/10.1080/01621459.2023.2277417  

   Wang, Xueqin; Zhu, Jin; Pan, Wenliang; Zhu, Junhao; Zhang, Heping (December 2023, Journal of the American Statistical Association)

   The distribution function is essential in statistical inference and connected with samples to form a directed closed loop by the correspondence theorem in measure theory and the Glivenko-Cantelli and Donsker properties. This connection creates a paradigm for statistical inference. However, existing distribution functions are defined in Euclidean spaces and are no longer convenient to use in rapidly evolving data objects of complex nature. It is imperative to develop the concept of the distribution function in a more general space to meet emerging needs. Note that the linearity allows us to use hypercubes to define the distribution function in a Euclidean space. Still, without the linearity in a metric space, we must work with the metric to investigate the probability measure. We introduce a class of metric distribution functions through the metric only. We overcome this challenging step by proving the correspondence theorem and the Glivenko-Cantelli theorem for metric distribution functions in metric spaces, laying the foundation for conducting rational statistical inference for metric space-valued data. Then, we develop a homogeneity test and a mutual independence test for non-Euclidean random objects and present comprehensive empirical evidence to support the performance of our proposed methods. Supplementary materials for this article are available online. 

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3. On the Consistency of Metric and Non-Metric K-medoids

   Jiang, H.; Arias-Castro, E. (January 2021, Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS) 2021)

   Banerjee, A.; Fukumizu, K. (Ed.)

   We establish the consistency of K-medoids in the context of metric spaces. We start by proving that K-medoids is asymptotically equivalent to K-means restricted to the support of the underlying distribution under general conditions, including a wide selection of loss functions. This asymptotic equivalence, in turn, enables us to apply the work of Pärna (1986) on the consistency of K-means. This general approach applies also to non-metric settings where only an ordering of the dissimilarities is available. We consider two types of ordinal information: one where all quadruple comparisons are available; and one where only triple comparisons are available. We provide some numerical experiments to illustrate our theory. 

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4. Metric Program Synthesis

   John Feser; Isil Dillig; Armando Solar-Lezama (January 2022, POPL 2022)

   We present a new domain-agnostic synthesis technique for generating programs from input-output examples. Our method, called metric program synthesis, relaxes the well-known observational equivalence idea (used widely in bottom-up enumerative synthesis) into a weaker notion of observational similarity, with the goal of reducing the search space that the synthesizer needs to explore. Our method clusters programs into equivalence classes based on a distance metric and constructs a version space that compactly represents ""approximately correct"" programs. Then, given a ""close enough"" program sampled from this version space, our approach uses a distance-guided repair algorithm to find a program that exactly matches the given input-output examples. We have implemented our proposed metric program synthesis technique in a tool called SyMetric and evaluate it in three different domains considered in prior work. Our evaluation shows that SyMetric outperforms other domain-agnostic synthesizers that use observational equivalence and that it achieves results competitive with domain-specific synthesizers that are either designed for or trained on those domains. 

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5. Generic Quantum Metric Rigidity

   https://doi.org/10.1093/imrn/rnaa028  

   Chirvasitu, Alexandru (March 2020, International Mathematics Research Notices)

   Abstract We introduce the coherent algebra of a compact metric measure space by analogy with the corresponding concept for a finite graph. As an application we show that upon topologizing the collection of isomorphism classes of compact metric measure spaces appropriately, the subset consisting of those with trivial compact quantum automorphism group is of 2nd Baire category. The latter result can be paraphrased as saying that “most” compact metric measure spaces have no (quantum) symmetries; in particular, they also have trivial ordinary (i.e., classical) automorphism group. 

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