This work presents a data-driven approach for the indoor localization of an observer on a 2D topological map of the environment. State-of-the-art techniques may yield accurate estimates only when they are tailor-made for a specific data modality like camera-based system that prevents their applicability to broader domains. Here, we establish a modality-agnostic framework (called OT-Isomap) and formulate the localization problem in the context of parametric manifold learning while leveraging optimal transportation. This framework allows jointly learning a lowdimensional embedding as well as correspondences with a topological map. We examine the generalizability of the proposed algorithm by applying it to data from diverse modalities such as image sequences and radio frequency signals. The experimental results demonstrate decimeter-level accuracy for localization using different sensory inputs.

t-SNE has gained popularity as a dimension reduction technique, especially for visualizing data. It is well-known that all dimension reduction techniques may lose important features of the data. We provide a mathematical framework for understanding this loss for t-SNE by establishing a number of results in different scenarios showing how important features of data are lost by using t-SNE.

Nevertheless, it remains elusive that whether the Transformer-based architecture can do molecular modeling as good as equivariant GNNs.

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Fitting T1-mGPLVM to the binned spike data, we found that the inferred latent state was highly correlated with the true head direction (Figure 5b). Here we make this connection more explicit. As described in the main text, the Lie algebrag of a groupG is a vector space tangent toG at its identity element. However,because the Lie algebra is isomorphic toRn, we have found it convenient in both our exposition and our implementation to work directly with the pair(Rn,ExpG), instead of(g,expG). We begin by noting thatSn is not a Lie group unlessn = 1 or n = 3, thus we can only apply the ReLie framework toS1 and S3.