We present NeuSE, a novel Neural SE(3)-Equivariant Embedding for objects, and illustrate how it supports object-based Simultaneous Localization and Mapping (SLAM) for consistent spatial understanding with long-term scene changes. NeuSE is a set of latent object embeddings created from partial object observations. It serves as a compact point cloud surrogate for complete object models, encoding the full shape, scale, and transform information about an object. In addition, the inferred latent code is both SE(3) and scale equivariant, enabling strong generalization to objects of both unseen sizes and different SE(3) poses. This makes NeuSE particularly effective in real-world scenarios where objects may vary in size or spatial configuration. With NeuSE, relative frame transforms can be directly derived from inferred latent codes. Our proposed SLAM paradigm, using NeuSE for object shape, size, and pose characterization, can operate independently or in conjunction with typical SLAM systems. It directly infers SE(3) camera pose constraints that are compatible with general SLAM pose graph optimization, while maintaining a lightweight, object-centric map that adapts to real-world changes. Our evaluation is conducted on synthetic and real-world sequences with changes in both controlled and uncontrolled settings, featuring multi-category objects of various shapes and sizes. Our approach demonstrates improved localization capability and change-aware mapping consistency when working either independently or as a complement to common SLAM pipelines. Additional results are presented in the supplementary video: https://youtu.be/fZ4_2P1fQBQ

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