Unlike distance metric learning where the subsequent tasks utilizing the estimated distance metric is the usual focus, the proposal focuses on the estimated metric characterizing the geometry structure. Despite the illustrated taxi and MNIST examples, it is still open to finding more compelling applications that target the data space geometry. Interpreting mathematical concepts such as Riemannian metric and geodesic in the context of potential application (e.g., cognition and perception research where similarity measures are common) could be inspiring. Our proposal requires sufficiently dense data, which could be demanding, especially for high-dimensional data due to the curse of dimensionality. Dimensional reduction (e.g., manifold embedding as in the MNIST example) can substantially alleviate the curse of dimensionality, and the dense data requirement will more likely hold true.

Despite the flexibility of these methods, their inference is based on the computationally intensive thinning algorithm.

Empirical Bayes (EB) improves the accuracy of simultaneous inference "by learning from the experience of others" (Efron, 2012). Classical EB theory focuses on latent variables that are iid draws from a fitted prior (Efron, 2019). Modern applications, however, feature complex structure, like arrays, spatial processes, or covariates. How can we apply EB ideas to these settings? We propose a generalized approach to empirical Bayes based on the notion of probabilistic symmetry. Our method pairs a simultaneous inference problem-with an unknown prior-to a symmetry assumption on the joint distribution of the latent variables. Each symmetry implies an ergodic decomposition, which we use to derive a corresponding empirical Bayes method. We call this methodBayesian empirical Bayes (BEB). We show how BEB recovers the classical methods of empirical Bayes, which implicitly assume exchangeability. We then use it to extend EB to other probabilistic symmetries: (i) EB matrix recovery for arrays and graphs; (ii) covariate-assisted EB for conditional data; (iii) EB spatial regression under shift invariance. We develop scalable algorithms based on variational inference and neural networks. In simulations, BEB outperforms existing approaches to denoising arrays and spatial data. On real data, we demonstrate BEB by denoising a cancer gene-expression matrix and analyzing spatial air-quality data from New York City.

The non-commutative nature of 3D rotations poses well-known challenges in generalizing planar problems to three-dimensional ones, even more so in contact-rich tasks where haptic information (i.e., forces/torques) is involved. In this sense, not all learning-based algorithms that are currently available generalize to 3D orientation estimation. Non-linear filters defined on $\mathbf{\mathbb{SO}(3)}$ are widely used with inertial measurement sensors; however, none of them have been used with haptic measurements. This paper presents a unique complementary filtering framework that interprets the geometric shape of objects in the form of superquadrics, exploits the symmetry of $\mathbf{\mathbb{SO}(3)}$, and uses force and vision sensors as measurements to provide an estimate of orientation. The framework's robustness and almost global stability are substantiated by a set of experiments on a dual-arm robotic setup.

Recent advances in large language models (LLMs) have transformed open-domain question answering, yet their effectiveness in music-related reasoning remains limited due to sparse music knowledge in pretraining data. While music information retrieval and computational musicology have explored structured and multimodal understanding, few resources support factual and contextual music question answering (MQA) grounded in artist metadata or historical context. We introduce MusWikiDB, a vector database of 3.2M passages from 144K music-related Wikipedia pages, and ArtistMus, a benchmark of 1,000 questions on 500 diverse artists with metadata such as genre, debut year, and topic. These resources enable systematic evaluation of retrieval-augmented generation (RAG) for MQA. Experiments show that RAG markedly improves factual accuracy; open-source models gain up to +56.8 percentage points (for example, Qwen3 8B improves from 35.0 to 91.8), approaching proprietary model performance. RAG-style fine-tuning further boosts both factual recall and contextual reasoning, improving results on both in-domain and out-of-domain benchmarks. MusWikiDB also yields approximately 6 percentage points higher accuracy and 40% faster retrieval than a general-purpose Wikipedia corpus. We release MusWikiDB and ArtistMus to advance research in music information retrieval and domain-specific question answering, establishing a foundation for retrieval-augmented reasoning in culturally rich domains such as music.