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Abstract Optical spectrometers are essential tools for analysing light‒matter interactions, but conventional spectrometers can be complicated and bulky. Recently, efforts have been made to develop miniaturized spectrometers. However, it is challenging to overcome the trade-off between miniaturizing size and retaining performance. Here, we present a complementary metal oxide semiconductor image sensor-based miniature computational spectrometer using a plasmonic nanoparticles-in-cavity microfilter array. Size-controlled silver nanoparticles are directly printed into cavity-length-varying Fabry‒Pérot microcavities, which leverage strong coupling between the localized surface plasmon resonance of the silver nanoparticles and the Fabry‒Pérot microcavity to regulate the transmission spectra and realize large-scale arrayed spectrum-disparate microfilters. Supported by a machine learning-based training process, the miniature computational spectrometer uses artificial intelligence and was demonstrated to measure visible-light spectra at subnanometre resolution. The high scalability of the technological approaches shown here may facilitate the development of high-performance miniature optical spectrometers for extensive applications. 

Replica exchange stochastic gradient Langevin dynamics (reSGLD) is an effective sampler for non-convex learning in large-scale datasets. However, the simulation may encounter stagnation issues when the high-temperature chain delves too deeply into the distribution tails. To tackle this issue, we propose reflected reSGLD (r2SGLD): an algorithm tailored for constrained non-convex exploration by utilizing reflection steps within a bounded domain. Theoretically, we observe that reducing the diameter of the domain enhances mixing rates, exhibiting a quadratic behavior. Empirically, we test its performance through extensive experiments, including identifying dynamical systems with physical constraints, simulations of constrained multi-modal distributions, and image classification tasks. The theoretical and empirical findings highlight the crucial role of constrained exploration in improving the simulation efficiency.