# Slack More, Predict Better: Proximal Relaxation for Probabilistic Latent Variable Model-based Soft Sensors

arXiv: https://arxiv.org/abs/2603.11473 · HTML: https://mlsanity.com/paper/2603.11473/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 62 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
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- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Large-Scale Wasserstein Gradient Flows](https://mlsanity.com/paper/2106.00736/) (2021)
- [Variational Wasserstein gradient flow](https://mlsanity.com/paper/2112.02424/) (2021)
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- [Particle-based Variational Inference with Preconditioned Functional Gradient Flow](https://mlsanity.com/paper/2211.13954/) (2022)
- [Amortized Variational Inference: A Systematic Review](https://mlsanity.com/paper/2209.10888/) (2022)
- [Proximal Diffusion Neural Sampler](https://mlsanity.com/paper/2510.03824/) (2025)
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- [Entire Space Counterfactual Learning for Reliable Content Recommendations](https://mlsanity.com/paper/2210.11039/) (2022)
- [DistDF: Time-Series Forecasting Needs Joint-Distribution Wasserstein Alignment](https://mlsanity.com/paper/2510.24574/) (2025)
- [Gradient Flow Algorithms for Density Propagation in Stochastic Systems](https://mlsanity.com/paper/1908.00533/) (2019)
- [Domain-Guided Conditional Diffusion Model for Unsupervised Domain Adaptation](https://mlsanity.com/paper/2309.14360/) (2023)
- [GAD-PVI: A General Accelerated Dynamic-Weight Particle-Based Variational Inference Framework](https://mlsanity.com/paper/2312.16429/) (2023)
- [Debiased Collaborative Filtering with Kernel-Based Causal Balancing](https://mlsanity.com/paper/2404.19596/) (2024)
- [Latent Variable Models in the Era of Industrial Big Data: Extension and Beyond](https://mlsanity.com/paper/2208.10847/) (2022)

## Abstract (arXiv)

Nonlinear Probabilistic Latent Variable Models (NPLVMs) are a cornerstone of soft sensor modeling due to their capacity for uncertainty delineation. However, conventional NPLVMs are trained using amortized variational inference, where neural networks parameterize the variational posterior. While facilitating model implementation, this parameterization converts the distributional optimization problem within an infinite-dimensional function space to parameter optimization within a finite-dimensional parameter space, which introduces an approximation error gap, thereby degrading soft sensor modeling accuracy. To alleviate this issue, we introduce KProxNPLVM, a novel NPLVM that pivots to relaxing the objective itself and improving the NPLVM's performance. Specifically, we first prove the approximation error induced by the conventional approach. Based on this, we design the Wasserstein distance as the proximal operator to relax the learning objective, yielding a new variational inference strategy derived from solving this relaxed optimization problem. Based on this foundation, we provide a rigorous derivation of KProxNPLVM's optimization implementation, prove the convergence of our algorithm can finally sidestep the approximation error, and propose the KProxNPLVM by summarizing the abovementioned content. Finally, extensive experiments on synthetic and real-world industrial datasets are conducted to demonstrate the efficacy of the proposed KProxNPLVM.