# Structured Nonparametric Variational Inference for Dependent Latent Modeling

arXiv: https://arxiv.org/abs/2606.15458 · HTML: https://mlsanity.com/paper/2606.15458/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 61 references that are in the map.
- First posted on arXiv 13 Jun 2026.

## Builds on (most cited first, 30 of 31)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [From Louvain to Leiden: guaranteeing well-connected communities](https://mlsanity.com/paper/1810.08473/) (2018)
- [Stochastic Latent Actor-Critic: Deep Reinforcement Learning with a Latent Variable Model](https://mlsanity.com/paper/1907.00953/) (2019)
- [FlowEdit: Inversion-Free Text-Based Editing Using Pre-Trained Flow Models](https://mlsanity.com/paper/2412.08629/) (2024)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference](https://mlsanity.com/paper/1703.09194/) (2017)
- [Concrete Autoencoders for Differentiable Feature Selection and Reconstruction](https://mlsanity.com/paper/1901.09346/) (2019)
- [Tighter Variational Bounds are Not Necessarily Better](https://mlsanity.com/paper/1802.04537/) (2018)
- [ExSpliNet: An interpretable and expressive spline-based neural network](https://mlsanity.com/paper/2205.01510/) (2022)
- [Joint Prompt Optimization of Stacked LLMs using Variational Inference](https://mlsanity.com/paper/2306.12509/) (2023)
- [Structured Stochastic Variational Inference](https://mlsanity.com/paper/1404.4114/) (2014)
- [Automatic Differentiation Variational Inference with Mixtures](https://mlsanity.com/paper/2003.01687/) (2020)
- [SHOPPER: A Probabilistic Model of Consumer Choice with Substitutes and Complements](https://mlsanity.com/paper/1711.03560/) (2017)
- [Variational Gaussian Copula Inference](https://mlsanity.com/paper/1506.05860/) (2015)
- [Scalable Higher-Order Tensor Product Spline Models](https://mlsanity.com/paper/2402.01090/) (2024)
- [Nonparametric Automatic Differentiation Variational Inference with Spline Approximation](https://mlsanity.com/paper/2403.06302/) (2024)
- [Bayesian Additive Adaptive Basis Tensor Product Models for Modeling High Dimensional Surfaces: An application to high-throughput toxicity testing](https://mlsanity.com/paper/1702.04775/) (2017)
- [Multivariate Spline Estimation and Inference for Image-On-Scalar Regression](https://mlsanity.com/paper/2106.01431/) (2021)

## Abstract (arXiv)

Variational inference (VI) is a core engine of modern AI, enabling scalable approximate Bayesian learning and uncertainty-aware training of large probabilistic and generative models. In this paper, we propose Structured Nonparametric Variational Inference (SN-VI), a novel framework for modeling complex dependencies among latent variables in posterior approximation, leveraging multivariate spline techniques. Unlike traditional methods that rely on the mean-field assumption, SN-VI preserves intricate latent variable dependencies, providing a flexible and accurate approximation of posteriors with arbitrary shapes. We establish rigorous theoretical guarantees, including the derivation of the lower bound for the variational objective and proof of asymptotic consistency in posterior estimation. To facilitate practical implementation, we develop an algorithm that automatically identifies dependent latent variables and their underlying dependence structure, without requiring manual specification. Simulation studies validate the effectiveness of SN-VI in approximating posterior distributions with bounded support and complex dependencies. The proposed method has been successfully applied to high-dimensional structured data, including computer vision datasets and spatial transcriptomics. In these applications, SN-VI demonstrates improved generative model performance and effectively uncovers coupled biological signals through the learned dependency structure.