# Non-Linear Spectral Dimensionality Reduction Under Uncertainty

arXiv: https://arxiv.org/abs/2202.04678 · HTML: https://mlsanity.com/paper/2202.04678/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 61 references that are in the map.
- First posted on arXiv 9 Feb 2022.

## Builds on (most cited first, 14 of 14)

- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Domain Generalization via Invariant Feature Representation](https://mlsanity.com/paper/1301.2115/) (2013)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Robust Graph Learning from Noisy Data](https://mlsanity.com/paper/1812.06673/) (2018)
- [Kernel Mean Shrinkage Estimators](https://mlsanity.com/paper/1405.5505/) (2014)
- [Adaptive Metric Dimensionality Reduction](https://mlsanity.com/paper/1302.2752/) (2013)
- [One-Class Support Measure Machines for Group Anomaly Detection](https://mlsanity.com/paper/1303.0309/) (2013)
- [Learning Sparse Feature Representations using Probabilistic Quadtrees and Deep Belief Nets](https://mlsanity.com/paper/1509.03413/) (2015)
- [Bayesian Deconditional Kernel Mean Embeddings](https://mlsanity.com/paper/1906.00199/) (2019)
- [Hyperparameter Learning for Conditional Kernel Mean Embeddings with Rademacher Complexity Bounds](https://mlsanity.com/paper/1809.00175/) (2018)
- [Speed-up and multi-view extensions to Subclass Discriminant Analysis](https://mlsanity.com/paper/1905.00794/) (2019)
- [Robust PCA: Optimization of the Robust Reconstruction Error over the Stiefel Manifold](https://mlsanity.com/paper/1506.00323/) (2015)
- [Graph Embedding with Data Uncertainty](https://mlsanity.com/paper/2009.00505/) (2020)
- [Incremental Fast Subclass Discriminant Analysis](https://mlsanity.com/paper/2002.04348/) (2020)

## Abstract (arXiv)

In this paper, we consider the problem of non-linear dimensionality reduction under uncertainty, both from a theoretical and algorithmic perspectives. Since real-world data usually contain measurements with uncertainties and artifacts, the input space in the proposed framework consists of probability distributions to model the uncertainties associated with each sample. We propose a new dimensionality reduction framework, called NGEU, which leverages uncertainty information and directly extends several traditional approaches, e.g., KPCA, MDA/KMFA, to receive as inputs the probability distributions instead of the original data. We show that the proposed NGEU formulation exhibits a global closed-form solution, and we analyze, based on the Rademacher complexity, how the underlying uncertainties theoretically affect the generalization ability of the framework. Empirical results on different datasets show the effectiveness of the proposed framework.