# Vector-Valued Least-Squares Regression under Output Regularity Assumptions

arXiv: https://arxiv.org/abs/2211.08958 · HTML: https://mlsanity.com/paper/2211.08958/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 46 references that are in the map.
- First posted on arXiv 16 Nov 2022.

## Builds on (most cited first, 12 of 12)

- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Less is More: Nystr"om Computational Regularization](https://mlsanity.com/paper/1507.04717/) (2015)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Structured Prediction Energy Networks](https://mlsanity.com/paper/1511.06350/) (2015)
- [Last iterate convergence of SGD for Least-Squares in the Interpolation regime](https://mlsanity.com/paper/2102.03183/) (2021)
- [A General Framework for Consistent Structured Prediction with Implicit Loss Embeddings](https://mlsanity.com/paper/2002.05424/) (2020)
- [A Consistent Regularization Approach for Structured Prediction](https://mlsanity.com/paper/1605.07588/) (2016)
- [A Structured Prediction Approach for Label Ranking](https://mlsanity.com/paper/1807.02374/) (2018)
- [Deep Value Networks Learn to Evaluate and Iteratively Refine Structured Outputs](https://mlsanity.com/paper/1703.04363/) (2017)
- [On the Sample Complexity of Subspace Learning](https://mlsanity.com/paper/1408.5032/) (2014)
- [Leveraging Low-Rank Relations Between Surrogate Tasks in Structured Prediction](https://mlsanity.com/paper/1903.00667/) (2019)
- [Nonparametric Reduced Rank Regression](https://mlsanity.com/paper/1301.1919/) (2013)

## Built on by (most cited first, 5 of 5)

- [Sketch In, Sketch Out: Accelerating both Learning and Inference for Structured Prediction with Kernels](https://mlsanity.com/paper/2302.10128/) (2023)
- [A randomized algorithm to solve reduced rank operator regression](https://mlsanity.com/paper/2312.17348/) (2023)
- [Recursive Estimation of Conditional Kernel Mean Embeddings](https://mlsanity.com/paper/2302.05955/) (2023)
- [Scalable variable selection for two-view learning tasks with projection operators](https://mlsanity.com/paper/2307.01558/) (2023)
- [A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learning](https://mlsanity.com/paper/2509.11070/) (2025)

## Abstract (arXiv)

We propose and analyse a reduced-rank method for solving least-squares regression problems with infinite dimensional output. We derive learning bounds for our method, and study under which setting statistical performance is improved in comparison to full-rank method. Our analysis extends the interest of reduced-rank regression beyond the standard low-rank setting to more general output regularity assumptions. We illustrate our theoretical insights on synthetic least-squares problems. Then, we propose a surrogate structured prediction method derived from this reduced-rank method. We assess its benefits on three different problems: image reconstruction, multi-label classification, and metabolite identification.