# Analyzing Upper Bounds on Mean Absolute Errors for Deep Neural Network Based Vector-to-Vector Regression

arXiv: https://arxiv.org/abs/2008.05459 · HTML: https://mlsanity.com/paper/2008.05459/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2020.

## Built on by (most cited first, 10 of 10)

- [Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression](https://mlsanity.com/paper/2206.04804/) (2022)
- [Diffusion Model is a Good Pose Estimator from 3D RF-Vision](https://mlsanity.com/paper/2403.16198/) (2024)
- [Classical-to-Quantum Transfer Learning for Spoken Command Recognition Based on Quantum Neural Networks](https://mlsanity.com/paper/2110.08689/) (2021)
- [Optimizing Quantum Federated Learning Based on Federated Quantum Natural Gradient Descent](https://mlsanity.com/paper/2303.08116/) (2023)
- [Exploiting Low-Rank Tensor-Train Deep Neural Networks Based on Riemannian Gradient Descent With Illustrations of Speech Processing](https://mlsanity.com/paper/2203.06031/) (2022)
- [Exploiting Hybrid Models of Tensor-Train Networks for Spoken Command Recognition](https://mlsanity.com/paper/2201.10609/) (2022)
- [Pre-training Tensor-Train Networks Facilitates Machine Learning with Variational Quantum Circuits](https://mlsanity.com/paper/2306.03741/) (2023)
- [Leveraging Pre-Trained Neural Networks to Enhance Machine Learning with Variational Quantum Circuits](https://mlsanity.com/paper/2411.08552/) (2024)
- [VQC-MLPNet: An Unconventional Hybrid Quantum-Classical Architecture for Scalable and Robust Quantum Machine Learning](https://mlsanity.com/paper/2506.10275/) (2025)
- [Quantum LEGO Learning: A Modular Design Principle for Hybrid Artificial Intelligence](https://mlsanity.com/paper/2601.21780/) (2026)

## Abstract (arXiv)

In this paper, we show that, in vector-to-vector regression utilizing deep neural networks (DNNs), a generalized loss of mean absolute error (MAE) between the predicted and expected feature vectors is upper bounded by the sum of an approximation error, an estimation error, and an optimization error. Leveraging upon error decomposition techniques in statistical learning theory and non-convex optimization theory, we derive upper bounds for each of the three aforementioned errors and impose necessary constraints on DNN models. Moreover, we assess our theoretical results through a set of image de-noising and speech enhancement experiments. Our proposed upper bounds of MAE for DNN based vector-to-vector regression are corroborated by the experimental results and the upper bounds are valid with and without the "over-parametrization" technique.