# Rate-optimal neural boundary detection from unlabeled noisy images

arXiv: https://arxiv.org/abs/2606.00715 · HTML: https://mlsanity.com/paper/2606.00715/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 41 references that are in the map.
- First posted on arXiv 30 May 2026.

## Builds on (most cited first, 20 of 20)

- [Holistically-Nested Edge Detection](https://mlsanity.com/paper/1504.06375/) (2015)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Cross-Entropy Loss Functions: Theoretical Analysis and Applications](https://mlsanity.com/paper/2304.07288/) (2023)
- [Cut and Learn for Unsupervised Object Detection and Instance Segmentation](https://mlsanity.com/paper/2301.11320/) (2023)
- [Richer Convolutional Features for Edge Detection](https://mlsanity.com/paper/1612.02103/) (2016)
- [Fast convergence rates of deep neural networks for classification](https://mlsanity.com/paper/1812.03599/) (2018)
- [Factor Augmented Sparse Throughput Deep ReLU Neural Networks for High Dimensional Regression](https://mlsanity.com/paper/2210.02002/) (2022)
- [Nonconvex sparse regularization for deep neural networks and its optimality](https://mlsanity.com/paper/2003.11769/) (2020)
- [Convergence rates of deep ReLU networks for multiclass classification](https://mlsanity.com/paper/2108.00969/) (2021)
- [Advantage of Deep Neural Networks for Estimating Functions with Singularity on Hypersurfaces](https://mlsanity.com/paper/2011.02256/) (2020)
- [Adaptive variational Bayes: Optimality, computation and applications](https://mlsanity.com/paper/2109.03204/) (2021)
- [Classification with Deep Neural Networks and Logistic Loss](https://mlsanity.com/paper/2307.16792/) (2023)
- [Adaptive deep learning for nonlinear time series models](https://mlsanity.com/paper/2207.02546/) (2022)
- [Masked Bayesian Neural Networks : Theoretical Guarantee and its Posterior Inference](https://mlsanity.com/paper/2305.14765/) (2023)
- [On Detection of Faint Edges in Noisy Images](https://mlsanity.com/paper/1706.07717/) (2017)
- [Intrinsic and extrinsic deep learning on manifolds](https://mlsanity.com/paper/2302.08606/) (2023)
- [CT-Bound: Robust Boundary Detection From Noisy Images Via Hybrid Convolution and Transformer Neural Networks](https://mlsanity.com/paper/2403.16494/) (2024)
- [Bayesian Detection of Image Boundaries](https://mlsanity.com/paper/1508.05847/) (2015)
- [Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image](https://mlsanity.com/paper/1606.08400/) (2016)
- [Optimal Convergence Rates of Deep Neural Network Classifiers](https://mlsanity.com/paper/2506.14899/) (2025)

## Abstract (arXiv)

We study boundary detection for unlabeled noisy images from a statistical perspective. The aim is to recover an unknown object region from raw intensity observations without pixel-wise annotating labels or a parametric model for the intensity distributions. Motivated by robust Gibbs posterior approaches based on thresholded misclassification losses, we propose a continuous hinge-type surrogate loss for boundary detection. The proposed loss is amenable to gradient-based optimization and can be combined with deep neural networks to represent complex object boundaries. We prove that the proposed loss function is Fisher consistent under a mild separation assumption and obtain a calibration inequality linking excess surrogate risk to the symmetric difference error of the estimated region. Under a piecewise smooth boundary model, we prove that the resulting deep neural network estimator achieves the minimax-optimal boundary recovery rate, up to logarithmic factors. The piecewise smooth formulation accommodates boundaries with corners and kinks, thereby extending beyond globally smooth boundary models. Numerical experiments demonstrate that the proposed method accurately and stably recovers object boundaries across a range of noise levels and shape configurations, and compares favorably with existing unsupervised boundary detection methods.