[Paper] Self-Supervised Learning from Noisy and Incomplete Data

Source: arXiv - 2601.03244v1

Overview

The paper Self‑Supervised Learning from Noisy and Incomplete Data by Julián Tachella and Mike Davies tackles a classic dilemma in signal processing: how to recover high‑quality signals when the only data you have are corrupted, partial measurements and you lack clean ground‑truth examples for training. By systematically reviewing and extending self‑supervised learning (SSL) techniques for inverse problems, the authors show that you can train powerful reconstruction solvers directly from the measurements themselves—opening the door to data‑driven solutions in domains where labeled data are prohibitively expensive.

Key Contributions

• Unified taxonomy of self‑supervised strategies for inverse problems (e.g., masking, noise‑to‑noise, cycle‑consistency, and equivariant losses).
• Theoretical analysis that clarifies when and why SSL yields unbiased or consistent estimators, linking the methods to classical regularization theory.
• Practical recipe for turning any known forward model (the measurement process) into a self‑supervised training pipeline with minimal engineering effort.
• Extensive empirical validation on several imaging inverse tasks (denoising, compressive sensing MRI, and limited‑angle tomography), demonstrating performance on par with or exceeding supervised baselines.
• Open‑source implementation and benchmark suite that developers can plug into existing deep‑learning frameworks (PyTorch/TensorFlow).

Methodology

1. Problem Setup – The forward model (y = \mathcal{A}(x) + \epsilon) is assumed known (e.g., a blur kernel, sampling mask, or sensor physics). The goal is to learn a reconstruction operator (\mathcal{R}_\theta) that maps noisy/incomplete observations (y) back to an estimate (\hat{x}).
2. Self‑Supervised Losses – Instead of pairing (y) with a clean (x), the authors generate pseudo‑targets from the data itself:
   • Mask‑based loss: Randomly hide a subset of measurements, reconstruct the full signal, then enforce consistency on the hidden part.
   • Noise2Noise‑style loss: Use two independent noisy realizations of the same underlying signal (when available) and train the network to map one to the other.
   • Cycle‑consistency: Apply the forward model to the network’s output and compare it to the original measurement, encouraging (\mathcal{A}(\mathcal{R}_\theta(y)) \approx y).
   • Equivariance regularization: Exploit known symmetries (e.g., rotations, translations) of the measurement process to create additional constraints.
3. Training Pipeline – The forward model (\mathcal{A}) is embedded as a differentiable layer, allowing end‑to‑end back‑propagation. The loss is a weighted sum of the above terms, with hyper‑parameters that can be tuned automatically via validation on a held‑out measurement set.
4. Theoretical Guarantees – By treating the SSL loss as a surrogate for the true risk, the authors prove that under mild assumptions (e.g., unbiased noise, linear forward operator) the learned reconstructor converges to the same solution as a supervised model trained on infinite data.

Results & Findings

Takeaway: Across diverse imaging modalities, the self‑supervised approaches close >80 % of the performance gap relative to fully supervised training, while requiring zero paired ground‑truth data. The experiments also reveal that combining complementary SSL objectives (e.g., masking + cycle‑consistency) yields the most robust reconstructions.

Practical Implications

• Rapid prototyping: Engineers can now train reconstruction networks directly on the raw sensor data they already collect, bypassing costly annotation pipelines.
• Edge deployment: Since the forward model is known, the same SSL pipeline can be executed on‑device (e.g., on a medical scanner or a smartphone camera) to fine‑tune a model for a specific hardware configuration or patient cohort.
• Domain adaptation: When moving a model from one imaging device to another, the SSL loss automatically aligns the reconstructor to the new measurement statistics without re‑collecting ground‑truth phantoms.
• Open‑source toolbox: The provided code integrates with popular DL libraries, exposing a simple API (train_ssl(reconstructor, forward_model, data_loader)) that developers can drop into existing pipelines.
• Regulatory friendliness: Because the method is grounded in a known physical forward model, the resulting reconstructions retain interpretability—a key factor for medical‑device approval.

Limitations & Future Work

• Assumption of a known forward model: The theory and experiments rely on an accurate (\mathcal{A}). In scenarios where the measurement physics are partially unknown or highly nonlinear, performance degrades.
• Noise model dependence: Guarantees hold for unbiased, zero‑mean noise; heavy‑tailed or signal‑dependent noise may require additional robustification.
• Scalability to ultra‑high‑resolution data: Training on gigapixel microscopy images still challenges GPU memory; the authors suggest patch‑wise training but note potential boundary artifacts.
• Future directions proposed include:
  1. Learning the forward model jointly with the reconstructor.
  2. Extending SSL to non‑linear inverse problems (e.g., phase retrieval).
  3. Integrating uncertainty quantification to flag reconstructions that fall outside the self‑supervised training distribution.

Authors

• Julián Tachella
• Mike Davies

Paper Information

• arXiv ID: 2601.03244v1
• Categories: stat.ML, cs.LG, eess.IV
• Published: January 6, 2026
• PDF: Download PDF