# mSQUID: Model-Based Leanred Modulo Recovery at Low Sampling Rates

arXiv: https://arxiv.org/abs/2510.18729 · HTML: https://mlsanity.com/paper/2510.18729/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 50 references that are in the map.
- First posted on arXiv 21 Oct 2025.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [Deep Unfolded Robust PCA with Application to Clutter Suppression in Ultrasound](https://mlsanity.com/paper/1811.08252/) (2018)
- [Deep learning in ultrasound imaging](https://mlsanity.com/paper/1907.02994/) (2019)
- [WINNet: Wavelet-inspired Invertible Network for Image Denoising](https://mlsanity.com/paper/2109.06381/) (2021)
- [On Unlimited Sampling and Reconstruction](https://mlsanity.com/paper/1905.03901/) (2019)
- [Unlimited Sampling from Theory to Practice: Fourier-Prony Recovery and Prototype ADC](https://mlsanity.com/paper/2105.05818/) (2021)
- [Deep Task-Based Quantization](https://mlsanity.com/paper/1908.06845/) (2019)
- [Deep Unrolled Recovery in Sparse Biological Imaging](https://mlsanity.com/paper/2109.14025/) (2021)
- [Above the Nyquist Rate, Modulo Folding Does Not Hurt](https://mlsanity.com/paper/1903.12289/) (2019)
- [Residual Recovery Algorithm For Modulo Sampling](https://mlsanity.com/paper/2110.03335/) (2021)
- [Modulo Sampling of FRI Signals](https://mlsanity.com/paper/2207.08774/) (2022)
- [Line Spectral Estimation via Unlimited Sampling](https://mlsanity.com/paper/2210.15811/) (2022)
- [Optimization Guarantees of Unfolded ISTA and ADMM Networks With Smooth Soft-Thresholding](https://mlsanity.com/paper/2309.06195/) (2023)
- [Unlimited Dynamic Range Analog-to-Digital Conversion](https://mlsanity.com/paper/1911.09371/) (2019)
- [Modulo Sampling in Shift-Invariant Spaces: Recovery and Stability Enhancement](https://mlsanity.com/paper/2406.10929/) (2024)
- [Compressed Sensing Based Residual Recovery Algorithms and Hardware for Modulo Sampling](https://mlsanity.com/paper/2412.12724/) (2024)
- [Practical Modulo Sampling: Mitigating High-Frequency Components](https://mlsanity.com/paper/2501.11330/) (2025)
- [Unlimited Sampling Theorem Based on Fractional Fourier Transform](https://mlsanity.com/paper/2110.13003/) (2021)
- [Modulo Sampling: Performance Guarantees in The Presence of Quantization](https://mlsanity.com/paper/2501.01506/) (2025)

## Built on by (most cited first, 1 of 1)

- [Lattice Modulo Sampling](https://mlsanity.com/paper/2605.24974/) (2026)

## Abstract (arXiv)

Modulo sampling enables acquisition of signals with unlimited dynamic range by folding the input into a bounded interval prior to sampling, thus eliminating the risk of signal clipping and preserving information without requiring highresolution ADCs. While this enables low-cost hardware, the nonlinear distortion introduced by folding presents recovery challenges, particularly under noise and quantization. We propose a model-based deep unfolding network tailored to this setting, combining the interpretability of classical compress sensing (CS) solvers with the flexibility of learning. A key innovation is a soft-quantization module that encodes the modulo prior by guiding the solution toward discrete multiples of the folding range in a differentiable and learnable way. Our method, modulo soft-quantized unfolded iterative decoder (mSQUID), achieves superior reconstruction performance at low sampling rates under additive Gaussian noise. We further demonstrate its utility in a challenging case where signals with vastly different amplitudes and disjoint frequency bands are acquired simultaneously and quantized. In this scenario, classical sampling often struggles due to weak signal distortion or strong signal clipping, while our approach is able to recover the input signals. Our method also offers significantly reduced runtimes, making it suitable for real-time, resource-limited systems.