# On Recovery Guarantees for One-Bit Compressed Sensing on Manifolds

@article{Iwen2018OnRG, title={On Recovery Guarantees for One-Bit Compressed Sensing on Manifolds}, author={Mark A. Iwen and Felix Krahmer and Sara Krause-Solberg and Johannes Maly}, journal={Discrete \& Computational Geometry}, year={2018}, volume={65}, pages={953 - 998} }

This paper studies the problem of recovering a signal from one-bit compressed sensing measurements under a manifold model; that is, assuming that the signal lies on or near a manifold of low intrinsic dimension. We provide a convex recovery method based on the Geometric Multi-Resolution Analysis and prove recovery guarantees with a near-optimal scaling in the intrinsic manifold dimension. Our method is the first tractable algorithm with such guarantees for this setting. The results are…

## 12 Citations

### New Algorithms and Improved Guarantees for One-Bit Compressed Sensing on Manifolds

- Computer Science, Mathematics2019 13th International conference on Sampling Theory and Applications (SampTA)
- 2019

An upper bound on the recovery error is proved which outperforms prior works that use memoryless scalar quantization, requires a simpler analysis, and extends the class of measurements beyond Gaussians.

### Robust One-bit Compressed Sensing With Manifold Data

- Computer Science, Mathematics2019 13th International conference on Sampling Theory and Applications (SampTA)
- 2019

Two computationally efficient reconstruction algorithms that only require access to a geometric multi-resolution analysis approximation of the manifold are introduced that are robust with respect to both pre- and post-quantization noise.

### One-Bit Compressed Sensing by Convex Relaxation of the Hamming Distance

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This work proposes to estimate the signals via a convex program based on rectified linear units (ReLUs) for two different quantization schemes, namely one-bit and uniform multi-bit quantization, and shows that the program is robust against adversarial bit corruptions as well as additive noise on the linear measurements.

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A rigorous proof is provided that the Igelnik and Pao construction is a universal approximator for continuous functions on compact domains, with \(\varepsilon\)-error convergence rate inversely proportional to the number of network nodes; this result is extended to the non-asymptotic setting using a concentration inequality for Monte-Carlo integral approximations.

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A new class of highly structured distributions on matrices which outperform prior structured matrix distributions for embedding suﬁciently low-dimensional submanifolds of R N with respect to both achievable embedding dimension, and computationally eﬃcient realizations is presented.

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The gaugep function is a simple generalization of the gauge function that can tightly control the sparsity of a statistical model within the learning alphabet and, perhaps surprisingly, draws further inspiration from the Burer-Monteiro factorization in computational mathematics.

### Higher order 1-bit Sigma-Delta modulation on a circle

- Computer Science2019 13th International conference on Sampling Theory and Applications (SampTA)
- 2019

The results show how to design an update for the second and third order Sigma-Delta schemes based on the reconstruction error analysis such that for the updated scheme the reconstructionerror is improved.

### Random Vector Functional Link Networks for Function Approximation on Manifolds

- Computer Science, MathematicsArXiv
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An upper bound on the recovery error is proved which outperforms prior works that use memoryless scalar quantization, requires a simpler analysis, and extends the class of measurements beyond Gaussians.

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Two computationally efficient reconstruction algorithms that only require access to a geometric multi-resolution analysis approximation of the manifold are introduced that are robust with respect to both pre- and post-quantization noise.

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