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}
}
  • M. IwenF. Krahmer J. Maly
  • Published 17 July 2018
  • Computer Science, Mathematics
  • Discrete & Computational Geometry
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… 
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12 Citations
Background Citations
7
Methods Citations
2

12 Citations

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

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Robust One-bit Compressed Sensing With Manifold Data

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

New theoretical analysis of a tractable recovery algorithm for one-bit compressed sensing which is based on minimizing averages of Rectified Linear Units forming a convex relaxation of the Hamming distance is provided.

Quantized Compressed Sensing by Rectified Linear Units

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Modern Problems in Mathematical Signal Processing: Quantized Compressed Sensing and Randomized Neural Networks

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.

One-Bit Sigma-Delta modulation on the circle

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

On Fast Johnson–Lindenstrauss Embeddings of Compact Submanifolds of $$\mathbbm {R}^N$$ with Boundary

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.

The Nonconvex Geometry of Linear Inverse Problems

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

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

A (corrected) rigorous proof that the Igelnik and Pao construction is a universal approximator for continuous functions on compact domains, with approximation error decaying asymptotically like $O(1/\sqrt{n})$ for the number of network nodes, is provided.

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Robust One-bit Compressed Sensing With Manifold Data

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