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 complemented by numerical experiments confirming the validity of our approach.
Many relevant applications of signal processing rely on the separation of sources from a mixture of signals without a prior knowledge about the mixing process. Given a mixture of signals f = i fi, the task of signal separation is to estimate the components fi by using specific assumptions on their timefrequency behaviour or statistical characteristics. Time-frequency data is often very high-dimensional which affects the performance of signal separation methods quite significantly. Therefore, the embedding dimension of the time-frequency representation of f should be reduced prior to the application of a decomposition strategy, such as independent component analysis (ICA) or nonnegative matrix factorization (NNMF). In other words, a suitable dimensionality reduction method should be applied, before the data is decomposed and then back-projected. But the choice of the dimensionality reduction method requires particular care, especially in combination with ICA and NNMF, since nonnegative input data are required. In this paper, we introduce a generic concept for the construction of suitable non-negative dimensionality reduction methods. Furthermore, we discuss the two different decomposition strategies NNMF and ICA for single channel signal separation in combination with non-negative principal component analysis (NNPCA), where our main interest is in acoustic signals with transitory components.
We study one-bit compressed sensing for signals on a low-dimensional manifold. We introduce two computationally efficient reconstruction algorithms that only require access to a geometric multi-resolution analysis approximation of the manifold. We derive rigorous reconstruction guarantees for these methods in the scenario that the measurements are subgaussian and show that they are robust with respect to both pre-and post-quantization noise. Our results substantially improve upon earlier work in this direction.
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