We study the problem of demixing a pair of sparse signals from nonlinear observations of their superposition. Mathematically, we consider a nonlinear signal observation model, yi = g(a T i x) + ei, i = 1, . . . , m, where x = Φw + Ψz denotes the superposition signal, Φ and Ψ are orthonormal bases in R n , and w, z ∈ R n are sparse coefficient vectors of the constituent signals. Further, we assume that the observations are corrupted by a subgaussian additive noise. Within this model, g represents a nonlinear link function, and ai ∈ R n is the i-th row of the measurement matrix, A ∈ R m×n . Problems of this nature arise in several applications ranging from astronomy, computer vision, and machine learning.In this paper, we make some concrete algorithmic progress for the above demixing problem. Specifically, we consider two scenarios: (i) the case when the demixing procedure has no knowledge of the link function, and (ii) the case when the demixing algorithm has perfect knowledge of the link function. In both cases, we provide fast algorithms for recovery of the constituents w and z from the observations. Moreover, we support these algorithms with a rigorous theoretical analysis, and derive (nearly) tight upper bounds on the sample complexity of the proposed algorithms for achieving stable recovery of the component signals. Our analysis also shows that the running time of our algorithms is essentially as good as the best possible.We also provide a range of numerical simulations to illustrate the performance of the proposed algorithms on both real and synthetic signals and images. Our simulations show the superior performance of our algorithms compared to existing methods for demixing signals and images based on convex optimization. In particular, our proposed methods yield demonstrably better sample complexities as well as improved running times, thereby enabling their applicability to large-scale problems.

Random sinusoidal features are a popular approach for speeding up kernel-based inference in large datasets. Prior to the inference stage, the approach suggests performing dimensionality reduction by first multiplying each data vector by a random Gaussian matrix, and then computing an element-wise sinusoid. Theoretical analysis shows that collecting a sufficient number of such features can be reliably used for subsequent inference in kernel classification and regression.In this work, we demonstrate that with a mild increase in the dimension of the embedding, it is also possible to reconstruct the data vector from such random sinusoidal features, provided that the underlying data is sparse enough. In particular, we propose a numerically stable algorithm for reconstructing the data vector given the nonlinear features, and analyze its sample complexity. Our algorithm can be extended to other types of structured inverse problems, such as demixing a pair of sparse (but incoherent) vectors. We support the efficacy of our approach via numerical experiments.

Catalyzed by the recent emergence of site-specific, high-fidelity radio frequency (RF) modeling and simulation tools purposed for radar, data-driven formulations of classical methods in radar have rapidly grown in popularity over the past decade. Despite this surge, limited focus has been directed toward the theoretical foundations of these classical methods. In this regard, as part of our ongoing datadriven approach to radar space-time adaptive processing (STAP), we analyze the asymptotic performance guarantees of select subspace separation methods in the context of radar target localization, and augment this analysis through a proposed deep learning framework for target location estimation. In our approach, we generate comprehensive datasets by randomly placing targets of variable strengths in predetermined constrained areas using RFView ® , a site-specific RF modeling and simulation tool developed by ISL Inc.For each radar return signal from these constrained areas, we generate heatmap tensors in range, azimuth, and elevation of the normalized adaptive matched filter (NAMF) test statistic, and of the output power of a generalized sidelobe canceller (GSC). Using our deep learning framework, we estimate target locations from these heatmap tensors to demonstrate the feasibility of and significant improvements provided by our data-driven approach in matched and mismatched settings.

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