The iterations of many sparse estimation algorithms are comprised of a fixed
linear filter cascaded with a thresholding nonlinearity, which collectively
resemble a typical neural network layer. Consequently, a lengthy sequence of
algorithm iterations can be viewed as a deep network with shared, hand-crafted
layer weights. It is therefore quite natural to examine the degree to which a
learned network model might act as a viable surrogate for traditional sparse
estimation in domains where ample training data is available. While the
possibility of a reduced computational budget is readily apparent when a
ceiling is imposed on the number of layers, our work primarily focuses on
estimation accuracy. In particular, it is well-known that when a signal
dictionary has coherent columns, as quantified by a large RIP constant, then
most tractable iterative algorithms are unable to find maximally sparse
representations. In contrast, we demonstrate both theoretically and empirically
the potential for a trained deep network to recover minimal $\ell_0$-norm
representations in regimes where existing methods fail. The resulting system is
deployed on a practical photometric stereo estimation problem, where the goal
is to remove sparse outliers that can disrupt the estimation of surface normals
from a 3D scene.

论文研究了稀疏估计算法、神经网络层和深度网络学习在估计准确性方面的应用，展示了深度网络学习可以在稀疏字典具有连贯列时恢复最小 L0 范数表示的潜力，并应用于三维场景表面法线的实际测量问题中。

深度网络中的最大稀疏性？

Maximal Sparsity with Deep Networks?

We consider the problem of estimating a deterministic sparse vector x from
underdetermined measurements Ax+w, where w represents white Gaussian noise and
A is a given deterministic dictionary. We analyze the performance of three
sparse estimation algorithms: basis pursuit denoising (BPDN), orthogonal
matching pursuit (OMP), and thresholding. These algorithms are shown to achieve
near-oracle performance with high probability, assuming that x is sufficiently
sparse. Our results are non-asymptotic and are based only on the coherence of
A, so that they are applicable to arbitrary dictionaries. Differences in the
precise conditions required for the performance guarantees of each algorithm
are manifested in the observed performance at high and low signal-to-noise
ratios. This provides insight on the advantages and drawbacks of convex
relaxation techniques such as BPDN as opposed to greedy approaches such as OMP
and thresholding.

研究在给定的确定性字典下，从未确定测量中估计固定稀疏向量的性能，提出基于几种算法的稀疏估计技术，包括基 Pursuit Denoising，正交匹配追踪和阈值，这些算法都能以高概率接近 Oracle 性能， 基于 A 的连贯性的研究结果是非渐进的，可以应用于任意词典，并在高低信噪比下表现出差异，提供了对 BPDN 等凸松弛技术和贪婪方法 OMP 和阈值的优缺点的洞察。