111 research outputs found
Stable Recovery Of Sparse Vectors From Random Sinusoidal Feature Maps
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
Sampling and Recovery of Pulse Streams
Compressive Sensing (CS) is a new technique for the efficient acquisition of
signals, images, and other data that have a sparse representation in some
basis, frame, or dictionary. By sparse we mean that the N-dimensional basis
representation has just K<<N significant coefficients; in this case, the CS
theory maintains that just M = K log N random linear signal measurements will
both preserve all of the signal information and enable robust signal
reconstruction in polynomial time. In this paper, we extend the CS theory to
pulse stream data, which correspond to S-sparse signals/images that are
convolved with an unknown F-sparse pulse shape. Ignoring their convolutional
structure, a pulse stream signal is K=SF sparse. Such signals figure
prominently in a number of applications, from neuroscience to astronomy. Our
specific contributions are threefold. First, we propose a pulse stream signal
model and show that it is equivalent to an infinite union of subspaces. Second,
we derive a lower bound on the number of measurements M required to preserve
the essential information present in pulse streams. The bound is linear in the
total number of degrees of freedom S + F, which is significantly smaller than
the naive bound based on the total signal sparsity K=SF. Third, we develop an
efficient signal recovery algorithm that infers both the shape of the impulse
response as well as the locations and amplitudes of the pulses. The algorithm
alternatively estimates the pulse locations and the pulse shape in a manner
reminiscent of classical deconvolution algorithms. Numerical experiments on
synthetic and real data demonstrate the advantages of our approach over
standard CS
Reconstruction from Periodic Nonlinearities, With Applications to HDR Imaging
We consider the problem of reconstructing signals and images from periodic
nonlinearities. For such problems, we design a measurement scheme that supports
efficient reconstruction; moreover, our method can be adapted to extend to
compressive sensing-based signal and image acquisition systems. Our techniques
can be potentially useful for reducing the measurement complexity of high
dynamic range (HDR) imaging systems, with little loss in reconstruction
quality. Several numerical experiments on real data demonstrate the
effectiveness of our approach
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