116 research outputs found
Noisy Subspace Clustering via Thresholding
We consider the problem of clustering noisy high-dimensional data points into
a union of low-dimensional subspaces and a set of outliers. The number of
subspaces, their dimensions, and their orientations are unknown. A
probabilistic performance analysis of the thresholding-based subspace
clustering (TSC) algorithm introduced recently in [1] shows that TSC succeeds
in the noisy case, even when the subspaces intersect. Our results reveal an
explicit tradeoff between the allowed noise level and the affinity of the
subspaces. We furthermore find that the simple outlier detection scheme
introduced in [1] provably succeeds in the noisy case.Comment: Presented at the IEEE Int. Symp. Inf. Theory (ISIT) 2013, Istanbul,
Turkey. The version posted here corrects a minor error in the published
version. Specifically, the exponent -c n_l in the success probability of
Theorem 1 and in the corresponding proof outline has been corrected to
-c(n_l-1
Deep Convolutional Neural Networks Based on Semi-Discrete Frames
Deep convolutional neural networks have led to breakthrough results in
practical feature extraction applications. The mathematical analysis of these
networks was pioneered by Mallat, 2012. Specifically, Mallat considered
so-called scattering networks based on identical semi-discrete wavelet frames
in each network layer, and proved translation-invariance as well as deformation
stability of the resulting feature extractor. The purpose of this paper is to
develop Mallat's theory further by allowing for different and, most
importantly, general semi-discrete frames (such as, e.g., Gabor frames,
wavelets, curvelets, shearlets, ridgelets) in distinct network layers. This
allows to extract wider classes of features than point singularities resolved
by the wavelet transform. Our generalized feature extractor is proven to be
translation-invariant, and we develop deformation stability results for a
larger class of deformations than those considered by Mallat. For Mallat's
wavelet-based feature extractor, we get rid of a number of technical
conditions. The mathematical engine behind our results is continuous frame
theory, which allows us to completely detach the invariance and deformation
stability proofs from the particular algebraic structure of the underlying
frames.Comment: Proc. of IEEE International Symposium on Information Theory (ISIT),
Hong Kong, China, June 2015, to appea
Explicit and almost sure conditions for K/2 degrees of freedom
It is well known that in K-user constant single-antenna interference channels
K/2 degrees of freedom (DoF) can be achieved for almost all channel matrices.
Explicit conditions on the channel matrix to admit K/2 DoF are, however, not
available. The purpose of this paper is to identify such explicit conditions,
which are satisfied for almost all channel matrices. We also provide a
construction of corresponding asymptotically DoF-optimal input distributions.
The main technical tool used is a recent breakthrough result by Hochman in
fractal geometry.Comment: To be presented at IEEE Int. Symp. Inf. Theory 2014, Honolulu, H
High-SNR Capacity of Wireless Communication Channels in the Noncoherent Setting: A Primer
This paper, mostly tutorial in nature, deals with the problem of
characterizing the capacity of fading channels in the high signal-to-noise
ratio (SNR) regime. We focus on the practically relevant noncoherent setting,
where neither transmitter nor receiver know the channel realizations, but both
are aware of the channel law. We present, in an intuitive and accessible form,
two tools, first proposed by Lapidoth & Moser (2003), of fundamental importance
to high-SNR capacity analysis: the duality approach and the escape-to-infinity
property of capacity-achieving distributions. Furthermore, we apply these tools
to refine some of the results that appeared previously in the literature and to
simplify the corresponding proofs.Comment: To appear in Int. J. Electron. Commun. (AE\"U), Aug. 201
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