6,208 research outputs found
Approximate kernel clustering
In the kernel clustering problem we are given a large positive
semi-definite matrix with and a small
positive semi-definite matrix . The goal is to find a
partition of which maximizes the quantity We study the
computational complexity of this generic clustering problem which originates in
the theory of machine learning. We design a constant factor polynomial time
approximation algorithm for this problem, answering a question posed by Song,
Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp
approximation threshold for this problem assuming the Unique Games Conjecture
(UGC). In particular, when is the identity matrix the UGC
hardness threshold of this problem is exactly . We present
and study a geometric conjecture of independent interest which we show would
imply that the UGC threshold when is the identity matrix is
for every
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Beamforming and Rate Allocation in MISO Cognitive Radio Networks
We consider decentralized multi-antenna cognitive radio networks where
secondary (cognitive) users are granted simultaneous spectrum access along with
license-holding (primary) users. We treat the problem of distributed
beamforming and rate allocation for the secondary users such that the minimum
weighted secondary rate is maximized. Such an optimization is subject to (1) a
limited weighted sum-power budget for the secondary users and (2) guaranteed
protection for the primary users in the sense that the interference level
imposed on each primary receiver does not exceed a specified level. Based on
the decoding method deployed by the secondary receivers, we consider three
scenarios for solving this problem. In the first scenario each secondary
receiver decodes only its designated transmitter while suppressing the rest as
Gaussian interferers (single-user decoding). In the second case each secondary
receiver employs the maximum likelihood decoder (MLD) to jointly decode all
secondary transmissions, and in the third one each secondary receiver uses the
unconstrained group decoder (UGD). By deploying the UGD, each secondary user is
allowed to decode any arbitrary subset of users (which contains its designated
user) after suppressing or canceling the remaining users.Comment: 32 pages, 6 figure
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