3,923 research outputs found
A quasi-Newton proximal splitting method
A new result in convex analysis on the calculation of proximity operators in
certain scaled norms is derived. We describe efficient implementations of the
proximity calculation for a useful class of functions; the implementations
exploit the piece-wise linear nature of the dual problem. The second part of
the paper applies the previous result to acceleration of convex minimization
problems, and leads to an elegant quasi-Newton method. The optimization method
compares favorably against state-of-the-art alternatives. The algorithm has
extensive applications including signal processing, sparse recovery and machine
learning and classification
Preconditioned Data Sparsification for Big Data with Applications to PCA and K-means
We analyze a compression scheme for large data sets that randomly keeps a
small percentage of the components of each data sample. The benefit is that the
output is a sparse matrix and therefore subsequent processing, such as PCA or
K-means, is significantly faster, especially in a distributed-data setting.
Furthermore, the sampling is single-pass and applicable to streaming data. The
sampling mechanism is a variant of previous methods proposed in the literature
combined with a randomized preconditioning to smooth the data. We provide
guarantees for PCA in terms of the covariance matrix, and guarantees for
K-means in terms of the error in the center estimators at a given step. We
present numerical evidence to show both that our bounds are nearly tight and
that our algorithms provide a real benefit when applied to standard test data
sets, as well as providing certain benefits over related sampling approaches.Comment: 28 pages, 10 figure
Randomized Low-Memory Singular Value Projection
Affine rank minimization algorithms typically rely on calculating the
gradient of a data error followed by a singular value decomposition at every
iteration. Because these two steps are expensive, heuristic approximations are
often used to reduce computational burden. To this end, we propose a recovery
scheme that merges the two steps with randomized approximations, and as a
result, operates on space proportional to the degrees of freedom in the
problem. We theoretically establish the estimation guarantees of the algorithm
as a function of approximation tolerance. While the theoretical approximation
requirements are overly pessimistic, we demonstrate that in practice the
algorithm performs well on the quantum tomography recovery problem.Comment: 13 pages. This version has a revised theorem and new numerical
experiment
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