134,190 research outputs found
Top Rank Optimization in Linear Time
Bipartite ranking aims to learn a real-valued ranking function that orders
positive instances before negative instances. Recent efforts of bipartite
ranking are focused on optimizing ranking accuracy at the top of the ranked
list. Most existing approaches are either to optimize task specific metrics or
to extend the ranking loss by emphasizing more on the error associated with the
top ranked instances, leading to a high computational cost that is super-linear
in the number of training instances. We propose a highly efficient approach,
titled TopPush, for optimizing accuracy at the top that has computational
complexity linear in the number of training instances. We present a novel
analysis that bounds the generalization error for the top ranked instances for
the proposed approach. Empirical study shows that the proposed approach is
highly competitive to the state-of-the-art approaches and is 10-100 times
faster
Linear Convergence of a Frank-Wolfe Type Algorithm over Trace-Norm Balls
We propose a rank- variant of the classical Frank-Wolfe algorithm to solve
convex optimization over a trace-norm ball. Our algorithm replaces the top
singular-vector computation (-SVD) in Frank-Wolfe with a top-
singular-vector computation (-SVD), which can be done by repeatedly applying
-SVD times. Alternatively, our algorithm can be viewed as a rank-
restricted version of projected gradient descent. We show that our algorithm
has a linear convergence rate when the objective function is smooth and
strongly convex, and the optimal solution has rank at most . This improves
the convergence rate and the total time complexity of the Frank-Wolfe method
and its variants.Comment: In NIPS 201
Faster Eigenvector Computation via Shift-and-Invert Preconditioning
We give faster algorithms and improved sample complexities for estimating the
top eigenvector of a matrix -- i.e. computing a unit vector such
that :
Offline Eigenvector Estimation: Given an explicit with , we show how to compute an approximate top
eigenvector in time and . Here is the number of nonzeros in ,
is the stable rank, is the relative eigengap. By separating the
dependence from the term, our first runtime improves upon the
classical power and Lanczos methods. It also improves prior work using fast
subspace embeddings [AC09, CW13] and stochastic optimization [Sha15c], giving
significantly better dependencies on and . Our second running
time improves these further when .
Online Eigenvector Estimation: Given a distribution with covariance
matrix and a vector which is an approximate top
eigenvector for , we show how to refine to an approximation
using samples from . Here is a
natural notion of variance. Combining our algorithm with previous work to
initialize , we obtain improved sample complexity and runtime results
under a variety of assumptions on .
We achieve our results using a general framework that we believe is of
independent interest. We give a robust analysis of the classic method of
shift-and-invert preconditioning to reduce eigenvector computation to
approximately solving a sequence of linear systems. We then apply fast
stochastic variance reduced gradient (SVRG) based system solvers to achieve our
claims.Comment: Appearing in ICML 2016. Combination of work in arXiv:1509.05647 and
arXiv:1510.0889
A Spectral Learning Approach to Range-Only SLAM
We present a novel spectral learning algorithm for simultaneous localization
and mapping (SLAM) from range data with known correspondences. This algorithm
is an instance of a general spectral system identification framework, from
which it inherits several desirable properties, including statistical
consistency and no local optima. Compared with popular batch optimization or
multiple-hypothesis tracking (MHT) methods for range-only SLAM, our spectral
approach offers guaranteed low computational requirements and good tracking
performance. Compared with popular extended Kalman filter (EKF) or extended
information filter (EIF) approaches, and many MHT ones, our approach does not
need to linearize a transition or measurement model; such linearizations can
cause severe errors in EKFs and EIFs, and to a lesser extent MHT, particularly
for the highly non-Gaussian posteriors encountered in range-only SLAM. We
provide a theoretical analysis of our method, including finite-sample error
bounds. Finally, we demonstrate on a real-world robotic SLAM problem that our
algorithm is not only theoretically justified, but works well in practice: in a
comparison of multiple methods, the lowest errors come from a combination of
our algorithm with batch optimization, but our method alone produces nearly as
good a result at far lower computational cost
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