330 research outputs found
Relative Errors for Deterministic Low-Rank Matrix Approximations
We consider processing an n x d matrix A in a stream with row-wise updates
according to a recent algorithm called Frequent Directions (Liberty, KDD 2013).
This algorithm maintains an l x d matrix Q deterministically, processing each
row in O(d l^2) time; the processing time can be decreased to O(d l) with a
slight modification in the algorithm and a constant increase in space. We show
that if one sets l = k+ k/eps and returns Q_k, a k x d matrix that is the best
rank k approximation to Q, then we achieve the following properties: ||A -
A_k||_F^2 <= ||A||_F^2 - ||Q_k||_F^2 <= (1+eps) ||A - A_k||_F^2 and where
pi_{Q_k}(A) is the projection of A onto the rowspace of Q_k then ||A -
pi_{Q_k}(A)||_F^2 <= (1+eps) ||A - A_k||_F^2.
We also show that Frequent Directions cannot be adapted to a sparse version
in an obvious way that retains the l original rows of the matrix, as opposed to
a linear combination or sketch of the rows.Comment: 16 pages, 0 figure
Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data
Robust estimators, like the median of a point set, are important for data
analysis in the presence of outliers. We study robust estimators for
locationally uncertain points with discrete distributions. That is, each point
in a data set has a discrete probability distribution describing its location.
The probabilistic nature of uncertain data makes it challenging to compute such
estimators, since the true value of the estimator is now described by a
distribution rather than a single point. We show how to construct and estimate
the distribution of the median of a point set. Building the approximate support
of the distribution takes near-linear time, and assigning probability to that
support takes quadratic time. We also develop a general approximation technique
for distributions of robust estimators with respect to ranges with bounded VC
dimension. This includes the geometric median for high dimensions and the
Siegel estimator for linear regression.Comment: Full version of a paper to appear at SoCG 201
Geometric Inference on Kernel Density Estimates
We show that geometric inference of a point cloud can be calculated by
examining its kernel density estimate with a Gaussian kernel. This allows one
to consider kernel density estimates, which are robust to spatial noise,
subsampling, and approximate computation in comparison to raw point sets. This
is achieved by examining the sublevel sets of the kernel distance, which
isomorphically map to superlevel sets of the kernel density estimate. We prove
new properties about the kernel distance, demonstrating stability results and
allowing it to inherit reconstruction results from recent advances in
distance-based topological reconstruction. Moreover, we provide an algorithm to
estimate its topology using weighted Vietoris-Rips complexes.Comment: To appear in SoCG 2015. 36 pages, 5 figure
Improved Practical Matrix Sketching with Guarantees
Matrices have become essential data representations for many large-scale
problems in data analytics, and hence matrix sketching is a critical task.
Although much research has focused on improving the error/size tradeoff under
various sketching paradigms, the many forms of error bounds make these
approaches hard to compare in theory and in practice. This paper attempts to
categorize and compare most known methods under row-wise streaming updates with
provable guarantees, and then to tweak some of these methods to gain practical
improvements while retaining guarantees.
For instance, we observe that a simple heuristic iSVD, with no guarantees,
tends to outperform all known approaches in terms of size/error trade-off. We
modify the best performing method with guarantees FrequentDirections under the
size/error trade-off to match the performance of iSVD and retain its
guarantees. We also demonstrate some adversarial datasets where iSVD performs
quite poorly. In comparing techniques in the time/error trade-off, techniques
based on hashing or sampling tend to perform better. In this setting we modify
the most studied sampling regime to retain error guarantee but obtain dramatic
improvements in the time/error trade-off.
Finally, we provide easy replication of our studies on APT, a new testbed
which makes available not only code and datasets, but also a computing platform
with fixed environmental settings.Comment: 27 page
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