5,809 research outputs found
Composition of nested embeddings with an application to outlier removal
We study the design of embeddings into Euclidean space with outliers. Given a
metric space and an integer , the goal is to embed all but
points in (called the "outliers") into with the smallest possible
distortion . Finding the optimal distortion for a given outlier set size
, or alternately the smallest for a given target distortion are both
NP-hard problems. In fact, it is UGC-hard to approximate to within a factor
smaller than even when the metric sans outliers is isometrically embeddable
into . We consider bi-criteria approximations. Our main result is a
polynomial time algorithm that approximates the outlier set size to within an
factor and the distortion to within a constant factor.
The main technical component in our result is an approach for constructing a
composition of two given embeddings from subsets of into which
inherits the distortions of each to within small multiplicative factors.
Specifically, given a low distortion embedding from into
and a high(er) distortion embedding from the entire set into
, we construct a single embedding that achieves the same distortion
over pairs of points in and an expansion of at most over the remaining pairs of points, where . Our
composition theorem extends to embeddings into arbitrary metrics for
, and may be of independent interest. While unions of embeddings over
disjoint sets have been studied previously, to our knowledge, this is the first
work to consider compositions of nested embeddings.Comment: 25 pages (including 2 appendices), 5 figure
Computing Bi-Lipschitz Outlier Embeddings into the Line
The problem of computing a bi-Lipschitz embedding of a graphical metric into
the line with minimum distortion has received a lot of attention. The
best-known approximation algorithm computes an embedding with distortion
, where denotes the optimal distortion [B\u{a}doiu \etal~2005]. We
present a bi-criteria approximation algorithm that extends the above results to
the setting of \emph{outliers}.
Specifically, we say that a metric space admits a
-embedding if there exists , with , such that
admits an embedding into the line with distortion at
most . Given , and a metric space that admits a -embedding,
for some , our algorithm computes a -embedding in polynomial time. This is the first algorithmic result for
outlier bi-Lipschitz embeddings. Prior to our work, comparable outlier
embeddings where known only for the case of additive distortion
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