4 research outputs found
Online Embedding of Metrics
We study deterministic online embeddings of metric spaces into normed spaces of various dimensions and into trees. We establish some upper and lower bounds on the distortion of such embedding, and pose some challenging open questions
Online embedding of metrics
We study deterministic online embeddings of metrics spaces into normed spaces
and into trees against an adaptive adversary. Main results include a polynomial
lower bound on the (multiplicative) distortion of embedding into Euclidean
spaces, a tight exponential upper bound on embedding into the line, and a
-distortion embedding in of a suitably high
dimension.Comment: 15 pages, no figure
Capacity Bounds for Hyperbolic Neural Network Representations of Latent Tree Structures
We study the representation capacity of deep hyperbolic neural networks
(HNNs) with a ReLU activation function. We establish the first proof that HNNs
can -isometrically embed any finite weighted tree into a
hyperbolic space of dimension at least equal to with prescribed
sectional curvature (where
being optimal). We establish rigorous upper bounds for the network complexity
on an HNN implementing the embedding. We find that the network complexity of
HNN implementing the graph representation is independent of the representation
fidelity/distortion. We contrast this result against our lower bounds on
distortion which any ReLU multi-layer perceptron (MLP) must exert when
embedding a tree with leaves into a -dimensional Euclidean space,
which we show at least ; independently of the depth, width,
and (possibly discontinuous) activation function defining the MLP.Comment: 22 Pages + References, 1 Table, 4 Figure
Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings
Low-distortional metric embeddings are a crucial component in the modern
algorithmic toolkit. In an online metric embedding, points arrive sequentially
and the goal is to embed them into a simple space irrevocably, while minimizing
the distortion. Our first result is a deterministic online embedding of a
general metric into Euclidean space with distortion (or,
if the metric has doubling
dimension ), solving a conjecture by Newman and Rabinovich (2020), and
quadratically improving the dependence on the aspect ratio from Indyk et
al.\ (2010). Our second result is a stochastic embedding of a metric space into
trees with expected distortion , generalizing previous
results (Indyk et al.\ (2010), Bartal et al.\ (2020)).
Next, we study the \emph{online minimum-weight perfect matching} problem,
where a sequence of metric points arrive in pairs, and one has to maintain
a perfect matching at all times. We allow recourse (as otherwise the order of
arrival determines the matching). The goal is to return a perfect matching that
approximates the \emph{minimum-weight} perfect matching at all times, while
minimizing the recourse. Our third result is a randomized algorithm with
competitive ratio and recourse against an
oblivious adversary, this result is obtained via our new stochastic online
embedding. Our fourth result is a deterministic algorithm against an adaptive
adversary, using recourse, that maintains a matching of weight at
most times the weight of the MST, i.e., a matching of lightness
. We complement our upper bounds with a strategy for an oblivious
adversary that, with recourse , establishes a lower bound of
for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on
Discrete Algorithms (SODA24