1,707 research outputs found
Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back
We present an -time algorithm for
the maximum s-t flow and the minimum s-t cut problems in directed graphs with
unit capacities. This is the first improvement over the sparse-graph case of
the long-standing time bound due to Even and
Tarjan [EvenT75]. By well-known reductions, this also establishes an
-time algorithm for the maximum-cardinality bipartite
matching problem. That, in turn, gives an improvement over the celebrated
celebrated time bound of Hopcroft and Karp [HK73] whenever the
input graph is sufficiently sparse
Faster generation of random spanning trees
In this paper, we set forth a new algorithm for generating approximately
uniformly random spanning trees in undirected graphs. We show how to sample
from a distribution that is within a multiplicative of uniform in
expected time \TO(m\sqrt{n}\log 1/\delta). This improves the sparse graph
case of the best previously known worst-case bound of , which has stood for twenty years.
To achieve this goal, we exploit the connection between random walks on
graphs and electrical networks, and we use this to introduce a new approach to
the problem that integrates discrete random walk-based techniques with
continuous linear algebraic methods. We believe that our use of electrical
networks and sparse linear system solvers in conjunction with random walks and
combinatorial partitioning techniques is a useful paradigm that will find
further applications in algorithmic graph theory
Fast Generation of Random Spanning Trees and the Effective Resistance Metric
We present a new algorithm for generating a uniformly random spanning tree in
an undirected graph. Our algorithm samples such a tree in expected
time. This improves over the best previously known bound
of -- that follows from the work of
Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] --
whenever the input graph is sufficiently sparse.
At a high level, our result stems from carefully exploiting the interplay of
random spanning trees, random walks, and the notion of effective resistance, as
well as from devising a way to algorithmically relate these concepts to the
combinatorial structure of the graph. This involves, in particular,
establishing a new connection between the effective resistance metric and the
cut structure of the underlying graph
Spectral Signatures in Backdoor Attacks
A recent line of work has uncovered a new form of data poisoning: so-called
\emph{backdoor} attacks. These attacks are particularly dangerous because they
do not affect a network's behavior on typical, benign data. Rather, the network
only deviates from its expected output when triggered by a perturbation planted
by an adversary.
In this paper, we identify a new property of all known backdoor attacks,
which we call \emph{spectral signatures}. This property allows us to utilize
tools from robust statistics to thwart the attacks. We demonstrate the efficacy
of these signatures in detecting and removing poisoned examples on real image
sets and state of the art neural network architectures. We believe that
understanding spectral signatures is a crucial first step towards designing ML
systems secure against such backdoor attacksComment: 16 pages, accepted to NIPS 201
Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods
In this paper, we study matrix scaling and balancing, which are fundamental
problems in scientific computing, with a long line of work on them that dates
back to the 1960s. We provide algorithms for both these problems that, ignoring
logarithmic factors involving the dimension of the input matrix and the size of
its entries, both run in time where is the amount of error we are willing to
tolerate. Here, represents the ratio between the largest and the
smallest entries of the optimal scalings. This implies that our algorithms run
in nearly-linear time whenever is quasi-polynomial, which includes, in
particular, the case of strictly positive matrices. We complement our results
by providing a separate algorithm that uses an interior-point method and runs
in time .
In order to establish these results, we develop a new second-order
optimization framework that enables us to treat both problems in a unified and
principled manner. This framework identifies a certain generalization of linear
system solving that we can use to efficiently minimize a broad class of
functions, which we call second-order robust. We then show that in the context
of the specific functions capturing matrix scaling and balancing, we can
leverage and generalize the work on Laplacian system solving to make the
algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201
- …