1,140 research outputs found
Error-Tolerant Non-Adaptive Learning of a Hidden Hypergraph
We consider the problem of learning the hypergraph using edge-detecting queries. In this model, the learner is allowed to query whether a set of vertices includes an edge from a hidden hypergraph. Except a few, all previous algorithms assume that a query\u27s result is always correct. In this paper we study the problem of learning a hypergraph where alpha -fraction of the queries are incorrect. The main contribution of this paper is generalizing the well-known structure CFF (Cover Free Family) to be Dense (we will call it DCFF - Dense Cover Free Family) while presenting three different constructions for DCFF. Later, we use these constructions wisely to give a polynomial time non-adaptive learning algorithm for a hypergraph problem with at most alpha-fracion incorrect queries. The hypergraph problem is also known as both monotone DNF learning problem, and complexes group testing problem
On Multistage Learning a Hidden Hypergraph
Learning a hidden hypergraph is a natural generalization of the classical
group testing problem that consists in detecting unknown hypergraph
by carrying out edge-detecting tests. In the given paper we
focus our attention only on a specific family of localized
hypergraphs for which the total number of vertices , the number of
edges , , and the cardinality of any edge ,
. Our goal is to identify all edges of by
using the minimal number of tests. We develop an adaptive algorithm that
matches the information theory bound, i.e., the total number of tests of the
algorithm in the worst case is at most . We also discuss
a probabilistic generalization of the problem.Comment: 5 pages, IEEE conferenc
Optimal Query Complexity for Reconstructing Hypergraphs
In this paper we consider the problem of reconstructing a hidden weighted
hypergraph of constant rank using additive queries. We prove the following: Let
be a weighted hidden hypergraph of constant rank with n vertices and
hyperedges. For any there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using
additive queries. This solves the open problem in [S. Choi, J. H. Kim. Optimal
Query Complexity Bounds for Finding Graphs. {\em STOC}, 749--758,~2008].
When the weights of the hypergraph are integers that are less than
where is the rank of the hypergraph (and therefore for
unweighted hypergraphs) there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using additive queries.
Using the information theoretic bound the above query complexities are tight
Learning and Testing Variable Partitions
Let be a multivariate function from a product set to an
Abelian group . A -partition of with cost is a partition of
the set of variables into non-empty subsets such that is -close to
for some with
respect to a given error metric. We study algorithms for agnostically learning
partitions and testing -partitionability over various groups and error
metrics given query access to . In particular we show that
Given a function that has a -partition of cost , a partition
of cost can be learned in time
for any .
In contrast, for and learning a partition of cost is NP-hard.
When is real-valued and the error metric is the 2-norm, a
2-partition of cost can be learned in time
.
When is -valued and the error metric is Hamming
weight, -partitionability is testable with one-sided error and
non-adaptive queries. We also show that even
two-sided testers require queries when .
This work was motivated by reinforcement learning control tasks in which the
set of control variables can be partitioned. The partitioning reduces the task
into multiple lower-dimensional ones that are relatively easier to learn. Our
second algorithm empirically increases the scores attained over previous
heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202
Spectral Detection on Sparse Hypergraphs
We consider the problem of the assignment of nodes into communities from a
set of hyperedges, where every hyperedge is a noisy observation of the
community assignment of the adjacent nodes. We focus in particular on the
sparse regime where the number of edges is of the same order as the number of
vertices. We propose a spectral method based on a generalization of the
non-backtracking Hashimoto matrix into hypergraphs. We analyze its performance
on a planted generative model and compare it with other spectral methods and
with Bayesian belief propagation (which was conjectured to be asymptotically
optimal for this model). We conclude that the proposed spectral method detects
communities whenever belief propagation does, while having the important
advantages to be simpler, entirely nonparametric, and to be able to learn the
rule according to which the hyperedges were generated without prior
information.Comment: 8 pages, 5 figure
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