11,497 research outputs found
Scalable Boolean Tensor Factorizations using Random Walks
Tensors are becoming increasingly common in data mining, and consequently,
tensor factorizations are becoming more and more important tools for data
miners. When the data is binary, it is natural to ask if we can factorize it
into binary factors while simultaneously making sure that the reconstructed
tensor is still binary. Such factorizations, called Boolean tensor
factorizations, can provide improved interpretability and find Boolean
structure that is hard to express using normal factorizations. Unfortunately
the algorithms for computing Boolean tensor factorizations do not usually scale
well. In this paper we present a novel algorithm for finding Boolean CP and
Tucker decompositions of large and sparse binary tensors. In our experimental
evaluation we show that our algorithm can handle large tensors and accurately
reconstructs the latent Boolean structure
Algorithmic Aspects of a General Modular Decomposition Theory
A new general decomposition theory inspired from modular graph decomposition
is presented. This helps unifying modular decomposition on different
structures, including (but not restricted to) graphs. Moreover, even in the
case of graphs, the terminology ``module'' not only captures the classical
graph modules but also allows to handle 2-connected components, star-cutsets,
and other vertex subsets. The main result is that most of the nice algorithmic
tools developed for modular decomposition of graphs still apply efficiently on
our generalisation of modules. Besides, when an essential axiom is satisfied,
almost all the important properties can be retrieved. For this case, an
algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is
generalised and yields a very efficient solution to the associated
decomposition problem
Clustering Boolean Tensors
Tensor factorizations are computationally hard problems, and in particular,
are often significantly harder than their matrix counterparts. In case of
Boolean tensor factorizations -- where the input tensor and all the factors are
required to be binary and we use Boolean algebra -- much of that hardness comes
from the possibility of overlapping components. Yet, in many applications we
are perfectly happy to partition at least one of the modes. In this paper we
investigate what consequences does this partitioning have on the computational
complexity of the Boolean tensor factorizations and present a new algorithm for
the resulting clustering problem. This algorithm can alternatively be seen as a
particularly regularized clustering algorithm that can handle extremely
high-dimensional observations. We analyse our algorithms with the goal of
maximizing the similarity and argue that this is more meaningful than
minimizing the dissimilarity. As a by-product we obtain a PTAS and an efficient
0.828-approximation algorithm for rank-1 binary factorizations. Our algorithm
for Boolean tensor clustering achieves high scalability, high similarity, and
good generalization to unseen data with both synthetic and real-world data
sets
Satisfiability in multi-valued circuits
Satisfiability of Boolean circuits is among the most known and important
problems in theoretical computer science. This problem is NP-complete in
general but becomes polynomial time when restricted either to monotone gates or
linear gates. We go outside Boolean realm and consider circuits built of any
fixed set of gates on an arbitrary large finite domain. From the complexity
point of view this is strictly connected with the problems of solving equations
(or systems of equations) over finite algebras.
The research reported in this work was motivated by a desire to know for
which finite algebras there is a polynomial time algorithm that
decides if an equation over has a solution. We are also looking for
polynomial time algorithms that decide if two circuits over a finite algebra
compute the same function. Although we have not managed to solve these problems
in the most general setting we have obtained such a characterization for a very
broad class of algebras from congruence modular varieties. This class includes
most known and well-studied algebras such as groups, rings, modules (and their
generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie
algebras), lattices (and their extensions like Boolean algebras, Heyting
algebras or other algebras connected with multi-valued logics including
MV-algebras).
This paper seems to be the first systematic study of the computational
complexity of satisfiability of non-Boolean circuits and solving equations over
finite algebras. The characterization results provided by the paper is given in
terms of nice structural properties of algebras for which the problems are
solvable in polynomial time.Comment: 50 page
On Multi-Relational Link Prediction with Bilinear Models
We study bilinear embedding models for the task of multi-relational link
prediction and knowledge graph completion. Bilinear models belong to the most
basic models for this task, they are comparably efficient to train and use, and
they can provide good prediction performance. The main goal of this paper is to
explore the expressiveness of and the connections between various bilinear
models proposed in the literature. In particular, a substantial number of
models can be represented as bilinear models with certain additional
constraints enforced on the embeddings. We explore whether or not these
constraints lead to universal models, which can in principle represent every
set of relations, and whether or not there are subsumption relationships
between various models. We report results of an independent experimental study
that evaluates recent bilinear models in a common experimental setup. Finally,
we provide evidence that relation-level ensembles of multiple bilinear models
can achieve state-of-the art prediction performance
- …