7 research outputs found
Nearly-linear monotone paths in edge-ordered graphs
How long a monotone path can one always find in any edge-ordering of the complete graph Kn?
This appealing question was first asked by Chv´atal and Koml´os in 1971, and has since attracted the
attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that
one can always find a monotone path of linear length, but until now the best known lower bound was
n
2/3−o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph
contains a monotone path of length n
1−o(1
Nearly-linear monotone paths in edge-ordered graphs
How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n^2/3−o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n^1−o(1)
A theory of relation learning and cross-domain generalization
People readily generalize knowledge to novel domains and stimuli. We present
a theory, instantiated in a computational model, based on the idea that
cross-domain generalization in humans is a case of analogical inference over
structured (i.e., symbolic) relational representations. The model is an
extension of the LISA and DORA models of relational inference and learning. The
resulting model learns both the content and format (i.e., structure) of
relational representations from non-relational inputs without supervision, when
augmented with the capacity for reinforcement learning, leverages these
representations to learn individual domains, and then generalizes to new
domains on the first exposure (i.e., zero-shot learning) via analogical
inference. We demonstrate the capacity of the model to learn structured
relational representations from a variety of simple visual stimuli, and to
perform cross-domain generalization between video games (Breakout and Pong) and
between several psychological tasks. We demonstrate that the model's trajectory
closely mirrors the trajectory of children as they learn about relations,
accounting for phenomena from the literature on the development of children's
reasoning and analogy making. The model's ability to generalize between domains
demonstrates the flexibility afforded by representing domains in terms of their
underlying relational structure, rather than simply in terms of the statistical
relations between their inputs and outputs.Comment: Includes supplemental materia