27,248 research outputs found
Tight and simple Web graph compression
Analysing Web graphs has applications in determining page ranks, fighting Web
spam, detecting communities and mirror sites, and more. This study is however
hampered by the necessity of storing a major part of huge graphs in the
external memory, which prevents efficient random access to edge (hyperlink)
lists. A number of algorithms involving compression techniques have thus been
presented, to represent Web graphs succinctly but also providing random access.
Those techniques are usually based on differential encodings of the adjacency
lists, finding repeating nodes or node regions in the successive lists, more
general grammar-based transformations or 2-dimensional representations of the
binary matrix of the graph. In this paper we present two Web graph compression
algorithms. The first can be seen as engineering of the Boldi and Vigna (2004)
method. We extend the notion of similarity between link lists, and use a more
compact encoding of residuals. The algorithm works on blocks of varying size
(in the number of input lines) and sacrifices access time for better
compression ratio, achieving more succinct graph representation than other
algorithms reported in the literature. The second algorithm works on blocks of
the same size, in the number of input lines, and its key mechanism is merging
the block into a single ordered list. This method achieves much more attractive
space-time tradeoffs.Comment: 15 page
TorusE: Knowledge Graph Embedding on a Lie Group
Knowledge graphs are useful for many artificial intelligence (AI) tasks.
However, knowledge graphs often have missing facts. To populate the graphs,
knowledge graph embedding models have been developed. Knowledge graph embedding
models map entities and relations in a knowledge graph to a vector space and
predict unknown triples by scoring candidate triples. TransE is the first
translation-based method and it is well known because of its simplicity and
efficiency for knowledge graph completion. It employs the principle that the
differences between entity embeddings represent their relations. The principle
seems very simple, but it can effectively capture the rules of a knowledge
graph. However, TransE has a problem with its regularization. TransE forces
entity embeddings to be on a sphere in the embedding vector space. This
regularization warps the embeddings and makes it difficult for them to fulfill
the abovementioned principle. The regularization also affects adversely the
accuracies of the link predictions. On the other hand, regularization is
important because entity embeddings diverge by negative sampling without it.
This paper proposes a novel embedding model, TorusE, to solve the
regularization problem. The principle of TransE can be defined on any Lie
group. A torus, which is one of the compact Lie groups, can be chosen for the
embedding space to avoid regularization. To the best of our knowledge, TorusE
is the first model that embeds objects on other than a real or complex vector
space, and this paper is the first to formally discuss the problem of
regularization of TransE. Our approach outperforms other state-of-the-art
approaches such as TransE, DistMult and ComplEx on a standard link prediction
task. We show that TorusE is scalable to large-size knowledge graphs and is
faster than the original TransE.Comment: accepted for AAAI-1
Buildings, spiders, and geometric Satake
Let G be a simple algebraic group. Labelled trivalent graphs called webs can
be used to product invariants in tensor products of minuscule representations.
For each web, we construct a configuration space of points in the affine
Grassmannian. Via the geometric Satake correspondence, we relate these
configuration spaces to the invariant vectors coming from webs. In the case G =
SL(3), non-elliptic webs yield a basis for the invariant spaces. The
non-elliptic condition, which is equivalent to the condition that the dual
diskoid of the web is CAT(0), is explained by the fact that affine buildings
are CAT(0).Comment: 49 pages; revised and to appear in Compositio Mathematic
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