50,232 research outputs found
Elliptic systems with measurable coefficients of the type of Lam\'e system in three dimensions
We study the elliptic systems , where the coefficients and
are positive scalar functions that are measurable and bounded away from zero
and infinity. We prove that weak solutions of the above system are H\"older
continuous under some minimal conditions on the inhomogeneous term . We also
present some applications and discuss several related topics including
estimates of the Green's functions and the heat kernels of the above systems.Comment: Proof of Theorem 3.1 is correcte
ExplaiNE: An Approach for Explaining Network Embedding-based Link Predictions
Networks are powerful data structures, but are challenging to work with for
conventional machine learning methods. Network Embedding (NE) methods attempt
to resolve this by learning vector representations for the nodes, for
subsequent use in downstream machine learning tasks.
Link Prediction (LP) is one such downstream machine learning task that is an
important use case and popular benchmark for NE methods. Unfortunately, while
NE methods perform exceedingly well at this task, they are lacking in
transparency as compared to simpler LP approaches.
We introduce ExplaiNE, an approach to offer counterfactual explanations for
NE-based LP methods, by identifying existing links in the network that explain
the predicted links. ExplaiNE is applicable to a broad class of NE algorithms.
An extensive empirical evaluation for the NE method `Conditional Network
Embedding' in particular demonstrates its accuracy and scalability
Conditional network embeddings
Network Embeddings (NEs) map the nodes of a given network into -dimensional Euclidean space . Ideally, this mapping is such that 'similar' nodes are mapped onto nearby points, such that the NE can be used for purposes such as link prediction (if 'similar' means being 'more likely to be connected') or classification (if 'similar' means 'being more likely to have the same label'). In recent years various methods for NE have been introduced, all following a similar strategy: defining a notion of similarity between nodes (typically some distance measure within the network), a distance measure in the embedding space, and a loss function that penalizes large distances for similar nodes and small distances for dissimilar nodes.
A difficulty faced by existing methods is that certain networks are fundamentally hard to embed due to their structural properties: (approximate) multipartiteness, certain degree distributions, assortativity, etc. To overcome this, we introduce a conceptual innovation to the NE literature and propose to create \emph{Conditional Network Embeddings} (CNEs); embeddings that maximally add information with respect to given structural properties (e.g. node degrees, block densities, etc.). We use a simple Bayesian approach to achieve this, and propose a block stochastic gradient descent algorithm for fitting it efficiently.
We demonstrate that CNEs are superior for link prediction and multi-label classification when compared to state-of-the-art methods, and this without adding significant mathematical or computational complexity. Finally, we illustrate the potential of CNE for network visualization
ALPINE : Active Link Prediction using Network Embedding
Many real-world problems can be formalized as predicting links in a partially observed network. Examples include Facebook friendship suggestions, consumer-product recommendations, and the identification of hidden interactions between actors in a crime network. Several link prediction algorithms, notably those recently introduced using network embedding, are capable of doing this by just relying on the observed part of the network.
Often, the link status of a node pair can be queried, which can be used as additional information by the link prediction algorithm. Unfortunately, such queries can be expensive or time-consuming, mandating the careful consideration of which node pairs to query. In this paper we estimate the improvement in link prediction accuracy after querying any particular node pair, to use in an active learning setup.
Specifically, we propose ALPINE (Active Link Prediction usIng Network Embedding), the first method to achieve this for link prediction based on network embedding. To this end, we generalized the notion of V-optimality from experimental design to this setting, as well as more basic active learning heuristics originally developed in standard classification settings. Empirical results on real data show that ALPINE is scalable, and boosts link prediction accuracy with far fewer queries
Explainable subgraphs with surprising densities : a subgroup discovery approach
The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between any pair of people depends on a range of attributes, such as their age, residence location, workplace, and hobbies. The high-level structure of a graph can thus possibly be described well by means of patterns of the form `the subgroup of all individuals with a certain properties X are often (or rarely) friends with individuals in another subgroup defined by properties Y', in comparison to what is expected. Such rules present potentially actionable and generalizable insight into the graph.
We present a method that finds node subgroup pairs between which the edge density is interestingly high or low, using an information-theoretic definition of interestingness. Additionally, the interestingness is quantified subjectively, to contrast with prior information an analyst may have about the connectivity. This view immediatly enables iterative mining of such patterns. This is the first method aimed at graph connectivity relations between different subgroups. Our method generalizes prior work on dense subgraphs induced by a subgroup description. Although this setting has been studied already, we demonstrate for this special case considerable practical advantages of our subjective interestingness measure with respect to a wide range of (objective) interestingness measures
A perceptual hash function to store and retrieve large scale DNA sequences
This paper proposes a novel approach for storing and retrieving massive DNA
sequences.. The method is based on a perceptual hash function, commonly used to
determine the similarity between digital images, that we adapted for DNA
sequences. Perceptual hash function presented here is based on a Discrete
Cosine Transform Sign Only (DCT-SO). Each nucleotide is encoded as a fixed gray
level intensity pixel and the hash is calculated from its significant frequency
characteristics. This results to a drastic data reduction between the sequence
and the perceptual hash. Unlike cryptographic hash functions, perceptual hashes
are not affected by "avalanche effect" and thus can be compared. The similarity
distance between two hashes is estimated with the Hamming Distance, which is
used to retrieve DNA sequences. Experiments that we conducted show that our
approach is relevant for storing massive DNA sequences, and retrieving them
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