15,680 research outputs found
Shifting numbers in triangulated categories via bounded t-structures
The shifting numbers measure the asymptotic amount by which an endofunctor of
a triangulated category translates inside the category, and are analogous to
Poincare translation numbers that are widely used in dynamical systems. One of
the ways to define these invariants is via the phase functions of Bridgeland
stability conditions. We show in this short note that the shifting numbers can
also be defined via the bounded t-structures. In particular, the full package
of stability conditions (a bounded t-structure, and a central charge on a
charge lattice) is not necessary for the purpose of computing the shifting
numbers
Systolic inequalities for K3 surfaces via stability conditions
We introduce the notions of categorical systoles and categorical volumes of
Bridgeland stability conditions on triangulated categories. We prove that for
any projective K3 surface, there exists a constant C depending only on the rank
and discriminant of its Picard group, such that holds for any stability condition on the derived
category of coherent sheaves on the K3 surface. This is an algebro-geometric
generalization of a classical systolic inequality on two-tori. We also discuss
applications of this inequality in symplectic geometry.Comment: 23 pages; major improvement: remove the condition of Picard rank on
Multilabel Consensus Classification
In the era of big data, a large amount of noisy and incomplete data can be
collected from multiple sources for prediction tasks. Combining multiple models
or data sources helps to counteract the effects of low data quality and the
bias of any single model or data source, and thus can improve the robustness
and the performance of predictive models. Out of privacy, storage and bandwidth
considerations, in certain circumstances one has to combine the predictions
from multiple models or data sources to obtain the final predictions without
accessing the raw data. Consensus-based prediction combination algorithms are
effective for such situations. However, current research on prediction
combination focuses on the single label setting, where an instance can have one
and only one label. Nonetheless, data nowadays are usually multilabeled, such
that more than one label have to be predicted at the same time. Direct
applications of existing prediction combination methods to multilabel settings
can lead to degenerated performance. In this paper, we address the challenges
of combining predictions from multiple multilabel classifiers and propose two
novel algorithms, MLCM-r (MultiLabel Consensus Maximization for ranking) and
MLCM-a (MLCM for microAUC). These algorithms can capture label correlations
that are common in multilabel classifications, and optimize corresponding
performance metrics. Experimental results on popular multilabel classification
tasks verify the theoretical analysis and effectiveness of the proposed
methods
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