9,673 research outputs found
Systematic Analysis of Cluster Similarity Indices: How to Validate Validation Measures
Many cluster similarity indices are used to evaluate clustering algorithms,
and choosing the best one for a particular task remains an open problem. We
demonstrate that this problem is crucial: there are many disagreements among
the indices, these disagreements do affect which algorithms are preferred in
applications, and this can lead to degraded performance in real-world systems.
We propose a theoretical framework to tackle this problem: we develop a list of
desirable properties and conduct an extensive theoretical analysis to verify
which indices satisfy them. This allows for making an informed choice: given a
particular application, one can first select properties that are desirable for
the task and then identify indices satisfying these. Our work unifies and
considerably extends existing attempts at analyzing cluster similarity indices:
we introduce new properties, formalize existing ones, and mathematically prove
or disprove each property for an extensive list of validation indices. This
broader and more rigorous approach leads to recommendations that considerably
differ from how validation indices are currently being chosen by practitioners.
Some of the most popular indices are even shown to be dominated by previously
overlooked ones
Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
Absolutely Maximally Entangled (AME) states are those multipartite quantum
states that carry absolute maximum entanglement in all possible partitions. AME
states are known to play a relevant role in multipartite teleportation, in
quantum secret sharing and they provide the basis novel tensor networks related
to holography. We present alternative constructions of AME states and show
their link with combinatorial designs. We also analyze a key property of AME,
namely their relation to tensors that can be understood as unitary
transformations in every of its bi-partitions. We call this property
multi-unitarity.Comment: 18 pages, 2 figures. Comments are very welcom
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