8,246 research outputs found
Piecewise Linear Manifold Clustering
This work studies the application of topological analysis to non-linear manifold clustering. A novel method, that exploits the data clustering structure, allows to generate a topological representation of the point dataset. An analysis of topological construction under different simulated conditions is performed to explore the capabilities and limitations of the method, and demonstrated statistically significant improvements in performance. Furthermore, we introduce a new information-theoretical validation measure for clustering, that exploits geometrical properties of clusters to estimate clustering compressibility, for evaluation of the clustering goodness-of-fit without any prior information about true class assignments. We show how the new validation measure, when used as regularization criteria, allows creation of clusters that are more informative. A final contribution is a new metaclustering technique that allows to create a model-based clustering beyond point and linear shaped structures. Driven by topological structure and our information-theoretical criteria, this technique provides structured view of the data on new comprehensive and interpretation level. Improvements of our clustering approach are demonstrated on a variety of synthetic and real datasets, including image and climatological data
Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems
We consider two-dimensional lattice models that support Ising anyonic
excitations and are coupled to a thermal bath. We propose a phenomenological
model for the resulting short-time dynamics that includes pair-creation,
hopping, braiding, and fusion of anyons. By explicitly constructing topological
quantum error-correcting codes for this class of system, we use our
thermalization model to estimate the lifetime of the quantum information stored
in the encoded spaces. To decode and correct errors in these codes, we adapt
several existing topological decoders to the non-Abelian setting. We perform
large-scale numerical simulations of these two-dimensional Ising anyon systems
and find that the thresholds of these models range between 13% to 25%. To our
knowledge, these are the first numerical threshold estimates for quantum codes
without explicit additive structure.Comment: 34 pages, 9 figures; v2 matches the journal version and corrects a
misstatement about the detailed balance condition of our Metropolis
simulations. All conclusions from v1 are unaffected by this correctio
A Topological Approach to Spectral Clustering
We propose two related unsupervised clustering algorithms which, for input,
take data assumed to be sampled from a uniform distribution supported on a
metric space , and output a clustering of the data based on the selection of
a topological model for the connected components of . Both algorithms work
by selecting a graph on the samples from a natural one-parameter family of
graphs, using a geometric criterion in the first case and an information
theoretic criterion in the second. The estimated connected components of
are identified with the kernel of the associated graph Laplacian, which allows
the algorithm to work without requiring the number of expected clusters or
other auxiliary data as input.Comment: 21 Page
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