Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Continuous Fuzzy Transform as Integral Operator   The Fuzzy transform is ubiquitous in different research fields and
applications, such as image and data compression, data mining, knowledge
discovery, and the analysis of linguistic expressions. As a generalisation of
the Fuzzy transform, we introduce the continuous Fuzzy transform and its
inverse, as an integral operator induced by a kernel function. Through the
relation between membership functions and integral kernels, we show that the
main properties (e.g., continuity, symmetry) of the membership functions are
inherited by the continuous Fuzzy transform. Then, the relation between the
continuous Fuzzy transform and integral operators is used to introduce a
data-driven Fuzzy transform, which encodes intrinsic information (e.g.,
structure, geometry, sampling density) about the input data. In this way, we
avoid coarse fuzzy partitions, which group data into large clusters that do not
adapt to their local behaviour, or a too dense fuzzy partition, which generally
has cells that are not covered by the data, thus being redundant and resulting
in a higher computational cost. To this end, the data-driven membership
functions are defined by properly filtering the spectrum of the
Laplace-Beltrami operator associated with the input data. Finally, we introduce
the space of continuous Fuzzy transforms, which is useful for the comparison of
different continuous Fuzzy transforms and for their efficient computation.

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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th