708 research outputs found
The logical encoding of Sugeno integrals
International audienceSugeno integrals are a well-known family of qualitative multiple criteria aggregation operators. The paper investigates how the behavior of these operators can be described in a prioritized propositional logic language, namely possibilistic logic. The case of binary-valued criteria, which amounts to providing a logical description of the fuzzy measure underlying the integral, is first considered. The general case of a Sugeno integral when criteria are valued on a discrete scale is then studied
5* Knowledge Graph Embeddings with Projective Transformations
Performing link prediction using knowledge graph embedding (KGE) models is a
popular approach for knowledge graph completion. Such link predictions are
performed by measuring the likelihood of links in the graph via a
transformation function that maps nodes via edges into a vector space. Since
the complex structure of the real world is reflected in multi-relational
knowledge graphs, the transformation functions need to be able to represent
this complexity. However, most of the existing transformation functions in
embedding models have been designed in Euclidean geometry and only cover one or
two simple transformations. Therefore, they are prone to underfitting and
limited in their ability to embed complex graph structures. The area of
projective geometry, however, fully covers inversion, reflection, translation,
rotation, and homothety transformations. We propose a novel KGE model, which
supports those transformations and subsumes other state-of-the-art models. The
model has several favorable theoretical properties and outperforms existing
approaches on widely used link prediction benchmarks
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
Quantum Mechanics Unscrambled
Is quantum mechanics about 'states'? Or is it basically another kind of
probability theory? It is argued that the elementary formalism of quantum
mechanics operates as a well-justified alternative to 'classical'
instantiations of a probability calculus. Its providing a general framework for
prediction accounts for its distinctive traits, which one should be careful not
to mistake for reflections of any strange ontology. The suggestion is also made
that quantum theory unwittingly emerged, in Schroedinger's formulation, as a
'lossy' by-product of a quantum-mechanical variant of the Hamilton-Jacobi
equation. As it turns out, the effectiveness of quantum theory qua predictive
algorithm makes up for the computational impracticability of that master
equation.Comment: 25 pages, no figures, final versio
On the realization of Symmetries in Quantum Mechanics
The aim of this paper is to give a simple, geometric proof of Wigner's
theorem on the realization of symmetries in quantum mechanics that clarifies
its relation to projective geometry. Although several proofs exist already, it
seems that the relevance of Wigner's theorem is not fully appreciated in
general. It is Wigner's theorem which allows the use of linear realizations of
symmetries and therefore guarantees that, in the end, quantum theory stays a
linear theory. In the present paper, we take a strictly geometrical point of
view in order to prove this theorem. It becomes apparent that Wigner's theorem
is nothing else but a corollary of the fundamental theorem of projective
geometry. In this sense, the proof presented here is simple, transparent and
therefore accessible even to elementary treatments in quantum mechanics.Comment: 8 page
Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds
Motivated by the vast success of deep convolutional networks, there is a
great interest in generalizing convolutions to non-Euclidean manifolds. A major
complication in comparison to flat spaces is that it is unclear in which
alignment a convolution kernel should be applied on a manifold. The underlying
reason for this ambiguity is that general manifolds do not come with a
canonical choice of reference frames (gauge). Kernels and features therefore
have to be expressed relative to arbitrary coordinates. We argue that the
particular choice of coordinatization should not affect a network's inference
-- it should be coordinate independent. A simultaneous demand for coordinate
independence and weight sharing is shown to result in a requirement on the
network to be equivariant under local gauge transformations (changes of local
reference frames). The ambiguity of reference frames depends thereby on the
G-structure of the manifold, such that the necessary level of gauge
equivariance is prescribed by the corresponding structure group G. Coordinate
independent convolutions are proven to be equivariant w.r.t. those isometries
that are symmetries of the G-structure. The resulting theory is formulated in a
coordinate free fashion in terms of fiber bundles. To exemplify the design of
coordinate independent convolutions, we implement a convolutional network on
the M\"obius strip. The generality of our differential geometric formulation of
convolutional networks is demonstrated by an extensive literature review which
explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general
surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions
is publicly available at https://github.com/mauriceweiler/MobiusCNN
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