4,288 research outputs found
Geometric deep learning: going beyond Euclidean data
Many scientific fields study data with an underlying structure that is a
non-Euclidean space. Some examples include social networks in computational
social sciences, sensor networks in communications, functional networks in
brain imaging, regulatory networks in genetics, and meshed surfaces in computer
graphics. In many applications, such geometric data are large and complex (in
the case of social networks, on the scale of billions), and are natural targets
for machine learning techniques. In particular, we would like to use deep
neural networks, which have recently proven to be powerful tools for a broad
range of problems from computer vision, natural language processing, and audio
analysis. However, these tools have been most successful on data with an
underlying Euclidean or grid-like structure, and in cases where the invariances
of these structures are built into networks used to model them. Geometric deep
learning is an umbrella term for emerging techniques attempting to generalize
(structured) deep neural models to non-Euclidean domains such as graphs and
manifolds. The purpose of this paper is to overview different examples of
geometric deep learning problems and present available solutions, key
difficulties, applications, and future research directions in this nascent
field
Aspherical gravitational monopoles
We show how to construct non-spherically-symmetric extended bodies of uniform
density behaving exactly as pointlike masses. These ``gravitational monopoles''
have the following equivalent properties: (i) they generate, outside them, a
spherically-symmetric gravitational potential ; (ii) their
interaction energy with an external gravitational potential is ; and (iii) all their multipole moments (of order ) with
respect to their center of mass vanish identically. The method applies for
any number of space dimensions. The free parameters entering the construction
are: (1) an arbitrary surface bounding a connected open subset
of ; (2) the arbitrary choice of the center of mass within
; and (3) the total volume of the body. An extension of the method
allows one to construct homogeneous bodies which are gravitationally equivalent
(in the sense of having exactly the same multipole moments) to any given body.Comment: 55 pages, Latex , submitted to Nucl.Phys.
Spectal Harmonics: Bridging Spectral Embedding and Matrix Completion in Self-Supervised Learning
Self-supervised methods received tremendous attention thanks to their
seemingly heuristic approach to learning representations that respect the
semantics of the data without any apparent supervision in the form of labels. A
growing body of literature is already being published in an attempt to build a
coherent and theoretically grounded understanding of the workings of a zoo of
losses used in modern self-supervised representation learning methods. In this
paper, we attempt to provide an understanding from the perspective of a Laplace
operator and connect the inductive bias stemming from the augmentation process
to a low-rank matrix completion problem. To this end, we leverage the results
from low-rank matrix completion to provide theoretical analysis on the
convergence of modern SSL methods and a key property that affects their
downstream performance.Comment: 12 pages, 3 figure
Diffusion Maps: Analysis and Applications
A lot of the data faced in science and engineering is not as complicated as it seems. There is the possibility of ¯nding low dimensional descriptions of this usually high dimensional data. One of the ways of achieving this is with the use of diffusion maps. Diffusion maps represent the dataset by a weighted graph in which points correspond to vertices and edges are weighted. The spectral properties of the graph Laplacian are then used to map the high dimensional data into a lower dimensional representation.
The algorithm is introduced on simple test examples for which the low dimensional description is known. Justification of the algorithm is given by showing its equivalence to a suitable minimisation problem and to random walks on graphs. The description of random walks in terms of partial di®erential equations is discussed. The heat equation for a probability density function is derived and used to further analyse the algorithm.
Applications of diffusion maps are presented at the end of this dissertation. The first application is clustering of data (i.e. partitioning of a data set into subsets so that the data points in each subset have similar characteristics). An approach based on di®usion maps (spectral clustering) is compared to the K-means clustering algorithm. We then discuss techniques for colour image quantization (reduction of distinct colours in an image). Finally, the diffusion maps are used to discover low dimensional description of high dimensional sets of images
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