24,815 research outputs found
A Transfer Principle: Universal Approximators Between Metric Spaces From Euclidean Universal Approximators
We build universal approximators of continuous maps between arbitrary Polish
metric spaces and using universal approximators
between Euclidean spaces as building blocks. Earlier results assume that the
output space is a topological vector space. We overcome this
limitation by "randomization": our approximators output discrete probability
measures over . When and are Polish
without additional structure, we prove very general qualitative guarantees;
when they have suitable combinatorial structure, we prove quantitative
guarantees for H\"older-like maps, including maps between finite graphs,
solution operators to rough differential equations between certain Carnot
groups, and continuous non-linear operators between Banach spaces arising in
inverse problems. In particular, we show that the required number of Dirac
measures is determined by the combinatorial structure of and
. For barycentric , including Banach spaces,
-trees, Hadamard manifolds, or Wasserstein spaces on Polish metric
spaces, our approximators reduce to -valued functions. When the
Euclidean approximators are neural networks, our constructions generalize
transformer networks, providing a new probabilistic viewpoint of geometric deep
learning.Comment: 14 Figures, 3 Tables, 78 Pages (Main 40, Proofs 26, Acknowledgments
and References 12
Change Detection in Graph Streams by Learning Graph Embeddings on Constant-Curvature Manifolds
The space of graphs is often characterised by a non-trivial geometry, which
complicates learning and inference in practical applications. A common approach
is to use embedding techniques to represent graphs as points in a conventional
Euclidean space, but non-Euclidean spaces have often been shown to be better
suited for embedding graphs. Among these, constant-curvature Riemannian
manifolds (CCMs) offer embedding spaces suitable for studying the statistical
properties of a graph distribution, as they provide ways to easily compute
metric geodesic distances. In this paper, we focus on the problem of detecting
changes in stationarity in a stream of attributed graphs. To this end, we
introduce a novel change detection framework based on neural networks and CCMs,
that takes into account the non-Euclidean nature of graphs. Our contribution in
this work is twofold. First, via a novel approach based on adversarial
learning, we compute graph embeddings by training an autoencoder to represent
graphs on CCMs. Second, we introduce two novel change detection tests operating
on CCMs. We perform experiments on synthetic data, as well as two real-world
application scenarios: the detection of epileptic seizures using functional
connectivity brain networks, and the detection of hostility between two
subjects, using human skeletal graphs. Results show that the proposed methods
are able to detect even small changes in a graph-generating process,
consistently outperforming approaches based on Euclidean embeddings.Comment: 14 pages, 8 figure
Learning Generative Models across Incomparable Spaces
Generative Adversarial Networks have shown remarkable success in learning a
distribution that faithfully recovers a reference distribution in its entirety.
However, in some cases, we may want to only learn some aspects (e.g., cluster
or manifold structure), while modifying others (e.g., style, orientation or
dimension). In this work, we propose an approach to learn generative models
across such incomparable spaces, and demonstrate how to steer the learned
distribution towards target properties. A key component of our model is the
Gromov-Wasserstein distance, a notion of discrepancy that compares
distributions relationally rather than absolutely. While this framework
subsumes current generative models in identically reproducing distributions,
its inherent flexibility allows application to tasks in manifold learning,
relational learning and cross-domain learning.Comment: International Conference on Machine Learning (ICML
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