Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown group elements
x
1
,
…
,
x
n
∈
SE
(
d
...)
given noisy measurements of a subset of their pairwise relative transforms
x
i
−
1
x
j
. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a non-convex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation (MLE) whose minimizer provides an exact maximum-likelihood estimate so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so significantly faster than the Gauss–Newton-based approach that forms the basis of current state-of-the-art techniques.
•A practical guideline on leveraging graph neural networks (GNNs) for realizing intelligent fault diagnostics and prognostics.•A novel intelligent fault diagnostics and prognostics framework based on ...GNNs is established to illustrate how the proposed guideline works.•In this framework, three types graph construction methods are provided, and seven kinds of graph convolutional networks with four graph pooling methods are investigated.•A benchmark study of these models is carried out on eight datasets, including six fault diagnosis datasets and two prognosis datasets.
Deep learning (DL)-based methods have advanced the field of Prognostics and Health Management (PHM) in recent years, because of their powerful feature representation ability. The data in PHM are typically regular data represented in the Euclidean space. Nevertheless, there are an increasing number of applications that consider the relationships and interdependencies of data and represent the data in the form of graphs. Such kind of irregular data in non-Euclidean space pose a huge challenge to the existing DL-based methods, making some important operations (e.g., convolutions) easily applied to Euclidean space but difficult to model graph data in non-Euclidean space. Recently, graph neural networks (GNNs), as the emerging neural networks, have been utilized to model and analyze the graph data. However, there still lacks a guideline on leveraging GNNs for realizing intelligent fault diagnostics and prognostics. To fill this research gap, a practical guideline is proposed in this paper, and a novel intelligent fault diagnostics and prognostics framework based on GNN is established to illustrate how the proposed guideline works. In this framework, three types of graph construction methods are provided, and seven kinds of graph convolutional networks (GCNs) with four different graph pooling methods are investigated. To afford benchmark results for helping further study, a comprehensive evaluation of these models is performed on eight datasets, including six fault diagnosis datasets and two prognosis datasets. Finally, four issues related to the performance of GCNs are discussed and potential research directions are provided. The code library is available at: https://github.com/HazeDT/PHMGNNBenchmark.
This article is a complement to a paper by Kramer et al. that proposes a new matrix product, which is like a mixture of the usual and the Hadamard (or Schur) product. This new product is then related ...to the transformation group in the hyperbolic plane. Actually variations in the definition of internal operations lead to an infinity of products that can be associated with any of the Euclidean, hyperbolic or elliptic geometries in quite different manners. By the way: This article is not a review. The review can be found at the end of the paper.
In this paper we consider complete noncompact Riemannian manifolds (
M
,
g
) with nonnegative Ricci curvature and Euclidean volume growth, of dimension
n
≥
3
. For every bounded open subset
Ω
⊂
M
...with smooth boundary, we prove that
∫
∂
Ω
H
n
-
1
n
-
1
d
σ
≥
AVR
(
g
)
|
S
n
-
1
|
,
where
H
is the mean curvature of
∂
Ω
and
AVR
(
g
)
is the asymptotic volume ratio of (
M
,
g
). Moreover, the equality holds true if and only if
(
M
\
Ω
,
g
)
is isometric to a truncated cone over
∂
Ω
. An optimal version of Huisken’s Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.
Convolutional neural network (CNN) has demonstrated impressive ability to represent hyperspectral images and to achieve promising results in hyperspectral image classification. However, traditional ...CNN models can only operate convolution on regular square image regions with fixed size and weights, and thus, they cannot universally adapt to the distinct local regions with various object distributions and geometric appearances. Therefore, their classification performances are still to be improved, especially in class boundaries. To alleviate this shortcoming, we consider employing the recently proposed graph convolutional network (GCN) for hyperspectral image classification, as it can conduct the convolution on arbitrarily structured non-Euclidean data and is applicable to the irregular image regions represented by graph topological information. Different from the commonly used GCN models that work on a fixed graph, we enable the graph to be dynamically updated along with the graph convolution process so that these two steps can be benefited from each other to gradually produce the discriminative embedded features as well as a refined graph. Moreover, to comprehensively deploy the multiscale information inherited by hyperspectral images, we establish multiple input graphs with different neighborhood scales to extensively exploit the diversified spectral-spatial correlations at multiple scales. Therefore, our method is termed multiscale dynamic GCN (MDGCN). The experimental results on three typical benchmark data sets firmly demonstrate the superiority of the proposed MDGCN to other state-of-the-art methods in both qualitative and quantitative aspects.
We introduce a new technique to bound the fluctuations exhibited by a physical system, based on the Euclidean geometry of the space of observables. Through a simple unifying argument, we derive a ...sweeping generalization of so-called thermodynamic uncertainty relations (TURs). We not only strengthen the bounds but extend their realm of applicability and in many cases prove their optimality, without resorting to large deviation theory or information-theoretic techniques. In particular, we find the best TUR based on entropy production alone. We also derive a periodic uncertainty principle of which previous known bounds for periodic or stationary Markov chains known in the literature appear as limit cases. From it a novel bound for stationary Markov processes is derived, which surpasses previous known bounds. Our results exploit the non-invariance of the system under a symmetry which can be other than time reversal and thus open a wide new spectrum of applications.
ABSTRACT
We present a holographic derivation of the entropy of supersymmetric asymptotically AdS
5
black holes. We define a BPS limit of black hole thermodynamics by first focussing on a ...supersymmetric family of complexified solutions and then reaching extremality. We show that in this limit the black hole entropy is the Legendre transform of the on-shell gravitational action with respect to three chemical potentials subject toa constraint. This constraint follows from supersymmetry and regularity in the Euclidean bulk geometry. Further, we calculate, using localization, the exact partition function of the dual
N
= 1 SCFT on a twisted
S
1
×
S
3
with complexified chemical potentials obeying this constraint. This defines a generalization of the supersymmetric Casimir energy, whose Legendre transform at large
N
exactly reproduces the Bekenstein-Hawking entropy of the black hole.
Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning
, owing to their reputed difficulty among the world's best talents in ...pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges
, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.
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