Non-avoided crossings for n-body balanced configurations in R^3 near a central configuration

The balanced configurations are those n-body configurations which admit a relative equilibrium motion in a Euclidean space E of high enough dimension 2p. They are characterized by the commutation of two symmetric endomorphisms of the (n-1)-dimensional Euclidean space of codispositions, the intrinsic inertia endomorphism B which encodes the shape and the Wintner-Conley endomorphism A which encodes the forces. In general, p is the dimension d of the configuration, which is also the rank of B. Lowering to 2(d-1) the dimension of E occurs when the restriction of A to the (invariant) image of B possesses a double eigenvalue. It is shown that, while in the space of all dxd-symmetric endomorphisms, having a double eigenvalue is a condition of codimension 2 (the avoided crossings of physicists), here it becomes of codimension 1 provided some condition (H) is satisfied. As the condition is always satisfied for configurations of the maximal dimension (i.e. if d=n-1), this implies in particular the existence, in the neighborhood of the regular tetrahedron configuration of 4 bodies with no three of the masses equal, of exactly 3 families of balanced configurations which admit relative equilibrium motion in a four dimensional space.Comment: 35 pages, 1 diagram, 6 figures Section 1.5.2 is new: it introduces the condition (H) which had been overlooked in the first versio

Generalized Multi-manifold Graph Ensemble Embedding for Multi-View Dimensionality Reduction

In this paper, we propose a new dimension reduction (DR) algorithm called ensemble graph-based locality preserving projections (EGLPP); to overcome the neighborhood size k sensitivity in locally preserving projections (LPP). EGLPP constructs a homogeneous ensemble of adjacency graphs by varying neighborhood size k and finally uses the integrated embedded graph to optimize the low-dimensional projections. Furthermore, to appropriately handle the intrinsic geometrical structure of the multi-view data and overcome the dimensionality curse, we propose a generalized multi-manifold graph ensemble embedding framework (MLGEE). MLGEE aims to utilize multi-manifold graphs for the adjacency estimation with automatically weight each manifold to derive the integrated heterogeneous graph. Experimental results on various computer vision databases verify the effectiveness of proposed EGLPP and MLGEE over existing comparative DR methods

Support Neighbor Loss for Person Re-Identification

Person re-identification (re-ID) has recently been tremendously boosted due to the advancement of deep convolutional neural networks (CNN). The majority of deep re-ID methods focus on designing new CNN architectures, while less attention is paid on investigating the loss functions. Verification loss and identification loss are two types of losses widely used to train various deep re-ID models, both of which however have limitations. Verification loss guides the networks to generate feature embeddings of which the intra-class variance is decreased while the inter-class ones is enlarged. However, training networks with verification loss tends to be of slow convergence and unstable performance when the number of training samples is large. On the other hand, identification loss has good separating and scalable property. But its neglect to explicitly reduce the intra-class variance limits its performance on re-ID, because the same person may have significant appearance disparity across different camera views. To avoid the limitations of the two types of losses, we propose a new loss, called support neighbor (SN) loss. Rather than being derived from data sample pairs or triplets, SN loss is calculated based on the positive and negative support neighbor sets of each anchor sample, which contain more valuable contextual information and neighborhood structure that are beneficial for more stable performance. To ensure scalability and separability, a softmax-like function is formulated to push apart the positive and negative support sets. To reduce intra-class variance, the distance between the anchor's nearest positive neighbor and furthest positive sample is penalized. Integrating SN loss on top of Resnet50, superior re-ID results to the state-of-the-art ones are obtained on several widely used datasets.Comment: Accepted by ACM Multimedia (ACM MM) 201