Statistical Learning in Tree-Based Tensor Format

Tensor methods are widely used tools for the approximation of high dimensional functions. Such problems are encountered in uncertainty quantification and statistical learning, where the high dimensionality imposes to use specific techniques, such as rank-structured approximations [1]. In this work, we introduce a statistical learning algorithm for the approximation in tree-based tensor format, which are tensor networks whose graphs are dimension partition trees. This tensor format includes the Tucker format, the Tensor-Train format, as well as the more general Hierarchical tensor formats [4]. It can be interpreted as a deep neural network with a particular architecture [2]. The proposed algorithm uses random evaluations of a function to provide a tree-based tensor approximation, with adaptation of the tree-based rank by using a heuristic criterion based on the higher-order singular values to select the ranks to increase, and of the approximation spaces of the leaves of the tree. We then present a learning algorithm for the approximation under the form u(x) ≈ v(z_1,...,z_m) where v is a tensor in tree-based format and the z_i = g_i(x), 1 ≤ i ≤ m, are new variables. A strategy based on the projection pursuit regression [3] is proposed to compute the mappings g_i and increase the effective dimension m. The methods are illustrated on different examples to show their efficiency and adaptability as well as the power of representation of the tree-based tensor format, possibly combined with changes of variables.

Pelabelan 0-Anti Ajaib dan 2-Anti Ajaib untuk Graf Tangga L

For a connected graph and a subset of . For a vertex the distance betwen and is . For an ordered k-partition of , the representation of with respect to is The k-partition is a resolving partition if are distinct for every The minimum k for which there is a resolving partition of is the partition dimension of In this paper will shown resolving partition of connected graph order where is a bipartite graph. Then it is shown dimension partition of bipartite graph, are pd(Kst)=n-1

A New Cooperative PSO Approach with Landscape Estimation, Dimension Partition, and Velocity Control

Particle swarm optimization (PSO) has been proposed as an alternative to traditional evolutionary algorithms. Yet, more efficient strategies are still needed to control the trade-off between exploitation and exploration in the search process for solving complex tasks with high dimensional and multimodal objective functions. In this work, the authors propose a new PSO approach to overcome the search difficulties. Their approach first predicts the landscape type of a function for initial search settings, and then focuses on two search strategies for multimodal functions. One is a two-swarm cooperative strategy that controls search region and integrates partial and full dimension PSO search. The other strategy is to control the velocity of the particles in an adaptive way, according to how they move in the space. To evaluate the proposed approach, extensive experiments have been conducted and comparisons to several popular PSO variants have been made. Our experiments prove that the proposed approach can have better performance than others in most of the test cases.