Spherically symmetric solutions to a model for phase transitions driven by configurational forces

We prove the global in time existence of spherically symmetric solutions to an initial-boundary value problem for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, non-uniformly parabolic equation of second order. The problem models the behavior in time of materials in which martensitic phase transitions, driven by configurational forces, take place, and can be considered to be a regularization of the corresponding sharp interface model. By assuming that the solutions are spherically symmetric, we reduce the original multidimensional problem to the one in one space dimension, then prove the existence of spherically symmetric solutions. Our proof is valid due to the essential feature that the reduced problem is one space dimensional.Comment: 25 page

Low Rank Optimization for Efficient Deep Learning: Making A Balance between Compact Architecture and Fast Training

Deep neural networks have achieved great success in many data processing applications. However, the high computational complexity and storage cost makes deep learning hard to be used on resource-constrained devices, and it is not environmental-friendly with much power cost. In this paper, we focus on low-rank optimization for efficient deep learning techniques. In the space domain, deep neural networks are compressed by low rank approximation of the network parameters, which directly reduces the storage requirement with a smaller number of network parameters. In the time domain, the network parameters can be trained in a few subspaces, which enables efficient training for fast convergence. The model compression in the spatial domain is summarized into three categories as pre-train, pre-set, and compression-aware methods, respectively. With a series of integrable techniques discussed, such as sparse pruning, quantization, and entropy coding, we can ensemble them in an integration framework with lower computational complexity and storage. Besides of summary of recent technical advances, we have two findings for motivating future works: one is that the effective rank outperforms other sparse measures for network compression. The other is a spatial and temporal balance for tensorized neural networks

Exploring the Learning Difficulty of Data Theory and Measure

As learning difficulty is crucial for machine learning (e.g., difficulty-based weighting learning strategies), previous literature has proposed a number of learning difficulty measures. However, no comprehensive investigation for learning difficulty is available to date, resulting in that nearly all existing measures are heuristically defined without a rigorous theoretical foundation. In addition, there is no formal definition of easy and hard samples even though they are crucial in many studies. This study attempts to conduct a pilot theoretical study for learning difficulty of samples. First, a theoretical definition of learning difficulty is proposed on the basis of the bias-variance trade-off theory on generalization error. Theoretical definitions of easy and hard samples are established on the basis of the proposed definition. A practical measure of learning difficulty is given as well inspired by the formal definition. Second, the properties for learning difficulty-based weighting strategies are explored. Subsequently, several classical weighting methods in machine learning can be well explained on account of explored properties. Third, the proposed measure is evaluated to verify its reasonability and superiority in terms of several main difficulty factors. The comparison in these experiments indicates that the proposed measure significantly outperforms the other measures throughout the experiments.Comment: Ou Wu is the corresponding author of this wor