Hypersurfaces of Prescribed Curvature Measure

We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the corresponding curvature equation on $\mathbb S^n$

Parallel growing and training of neural networks using output parallelism

In order to find an appropriate architecture for a large-scale real-world application automatically and efficiently, a natural method is to divide the original problem into a set of sub-problems. In this paper, we propose a simple neural network task decomposition method based on output parallelism. By using this method, a problem can be divided flexibly into several sub-problems as chosen, each of which is composed of the whole input vector and a fraction of the output vector. Each module (for one sub-problem) is responsible for producing a fraction of the output vector of the original problem. The hidden structure for the original problem’s output units are decoupled. These modules can be grown and trained in parallel on parallel processing elements. Incorporated with a constructive learning algorithm, our method does not require excessive computation and any prior knowledge concerning decomposition. The feasibility of output parallelism is analyzed and proved. Some benchmarks are implemented to test the validity of this method. Their results show that this method can reduce computational time, increase learning speed and improve generalization accuracy for both classification and regression problems

Incremental learning with respect to new incoming input attributes

Neural networks are generally exposed to a dynamic environment where the training patterns or the input attributes (features) will likely be introduced into the current domain incrementally. This paper considers the situation where a new set of input attributes must be considered and added into the existing neural network. The conventional method is to discard the existing network and redesign one from scratch. This approach wastes the old knowledge and the previous effort. In order to reduce computational time, improve generalization accuracy, and enhance intelligence of the learned models, we present ILIA algorithms (namely ILIA1, ILIA2, ILIA3, ILIA4 and ILIA5) capable of Incremental Learning in terms of Input Attributes. Using the ILIA algorithms, when new input attributes are introduced into the original problem, the existing neural network can be retained and a new sub-network is constructed and trained incrementally. The new sub-network and the old one are merged later to form a new network for the changed problem. In addition, ILIA algorithms have the ability to decide whether the new incoming input attributes are relevant to the output and consistent with the existing input attributes or not and suggest to accept or reject them. Experimental results show that the ILIA algorithms are efficient and effective both for the classification and regression problems