978-1-4799-0333-7/13/$31.00 © 2013 IEEE 453 Different ZFs Lead to Different Nets: Examples of Zhang Generalized Inverse Dongsheng Guo, Chen Peng, Long Jin, Yingbiao Ling and Yunong Zhang School of Information Science and Technology, Sun Yat-sen University Guangzhou 510006, Guangdong, China Email: gdongsh2008@126.com; zhynong@mail.sysu.edu.cn; ynzhang@ieee.org Abstract—This paper demonstrates the flexibility of Z-type methodology for generating multiple solution-models of time- varying problems. As a case study with examples, we investigate the solution of time-varying generalized inverse (termed Zhang generalized inverse, ZGI). Specifically, to solve for time-varying left generalized inverse (TVLGI or termed Zhang left generalized inverse, ZLGI), five different effective Z-type models are derived by using different Zhang functions (ZFs) as source of derivations and employing the Z-type model design method. In addition, a clear and direct link between model A and the Getz-Marsden (G-M) dynamic system is discovered; and model B with a linear activation function (AF) array has global exponential convergence. Furthermore, two different AFs are adopted for comparisons and verifications. Keywords—Z-type models; Zhang problem solving; Generalized inverse; Getz-Marsden dynamic system; Neural nets I. I NTRODUCTION For a decade, Z-type models (i.e., Zhang neural nets) have been proposed for various time-varying problems solving [1]– [6]. Differing from the traditional G-type models (i.e., gradient neural nets) using scalar-valued nonnegative or lower-bounded energy functions [1], [6], Z-type models are designed based on vector/matrix-valued indefinite error functions. Generally speaking, Z-type models are constructed in implicit dynamics with some exceptionally in explicit dynamics (of which the latter are usually associated with G-type models). While G- type models always have time-lag problems and thus always have some degrees of estimation errors, Z-type models have great advantage in that their solution errors usually converge to zero exponentially. Recently, the notion of Zhang problem solving (ZPS) has been proposed, which refers to the time-varying problem solving related to division (or, generalized division), where the divisor (or, generalized divisor) may vary and pass through zero. It is therefore a challenging yet promising future re- search direction. Numerous problems solving can be classified as ZPS. For example, time-varying matrix inversion [i.e., A(t)X (t)= I ] is potentially a ZPS, especially when the determinant of A(t) varies and passes through zero at some time instant(s) t ∈ [0, +∞). Besides, time-varying reciprocal computation [2], time-varying linear system solving [3], time- varying matrix inversion [4] and time-varying inverse square root solving [5] can all be viewed as special cases of ZPS. In studying the Z-type methodology, we have found that many different effective Z-type models can be derived to solve a particular problem. The key is to use different Zhang functions (ZFs) as source of derivations. In demonstrating this, we use generalized inversion (GI), more specifically, time- varying left GI (time-varying LGI, TVLGI), as an application, which is one of the basic issues encountered in a variety of science and engineering fields [7]–[11]. Note that, in this paper, we call time-varying GI treated as ZPS using Z-type models as Zhang generalized inversion (ZGI), and similarly TVLGI treated as ZPS using Z-type models as Zhang left generalized inversion (ZLGI). TABLE I. ACRONYMS USED IN THE PAPER AF(s) activation function(s) GI generalized inverse/inversion LGI left generalized inverse/inversion RGI right generalized inverse/inversion TVLGI time-varying left generalized inversion ZF(s) Zhang function(s) ZGI Zhang generalized inverse/inversion ZLGI Zhang left generalized inverse/inversion ZPS Zhang problem(s) solving For better understanding, Table I lists all the acronyms used in this paper. Besides, the major contributions of this paper can be listed as follows. • Five different effective Z-type models (i.e., models A through E) are derived for ZLGI by using different ZFs and employing the Z-type model design method. • A clear link between model A and the G-M dynamic system [12] is discovered. • Model B with linear activation functions (AFs) has global exponential convergence. • Two different AFs are adopted for further comparisons and verifications. • This generating paradigm of multiple solution-models shows great flexibility of the Z-type methodology. II. PROBLEM FORMULATION In this section, we introduce the concept of GI, especially LGI, and then present the problem formulation of ZLGI (for further investigation). A. Static Left Generalized Inverse In mathematics, a GI A + of a matrix A ∈ R m×n is a matrix that has some properties of the inverse matrix but not necessarily all of them [13]. One of the most famous examples