Outer-product Matrix Research Articles

Mathematical aspects of outer-product asynchronous content-addressable memories

We have given evidence by mathematical analysis and by example that the construction of CAM's as quasineural networks based on the twin principles of an outer-product weight matrix and a random asynchronous single-neuron dynamics encounters two obstacles to good performance which appear to be inherent. It is desirable to have the stored memory vectors of a CAM as mutually far apart as possible in order to have unambiguous retrieval with as large a fraction of initial errors (minrad) as possible. For a given number,m, of memory vectors to be stored this requires that their dimension,n, be larger thanm and that the memory vectors be nearly mutually orthogonal (except for complementary pairs). In HOCAM's, it does not seem possible to have both a large minrad and an efficient ratiom/n. Attempts to increasem/n are likely to introduce extraneous fixed-points which reduce minrad appreciably. We have demonstrated this phenomenon in several cases for a particular mode of constructing CAM's of arbitrary size which have a desirable spacing between memory vectors. We conjecture that it is present also in HOCAM's having a random selection of memory vectors. (A mathematical proof of this conjecture now seems possible.) This may account for the rather pessimistic results on capacity obtained by mathematical analysis here and in Cottrell (1988), by a probabilistic analysis in Posner (1987) and by simulation in Hopfield (1982). Further, in Cottrell (1988) there is evidence that outer-product weights are near optimal with respect to minrad, so that otherW may not improve matters. We have left to another paper a study of other approaches to content-addressable memories of which we are aware, but which are not focused on asynchronous dynamics; e.g. computer CAM's as in Kohonen (1977) and biological memory models as in Little (1974); Palm (1980) and Little and Shaw (1978). We have not considered the learning, or adaptive, aspects of CAM's. However, insofar as learning is Hebbian and leads to outer-product weights, our analysis has implications for the effectiveness of learned weights, as may be inferred from our results on ambiguous retrieval.

Digital Optical Matrix Multiplication Based On A Systolic Outer-Product Method

A hybrid optical architecture for digital matrix-matrix multiplication is discussed. The proposed system requires less hardware than outer-product optical matrix multipliers and offers a higher processing speed than systolic array matrix multipliers. The throughput and hardware requirements of the proposed architecture are discussed with reference to the availability of various electro-optic devices. The performance of the proposed system is compared to existing systolic array optical matrix multipliers and to outer-product-based optical matrix processors. The accuracy and error rate of the proposed architecture are evaluated.

Comparison of methods of matching protein structures

Two fast methods of superposing two sets of atomic coordinates by least-squares refinement are described and related to two earlier fast methods. A Newton method is applied to rotations of a 3 x 3 outer product matrix used previously by Ferro & Hermans [Acta Cryst. (1977), A33, 345-347] and by McLachlan [Acta Cryst. (1972), A28, 656-657]. Three of the methods work better if one molecule has its inertial matrix aligned with xyz. A Newton-Gauss method that rotates the coordinates can converge rapidly after a rough orientation using three strategic atoms. The average superposition takes about 0.003 s on a Cyber 175 with the best method, rotations about the xyz axes in turn. Experience with reliability is reported for large residuals.