论文标题
验证$ k $ - 征收而无需计算$ k $ -pounds
Verifying $k$-Contraction without Computing $k$-Compounds
论文作者
论文摘要
复合矩阵已经在许多科学领域中找到了应用，包括系统和控制理论。特别是，$ k $ - 征收的足够条件是，雅各比式的$ k $ - addive addive goblesound的对数标准（也称为矩阵度量）均匀为负。但是，这可能很难在实践中检查，因为$ n \ times n $矩阵的$ k $ - addive化合物具有尺寸$ \ binom {n} {k} {k} \ times \ binom \ binom {n} {k} {k} $。对于$ n \ times n $矩阵$ a $，我们证明了$ k $和$ a $ a $的$ k $和$ a $ a $的双重关系。我们使用这种双重性关系来得出$ k $ contraction的足够条件，该条件不需要计算任何$ k $ compounds。 我们通过得出$ k $ - 二维Hopfield网络的$ k $ contraction的足够条件来证明我们的结果，该网络不需要计算任何化合物。特别是，对于$ k = 2 $，这种足够的条件意味着网络是$ 2 $ - 收缩，这意味着一个强大的渐近属性：网络的每个有界解决方案都会收敛到平衡点，这可能不是唯一的。例如，当使用Hopfield网络作为一个将模式作为动力学平衡点存储的关联内存时，这是相关的。
Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for $k$-contraction is that a logarithmic norm (also called matrix measure) of the $k$-additive compound of the Jacobian is uniformly negative. However, this may be difficult to check in practice because the $k$-additive compound of an $n\times n$ matrix has dimensions $\binom{n}{k}\times \binom{n}{k}$. For an $n\times n$ matrix $A$, we prove a duality relation between the $k$ and $(n-k)$ compounds of $A$. We use this duality relation to derive a sufficient condition for $k$-contraction that does not require the computation of any $k$-compounds. We demonstrate our results by deriving a sufficient condition for $k$-contraction of an $n$-dimensional Hopfield network that does not require to compute any compounds. In particular, for $k=2$ this sufficient condition implies that the network is $2$-contracting and this implies a strong asymptotic property: every bounded solution of the network converges to an equilibrium point, that may not be unique. This is relevant, for example, when using the Hopfield network as an associative memory that stores patterns as equilibrium points of the dynamics.
