Given an m n matrix M with m > n, it is shown that there exists a permutation FI and an integer k such that the QR factorization MYI= Q(Ak ckBk) reveals the numerical rank of M: the k k upper-triangular matrix Ak is well conditioned, IlCkll2 is small, and Bk is linearly dependent on Ak with coefficients bounded by a low-degree polynomial in n. Existing rank-revealing QR (RRQR) algorithms are related to such factorizations and two algorithms are presented for computing them. The new algorithms are nearly as efficient as QR with column pivoting for most problems and take O (ran2) floating-point operations in the worst case.

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Abstract. In this paper we generalize the hierarchically semiseparable (HSS) representations and propose some fast algorithms for HSS matrices. We provide a new linear complexity U LV T factorization algorithm for symmetric positive definite HSS matrices with small off-diagonal ranks. The corresponding factors can be used to solve compact HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the solver. We also present fast algorithms including new HSS structure generation, HSS form Cholesky factorization, and model compression. These algorithms are useful for problems where off-diagonal blocks have small numerical ranks.

Summary Cortico-hippocampal interactions during sleep are believed to reorganize neural circuits in support of memory consolidation. However, spike timing relationships across cortico-hippocampal networks, which are key determinants of synaptic changes, are not well understood. Here we show that cells in prefrontal cortex fire consistently within 100 ms after hippocampal cells in naturally sleeping animals. This provides evidence at the single cell-pair level for highly consistent directional interactions between these areas within the window of plasticity. Moreover, these interactions are state-dependent: they are driven by hippocampal sharp-wave/ripple (SWR) bursts in slow-wave sleep (SWS) and are sharply reduced during REM sleep. Finally, prefrontal responses are non-linear: as the strength of hippocampal bursts rises, short-latency prefrontal responses are augmented by increased spindle band activity and a secondary peak ~100 ms later. These findings suggest that SWR events serve as an atomic unit of hippocampal-prefrontal communication during SWS, and that the coupling between prefrontal and hippocampal areas is highly attenuated during REM sleep.