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An efficient eigenspace updating scheme for high-dimensional systems

100%

Gangl S., Mongus D., Žalik B.

International Journal of Applied Mathematics and Computer Science

2014

tom 24

nr 1

123-131

Systems based on principal component analysis have developed from exploratory data analysis in the past to current data processing applications which encode and decode vectors of data using a changing projection space (eigenspace). Linear systems, which need to be solved to obtain a constantly updated eigenspace, have increased significantly in their dimensions during this evolution. The basic scheme used for updating the eigenspace, however, has remained basically the same: (re)computing the eigenspace whenever the error exceeds a predefined threshold. In this paper we propose a computationally efficient eigenspace updating scheme, which specifically supports high-dimensional systems from any domain. The key principle is a prior selection of the vectors used to update the eigenspace in combination with an optimized eigenspace computation. The presented theoretical analysis proves the superior reconstruction capability of the introduced scheme, and further provides an estimate of the achievable compression ratios.

Average convergence rate of the first return time

75%

Choe G. H., Kim D. H.

Colloquium Mathematicum

2000

tom 84/85

nr 1

159-171

The convergence rate of the expectation of the logarithm of the first return time $R_{n}$, after being properly normalized, is investigated for ergodic Markov chains. I. Kontoyiannis showed that for any β > 0 we have $log[R_{n}(x)P_{n}(x)] =o(n^{β})$ a.s. for aperiodic cases and A. J. Wyner proved that for any ε >0 we have $-(1 + ε)log n ≤ log[R_{n}(x)P_{n}(x)] ≤ loglog n$ eventually, a.s., where $P_{n}(x)$ is the probability of the initial n-block in x. In this paper we prove that $ E[log R_{(L,S)} - (L-1)h]$ converges to a constant depending only on the process where $R_{(L,S)}$ is the modified first return time with block length L and gap size S. In the last section a formula is proposed for measuring entropy sharply; it may detect periodicity of the process.

Ograniczanie wyników

1 Colloquium Mathematicum

1 International Journal of Applied Mathematics and Computer Science

1 Choe G. H.

1 Gangl S.

1 Kim D. H.

1 Mongus D.

1 Žalik B.

1 2014

1 2000

Wyniki wyszukiwania

An efficient eigenspace updating scheme for high-dimensional systems

Average convergence rate of the first return time