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RKHS subspace domain adaption via minimum distribution gap

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Abstract

Subspace learning of Reproducing Kernel Hilbert Space (RKHS) is most popular among domain adaption applications. The key goal is to embed the source and target domain samples into a common RKHS subspace where their distributions could match better. However, most existing domain adaption measures are either based on the first-order statistics that can’t accurately qualify the difference of distributions for non-Guassian distributions or complicated co-variance matrix that is difficult to be used and optimized. In this paper, we propose a neat and effective RKHS subspace domain adaption measure: Minimum Distribution Gap (MDG), where the rigorous mathematical formula can be derived to learn the weighting matrix of the optimized orthogonal Hilbert subspace basis via the Lagrange Multiplier Method. To show the efficiency of the proposed MDG measure, extensive numerical experiments with different datasets have been performed and the comparisons with four other state-of-the-art algorithms in the literature show that the proposed MDG measure is very promising.

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Notes

  1. https://github.com/jindongwang/transferlearning/tree/master/data.

  2. http://www.daviddlewis.com/resources/testcollections/reuters21578/.

  3. http://yann.lecun.com/exdb/mnist/index.html.

  4. http://www-i6.informatik.rwth-aachen.de/.

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Authors and Affiliations

  1. Sun Yat-Sen University, Guangzhou, China

    Yanzhen Qiu, Chuangfeng Zhang, Chenkui Xiong, Zhengming Ma & Shaolin Liao

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  1. Yanzhen Qiu
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  2. Chuangfeng Zhang
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  3. Chenkui Xiong
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  4. Zhengming Ma
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  5. Shaolin Liao
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Correspondence to Zhengming Ma or Shaolin Liao.

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Appendix A: Identical random variables

Appendix A: Identical random variables

Two second-order moment random variables are identical if and only if their statistical mean square error is zero,

$$\begin{aligned} E\left[ \left| Y-Y' \right| ^{2}\right] =0 \Leftrightarrow Y = Y'. \end{aligned}$$
(22)

To demonstrate this, the variance of Eq. (22) can be expressed as follows

$$\begin{aligned} E\left[ \left| Y-Y' \right| ^{2}\right] = \int \int _{\Omega (Y, Y')} \left( y-y'\right) ^2 p(y, y') dydy'. \end{aligned}$$
(23)

Because both \(\left( y-y'\right) ^2\) and \( p(y, y')\) are semi-definite or non-negative, Eq. (23) is zero when one of the following two conditions are met for all points in the probability domain \(\Omega \),

$$\begin{aligned} \left\{ \begin{matrix} \left( y-y'\right) ^2 = 0; \\ p(y, y') = 0. \end{matrix} \right. \end{aligned}$$
(24)

It can be shown that Eq. (24) is equivalent to the joint probability \(p(y, y') = f(y) \delta (y-y')\),

$$\begin{aligned}{} & {} p(y) = \int _{y'} p(y, y') dy' = \int _{y'} f(y) \delta (y-y') dy' = f(y), \nonumber \\{} & {} p(y') = \int _{y} p(y, y')dy = \int _{y} f(y) \delta (y-y')dy = f(y), \end{aligned}$$
(25)

from which the marginal probabilities of Y and \(Y'\) are identical and Eq. 22 is proved.

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Qiu, Y., Zhang, C., Xiong, C. et al. RKHS subspace domain adaption via minimum distribution gap. Pattern Anal Applic 26, 1425–1439 (2023). https://doi.org/10.1007/s10044-023-01170-y

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