Paper:
Domain adaptation focuses on the reuse of supervised learning models in a new context. Prominent applications can be found in robotics, image processing or web mining. In these areas, learning scenarios change by nature, but often remain related and motivate the reuse of existing supervised models. While the majority of symmetric and asymmetric domain adaptation algorithms utilize all available source and target domain data, we show that efficient domain adaptation requires only a substantially smaller subset from both domains. This makes it more suitable for real-world scenarios where target domain data is rare. The presented approach finds a target subspace representation for source and target data to address domain differences by orthogonal basis transfer. By employing a low-rank approximation, the approach remains low in computational time. The presented idea is evaluated in typical domain adaptation tasks with standard benchmark data.

Review:
In this work, the authors improve the work (Raab & Schleif, 2018) by (1) reducing the computational complexity, (2) neglecting the sample size requirement, and (3) achieving a low-rank projection through Nystrom approximation. More specifically, the feature dimensionality is reduced by using only s biggest eigenvalues and eigenvectors, and the sample size is coordinated through Nystrom approximation. Class-wise sampling is used for the source, and uniform sampling is used for the target. Experimental studies on three datasets have been done.
This paper should be rejected because (1) the paper lacks important latest references on domain adaptation, (2) the paper misuses the notations that makes the paper is not easy to follow, (3) the algorithm is not well justified either by theory or experiments, and (4) the presentation should be further polished. Here are some detailed comments:
(1)	A lot of recent deep domain adaptation methods are missing. These deep works achieve the state-of-the-art results on many transfer benchmark tasks. Without comparison with them (the paper does not mention any deep works), it is very unconvincing to conclude the paper makes new contributions to the transfer community. 
(2)	In line 2 of page 5, the authors claim that BT assumes S_Z ~ S_X, which is not true. The key idea of BT is to construct new source data using target basis and source eigenvalues. Similarly, the authors make the same claim in section 4.1 for NBT, which is also not valid. 
(3)	The notations in eq. (12) and (13) are very misleading. Eq. (12) follows the notations of the first line in page 5, but in the first line below eq. (12), why R_X \in R^{d \times s}, S_X^2 \in R^{s \times s}? I understand that X_s is the low dimensional X, then X_s = XR_s with R_s is the dimensionality reduction matrix whose size is d \times s (using biggest s eigenvectors is fine). With this, eq. (13) is incorrect as X should be L_X*S_X*R_X^T, but not L_s*S_s*R_s^T. Moreover, it is also unclear why X is decomposed into product of two matrices in this work, is there any benefit of doing so for transfer purpose? 
(4)	In section 4.1, A_Z and A_X have exactly same form with X and Z, i.e., L_Z*S_Z*R_Z^T and L_X*S_X*R_X^T, please clarify. How eq. (14) X_s = \tide{L_X}*S_X come from? Is it the same as eq. (13)? 
(5)	The title highlights low-rank, but it is not very clear how low-rank matters in the proposed method. I do not find contents stating the low-rank property of the proposed algorithm in the main technical sections.  
(6)	Regarding section 4.2, what is the benefit of using class-wise sampling for the source? Have you tried to use uniform sampling for both the source and target domains?
(7)	The bound in eq.(16) is not very meaningful as s << n, m, d. Moreover, I am also not convinced by the claim this reduces the distribution differences, please give more theoretical justifications.
(8)	Regarding the experimental studies, why not use accuracy as many existing works do? The baselines are all subspace-based papers, and are out-of-the-date. The latest subspace papers, e.g., JGSA and MEDA, should be included. Moreover, deep methods are completely missing, which makes the empirical evaluation much less convincing. The improvements of NBT to BT are very marginal, 0.6. 
(9)	Some typos and unclear points (please further polish the paper): 
(a)	The last sentence of para 2, X and Z should be data, not features/
(b)	Para 3, it is unclear what are the implicit alignment and explicit alignment of domain distributions.
(c)	The first sentence in section 2, it should be homogeneous. 
(d)	Page 5 above eq. (15), it should be S_Z ~ S_X.
(e)	Section 4.2 the second line, it should be “in the data matrix”
(f)	The second last line of page 6, it should be inequality (16).
(g)	Some references miss page information.