Deep banach space kernels
Authors: Mrityunjay Bhardwaj
Venue: ICLR 2022 Submitted (2022)
Keywords: RKBS
RKHS
concatenated kernel learning
representation learning
deep learning
MLMKL
Deep Gaussian Processes
gaussian processes
kernel machines

Abstract: The recent success of deep learning has encouraged many researchers to explore the deep/concatenated variants of classical kernel methods. Some of which includes MLMKL, DGP and DKL. Although, These methods have proven to be quite useful in various real-world settings. They still suffer from the limitations of only utilizing kernels from Hilbert spaces. In this paper, we address these shortcomings by introducing a new class of concatenated kernel learning methods that use the kernels from the reproducing kernel Banach spaces(RKBSs). These spaces turned out to be one of the most general spaces where a reproducing Kernel exists. We propose a framework of construction for these Deep RKBS models and then provide a representer theorem for regularized learning problems. We also describe the relationship with its deep RKHS variant as well as standard Deep Gaussian Processes. In the end, we construct and implement a two-layer deep RKBS model and demonstrate it on a range of machine learning tasks.

Under review as a conference paper at ICLR 2021
DEEP BANACH SPACE KERNELS
Anonymous authors Paper under double-blind review
ABSTRACT
The recent success of deep learning has encouraged many researchers to explore the deep/concatenateed variants of classical kernel methods. Some of which in- cludes MLMKL,, DGP and DKL. These methods have proven to he quite helpful in various real-world settings. But so far, have only been utilizing kernels from Hilbert spaces, which has their own limitations. In this paper, we address these shortcomings by introducing new concatenated kernel learning approach that uses the kernels from the reproducing kernel Banach Spaces(RKBSs) instead. We present general framework of construction for these Deep RKBS models and then provide representer theorem for regularized learning problems. We also deseribe the relationship with its deep RKHS variant as well as standard deep Gaussian processes. In the end, we construct and implement two-layer deep RKBS model and demonstrate it on a range of machine learning tasks.
INTRODUCTION
In recent years, promising new variants of kernel learning methods, namely deep ker-nel learning and multi-layer-MKL (MLMKL) algorithms have been developed. These concatenated kernel learn- ing approach includes, see e.g. Cho & Saul (2009); Damianou & Lawrence 2; ZZuaang et al. (201T). Although they have proven be very successful regression and classification tasks. we are still only utilizing kernel from Hilbert Space. There has be a recent interest in studying space which similar to RKHS but contains alot more functions i.e, reproducing kernel Banach space (RKBS). In this paper, will define general framework of concatenated RKBSs as well as pro- vide a concatenated representer theorem which can be used to develop a generalized variant of all the concatenated kernel learning (Bohn et al. (2019)) methods described earlier. We will now give brief overview of our paper.
In Section 2, we briefly review the interpolation problem in RKBS and discuss how we recast this to RKHS. In Section 3, we introduce the optimal concatenated approximation problem for arbitrary loss function and regularizers. We first present general framework to construct deep/concatenated RKBS and then propose a representer theorem for this problem in multi-layer case. we then examine the concrete example of two-layer RKBS kernel. Furthermore, method to classical concatenated kernel learning in RKHSs which describe the connect with multiple array of methods such as DeepGP, DKL, MLMKL etc. section we look at a simple experiment and compare our Deep RKBS kernels with Deep RKHS ones. we also briefly talk about our library we have built to easily perform experimentation with Deep RKBS Kernels. Finally, we conclude with some related works and a summary of our paper.
discuss the relation of our
REPRODUCING KERNEL BANACH SPACES
In this section, we briefly define RKBS and its reproducing kernel then we will present the repre- senter theorem for minimal norm interpolation problem. for more comprehensive overview we refer the readers to Lin et al. (019b)).
A reproducing kernel Banach space B on prescribed nonempty set X. is Banach space of certain functions on X such that every point evaluation functional d,, æ E X on continuous, that is, there exists positive constant C, such that
55 (z(( [f())) Callslls for all f E B.

---

Under review as a conference paper at ICLR 2021
this definition is the natural generalization of the classical definition of RKHS. Now before moving forward, we will present another definition as well as general framework construction for RKBS which we will be using throughout this paper.
(Construction of RKBS using feature maps) Let WX(),, W be two Banach spaces, and (1) be continuous bilinear form on WX1) x WM(2), Suppose there exist two nonempty sets 12 and Q(2), and mappings 4(1) (211) w(l), Φ(²), Q(2) 1x(2) such that with
spect to the bilinear form span ((1) is dense in wi)) span ((2) (f)(2)) is dense in W(?), We construct
B = {fo(x) = (5(')) www. W(2), x E 2
(2.1)
and
B = ( = (u, (12) (v)) : E W y E !
(2.2)
with norm [e ||lllaa and || Boo == |ullwa respectively.
Theorem 2.1 Let (0)) and B be constructed as above. Then with the bilinear form on BO) x BB2)
(u) B0) x x (u, o) xwa for all Su E B()) and all Iu E (3(2)
(2.3)
B()) is RKBS on (2(1) with the adjoint RKBS B(2), on Ω222) Moreover,
((x, 3) == (z)))))) 4 (y)) wi w
222), y E 12(2)
(2.4)
a reproducing kerne for (1)) Using the reproducing property we can rewrite f E B() as:
f(x) = (f, K ((,,(( (zx, )) for all x E (22) and all f E B1,
(2.5)
where, KII,)) E B(2)
2.1
INTERPOLATION PROBLEM IN RKBS
The minimal norm interpolation problem looks for the minimizer
fult arg inf [Ifllgo where Sx.t = { B()) f(x,) = j [Nm) fESx.t
(2.6)
Vinf := arg inf ||u||wes vEVet
(2.7)
with
= foe w(2), (xz)) u) =t,, , E Nm
Vx, =
(2.8)
Theorem 2.2 (Representer Theorem) Assume the same assumptions as in Theorem 2.1 ad- dition, suppose that W22) is reflexive, strictly convex and Gâteaux differentiable, and the set (0()) x j E }m) is linearly independent in W(1), Then the minimal norm interpolation prob- lem 2.7 has a unique solution Vint E W(() and it satisfies
g (Vin)) E ((8(1(())))+
(2.9)
( = 229, w(2) = W(1) and (2 reduced to a Hilbert Space then we can recover the classical representer Ancorem(Schölkee et al. (2002) for minimal norm interpolation in RKHS.
A REPRESENTER THEOREM FOR CONCATENATED KERNEL LEARNING IN BANACH SPACE
In this section, we will be deriving a concatenated RKBS representer theorem for an arbitrary number L E N of concatenations of vector-valued RKBS spaces. For more comprehensive treatment of vector- vector-valued RKBS we refer the readers to Chen et al. (2019). Zhang Zhang (2013) and Lin et al. (2019) respectively.
2

---

Under review as a conference paper at ICLR 2021
Let Bry..., Br. be reproducing kernel Banach Spaces with finite dimensional domain 2 and ranges Rí S R4 with dj E N for 1 = 1,... L such that R₁ S 9 for | = 2, L. = Ω and R₁ S C.
We consider learning function from a prescribed set of finite sampling data
1 = (Z t,): : N] _Ω C
Let furthermore C : 2 + [0, ooj be arbitrary continuous and convex loss function and let e,...., 0r 00, oc) be continuous, convex and strictly monotonically increasing functions. For each arbitrary function fi B vI 1, ...,, we set
J(f ... ft) Σ
fz())) Σ 1=1
(3.10)
and our objective is:
of.. xB.... J(1 o o ft)
(3.11)
Since, VI L, L....
fi(x!9) = Lon (2o10) (o w xw,n,, 100 E st or span( yy E (
Thus 3.1I to
(z:). C C(tss ( ve) (xx) W
Uin arginf inf
eww w..,w! "ww""...,
e (((lolll
(3.12)
Even if is a convex loss function 3.11 is still a highly non-lincar optimization problem. we therefore assume that, there are w N optimal composite functions which minimizes J. We Let Fi f fhr be a set of all the optimal functions in B; for all I L.......
Theorem 3.1 In addition to above assumption, suppose that w(2 reflexive, strictly convex and Gâteaux differentiable for all 1 I,...,,, and the set (12) (x²)) 23 Dr, Vj N is linearly independent in w("). Provided that FÝ is non-empty for all 1 = 1,... L then, there exist set of w minimizers where each Utint W!(2) 1....L satisfies
((vinnt E (Vir,))
where,
Vi (m W(?) (c 1, xw =0 E E N)
Proof: Suppose there exist minimizer ft E Fi E Bi then we create data set D₁:
Di (1), : NM
By theorem 2.2 there exists unique solution Van E W2 for the min norm interpolation prob- lem with the samples Dj and it satisfy that G(Vraat) E (Vi,,) It follows that fieiuf (z()) viat) wi"h xwee interpolates the sample data Dj and for all w!?)
|| 1l & ||olllm
Thus, (xin fu- (x)).
3

---

Under review as a conference paper at ICLR 2021
We can extend this result for all [ = 1,... L such that:
(( E (Viz,u)*
long F7 E Bi is non-empty. The proof is complete. 0
3.1 EXAMPLE
In this section, will define concrete example of layer RKBS using our framework of con- struction. Suppose, our inner vector-valued RKBS B2 is endowed by 4 norm and our outer vector- valued RKBSB, is endowed by Ly norm where, < p < sc. using the representer theorem (see, Lin et al. (019a) and Chen et al. (2019) we can rewrite a vector-valued fz(-) E B2 as:
d2
fa(1) =
Kg K2 (ai, ) Ck2
1=1
#2==1
for certain coefficients Cr E IR Furthermore, we have that fi E Sf span {Ki (f2 (a:;).) i 1,..., J} and thus
N 10
Σ
aiKi
Σ
Ci K2 (1is, 2z5) exa.
3=1
im
k==1
The concatenated function is then given by /1-)) == flof(()) = Ei== asK (x),.) with the following deep RKBS kernel
d2 Σ 1=
d2
K(x,, v) = K
K2 (I,, x)) Ckaz
Ci K2 (x,, 8) Cka
(3.13)
k2=1
Therefore, instead of considering the anfnite-dimensionall optimizatinn problem of finding fi B and f2 E B2 that minimize
J f2) = - CVr (2;))))) + (111ll(,)) O2 (152llee))
we can restrict ourselves to finding the N + d2 coefficients aj. Ci for i,j = 1,..., N and 11....;bb d2
la
3.2 RELATION TO DEEP KERNEL LEARNING
As briefly mentioned in 2.2 we can recover the classical representer theorem for interpolation prob- lem the detailed proof is given in the appendix ). If we assume B.. to he Hi,..., Hr, where Hi VI 1, L. set of RKHS of vector-valued function then we can find of minimizers fi...., fL fi E H, of 3.10 which satisfies that for all = 1,...L:
E span Ki(f41 o... fL( )ek 1,n..aandky 1,....d. de
(3.14)
where K; denotes the reproducing kernel of H4 and exs Rd is the ki-th unit vector.
which exactly the representer theorem for concatenated kernel learning in hilbert space as de- scribed in Bohn et al. (201)), if we define probability measure on each Hi above then we can recover the Deep gaussian processes. which means that the methods such as Cho & Saul (2009), Damianou & Lawrencel (201)), Zhuang et al. (2011). Wilson al. (2016) are the special cases of our framework. will be interesting to see the RKBS equivalent of these methods using our framework in future.

---

Under review as a conference paper at ICLR 2021
4 EXPERIMENT
Although, we are leaving more thorough experimentation on deep RKBS kernels for future works. Here, for the sake of completeness we simply perform some preliminary experiments syn- thetic data which is generated as follows:
We choose 5 [0, 1]²
h1 9 +
(1 (0.1 Jr y1) (2(2, if x y > 20
12 : Ω -2-RR R
else
We define our deep RKBS kernel similar to 3.13 where we use the following kemels:
K2(x, y) = - |x - yl
which is the reproducing kernel for C((O.1)) and for K1:
K1 (I, v) == min (ar, (z.y) y) - t my 0 <t<]
which is the generalization of Brownian bridge kernel. now, for deep RKHS we take the polynomial kernel of degree 2 as inner kernel and Matern kernel as its outer kernel.
We compare deep RKHS and deep RKBS in regularized regression setting which is regularized by 12, the results are presented in table below where the performance is calculated by averaging the MSE loss of 50 trials.
avg. MSE
Deep RKHS 0.00304
Deep RKBS
0.00112
The difference isn't significant, deep RKBS does slightly outperform deep RKHS in this setting. Since, Deep RKHS a subset of Deep RKBS we can see the practical significance this method. Although more thorough empirical study is required.
For this experiment we have also implemented a small library which contains implementation of multiple RKBS kernels, the library also allows to easily Create deep Kernels by simply specifying the kernel and its ranges.
5 RELATED WORKS
While concatenated kernel learning methods such as deepGP have been a well-established line of research, deep RKBS is still in this infency. although there some very promising recent works this direction such as Bartolucci et al.] (2021). but to the best authors knowledge, concatenated RKBS kernel learning is not been presented in the literature far.
CONCLUSION
We proposed general framework to construct deep RKBS Kernels, which give rise tion concatenated kernel learning literature. we started by presenting the general RKBS and its reproducing kernel. then define presented a representer theorem along with some examples of deep RKBS, we then derived the con- nection between the classical concatenated kernel learning methods which includes DKL, MLMKL, DeepGP among other, we finished our paper by describing our new Library to do experiments with Deep RKBS as well as showed some preliminary results of comparing deep RKHS and deep RKBS.
direc-
class of concatenated kernel learning RKBS spaces and
ACKNOWLEDGMENTS
We would like to thank ICLR committe for organising ICLR CoSubmitting Summer 2022 this year.
5

---

Under review as a conference paper at ICLR 2021
REFERENCES
Francesca Bartolucci, Emnesto de Vito, Lorenzo Rosasco, and Stefano Vigogna. Understanding neural networks with reproducing kernel banach spaces. ArXiv, abs/2109.09710, 2021.
Bastian Bohn, Christian Rieger, and Michael Griebel. A Representer Theorem for Deep Ker- nel Learning. Technical report, 2019. URL
html.
Liangzhi Chen, Haizhang Zhang, and Jun Zhang. valued Reproducing Kernel Banach Spaces with Group Lasso Norms* Technical report, 2019.
Youngmin Cho and Lawrence K. Saul. Kernel methods for deep learning. In NIPS, 2009.
Andreas C. Damianou and Neil Lawrence. Deep gaussian processes. ArXiv, 0tss/1211.0335 2013.
Rongrong Lin, Guohui Song, and Haizhang Zhang. Multi-task Learning in Vector-valued Repro- ducing Kernel Banach Spaces with the 1 Norm. Technical report, 2019a.
Rongrong Lin, Haizhang Zhang, and Jun Zhang. On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions. Technical report, 2019b.
Bernhard Schölkopf, Alex Smola, Alexander Smola, and A Smola. Support vector machines and kernel algorithms. Encyclopedia of Biostatistics, 5328-5335 (2005), 04 2002.
Andrew Gordon Wilson, Zhiting Hu, Ruslan Salakhutdinov, and Eric Xing. Deep kernel learning. ArXiv, abs/1511 1.0222,, 2016.
Haizhang Zhang and Jun Zhang. veatur- reproducing kernel Banach spaces with applications to multi-task learning. Technical Report 2, 2013.
Jinfeng Zhuang, Ivor Wai-Hung Tsang, and Steven C. H. Hoi. Two-layer multiple kernel learning. In AISTATS, 2011.
A
APPENDIX
You may include other additional sections here.


Reviews:
[{"review_id": "aPFTobiH9Qh", "reviewer": "Reviewer_57b6", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "This submission looks more like a working draft rather than a conference paper. In particular,\n\n**There are many typos**, making the reading very difficult (I am highlighting a few of them, only on page 1):\n- which *have* their own limitation (abstract)\n- that uses *functions* from Reproducing Kernel Banach Spaces (abstract)\n- and *test* it on a range on ML tasks (abstract)\n- the acronyms MLMKL, DGP, DKL are never introduced (abstract and later)\n- *W*e are still (1st paragraph)\n- *kernel from Hilbert Space* makes little sense (1st paragraph)\n- There has *been* (1st paragraph)\n- *W*e then examine (2nd paragraph)\n- *two layer RKBS kernel* makes little sense (2nd paragraph)\n- the connect*ion* with multiple (2nd paragraph)\n- *W*e also talk about *the* library (2nd paragraph)\n- the bibliography entry [Scholkopf and Smola] seems wrong\n\n**The maths are not rigorous**, to the extent they are almost incorrect:\n- the introduction of RKBSs is very confusing, as the proposed definition looks more like the RKHS definition to me, see e.g., Definition 4.18 in \"Support Vector Machines\" by Steinwart and Christmann, 2008\n- Theorems 2.1 and 2.2, that are not part of the contribution, should be explicitly linked to the paper they come from\n- undefined notation in eq. (2.9)\n- I would like to point out the paper \"Autoencoding any Data through Kernel Autoencoders\" by Laforgue et al., 2019 which considers a similar framework as the one introduced at the top of page 3. A Representer Theorem is also proved therein, as in [Bohn et al. 2019], except that it applies to potentially infinite dimensional output Hilbert spaces\n- the introduction of the $\\Theta_l$ is not understandable\n- In section 3.2 the proof is said to be given in the Appendix, but the latter is empty\n\n**The contribution is very limited**:\n- the Representer Theorem (RT) being mainly due to orthogonality properties, it is not surprising to recover it for a class of functions that share this property with RKHSs\n- I cannot see any novel idea in the proofs, so proving another RT seems not a sufficient contribution for acceptance\n- I cannot find any motivation for the presented results\n- the authors themselves acknowledge that their experimental contribution is very limited", "summary_of_the_paper": "This paper proves a Representer Theorem for the composition of functions from Reproducing Kernel Banach Spaces (RKBSs).", "summary_of_the_review": "This submission looks more like a working draft rather than a conference paper.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "XscGfGlCaxH", "reviewer": "Reviewer_5Wv8", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "There isn't a clear division between a background and contributions section, so it is difficult to see the distinction between the background and the contributions.  Nonetheless, it appears that the only contributions are to notice that deep kernel learning can be understood within their framework (Sec. 3.2) and to do a toy experiment on 2D data (Sec. 4).  This is simply not sufficient for a paper at a major machine learning conference, so I am forced to recommend rejection.", "summary_of_the_paper": "The paper proposes to use kernels from a reproducing kernel Banach space, instead of a reproducing Hilbert space in tasks such as deep kernel learning.", "summary_of_the_review": "Very limited theoretical and empirical results.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "M7PZ2k0ddP", "reviewer": "Reviewer_kbJr", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "While I like the idea underlying this paper I found the paper somewhat unsatisfying.  Specifically:\n\n- unclear/lack of attribution: particularly for the background material, several theorems and definitions are taken almost word-for-word from Lin et al, 2019b.  For example the definition of RKBS corresponds to definition 1.1 in Lin et al, the feature-map construction is the boxed text on page 4 of Lin et al, and theorem 2.1 is the same in both this and Lin et al.  While I understand that re-wording is often tedious, at the very least a simple parenthetical citation (for example \"Theorem 2.1 (see theorem 2.1 in Lin et al): ...\") seems to be called for.\n\n- Related to the above, theorem 2.2 uses notation $A^\\vdash$ that isn't (as far as I can tell) introduced at all in the submission, which makes understanding somewhat difficult.\n\n- Continuing on the question of notation, what is $e_{k_2}$ in section 3.1?  I'm working on the assumption that this is some form of basis vector set (after all this is vector-valued regression) but this needs to be clarified.\n\n- Perhaps I am wrong, but (3.13) looks like it would, in practice, act like an RKBS with a N hyper-parameters and \\mathcal{O} (N^2) computation time.  I am curious how this affects the performance of an algorithm using such a kernel - in my experience kernels this complicated can have a severe effect on training and evaluation times, particularly for large datasets!\n\n- The experimental section is unsatisfactory.  A single experiment is described, but the size of the training set is not specified.  50 trials are completed, but no error bars are included.  A link to code is given but the algorithm is not described and training issues (training times etc) are not discussed.\n\nTo be clear I think this avenue of research is fascinating and potentially important, but I feel that the deficits of the paper need to be addressed first.", "summary_of_the_paper": "The paper presents a means of constructing deep (concatenated) reproducing kernel Banach space kernels.  The main contribution appears to be a representor theory for concatenated Banach space kernels.  Some experimental results are provided to accompany this result.", "summary_of_the_review": "- The premise is interesting.\n- The clarity of attribution in the background section needs to be worked on.\n- \"Niche\" notations used but not defined.\n- Experimental results on a single dataset without important details (training set size, training times, details of training algorithm etc0.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper develops kernel functions in Banach spaces. However the results seem to be preliminary and further development is needed before
the manuscript can be published. Reviewers point out several errors and also author/authors have graciuously agree with the suggestion 
that they will incorporate all the feedback in future submissions.
Decision: Reject