Optimal projection method determination by Logdet Divergence and perturbed vonNeumann Divergence
 Hao Jiang^{1},
 WaiKi Ching^{2},
 Yushan Qiu^{3}Email author and
 XiaoQing Cheng^{4}
https://doi.org/10.1186/s1291801704790
© The Author(s) 2017
Published: 14 December 2017
Abstract
Background
Positive semidefiniteness is a critical property in kernel methods for Support Vector Machine (SVM) by which efficient solutions can be guaranteed through convex quadratic programming. However, a lot of similarity functions in applications do not produce positive semidefinite kernels.
Methods
We propose projection method by constructing projection matrix on indefinite kernels. As a generalization of the spectrum method (denoising method and flipping method), the projection method shows better or comparable performance comparing to the corresponding indefinite kernel methods on a number of real world data sets. Under the Bregman matrix divergence theory, we can find suggested optimal λ in projection method using unconstrained optimization in kernel learning. In this paper we focus on optimal λ determination, in the pursuit of precise optimal λ determination method in unconstrained optimization framework. We developed a perturbed vonNeumann divergence to measure kernel relationships.
Results
We compared optimal λ determination with Logdet Divergence and perturbed vonNeumann Divergence, aiming at finding better λ in projection method. Results on a number of real world data sets show that projection method with optimal λ by Logdet divergence demonstrate near optimal performance. And the perturbed vonNeumann Divergence can help determine a relatively better optimal projection method.
Conclusions
Projection method ia easy to use for dealing with indefinite kernels. And the parameter embedded in the method can be determined through unconstrained optimization under Bregman matrix divergence theory. This may provide a new way in kernel SVMs for varied objectives.
Keywords
Background
Support vector machines (SVMs), a supervised machine learning technique, have been introduced by Vapnik [1, 2]. In machine learning area, SVMs [3] are traditionally considered as one of the best algorithms in terms of structural risk minimization. Kernels in SVM work by data embedding in high dimensional feature space and one can construct an optimal separating hyperplane in this space [4]. Furthermore, kernel methods have wide applications in the field of bioinformatics. Authors in [5] have proposed incremental kernel ridge regression to predict soft tissue deformations after CMF surgery. In [6], researchers utilized the kernelbased linear discriminant analysis (LDA) method to address the problem of automatically tuning multiple kernel parameters. In order to address the nonlinear problem of nonnegative matrix factorization (NMF) and the seminonnegative problem of the existing kernel NMF methods, authors in [7] develop the nonlinear NMF based on a selfconstructed Mercer kernel which preserves the nonnegative constraints on both bases and coefficients in kernel feature space. Positive SemiDefiniteness (PSD) is crucial [8] for a kernel matrix in SVMs, which is required to guarantee the existence of a Reproducing Kernel Hilbert Space (RKHS). In RKHS, one can formulate a convex optimization problem to obtain an optimal solution. Sometimes however, similarity matrices generated for practical use cannot ensure such a PSD property. For example, in evaluation of pairwise similarity between DNA and protein sequences, popular functions like BLAST and Dynamic Time Warping generate indefinite kernel matrices [9–11]. The generalized histogram intersection kernel that is conditionally positive definite is not usually positive semidefinite [12]. Hyperbolic tangent kernels [13, 14] suitable for practice sometimes are indefinite as well. As far as we know, it is still not very clear how to effectively deal with indefinite kernels in the SVM framework. Training indefinite SVMs therefore becomes a challenging optimization problem since convex solutions are no longer valid for standard SVMs in this learning scenario [15].
To deal with indefinite kernel, a number of methods have been proposed in the literature [16]. Representatives in previous studies tackled such problem by altering the spectrum of an indefinite kernel matrix so as to create a PSD one. Authors in [17] developed the denoising method which deems negative eigenvalues as noise and replaces them with zero. The flipping method is another effective method for transforming indefinite kernel into PSD one by changing the sign of negative eigenvalues [18]. Authors in [19] proposed the diffusion method which considers the data distribution and replaces the eigenvalues with exponential form. The shifting method, i.e., shifts eigenvalues by introducing new parameters to ensure all the eigenvalues are nonnegative [20]. Authors in [13] developed a method in order to find stationary points under a nonconvex dual formulation of SVMs with sigmoid kernels. Authors considered indefinite kernel learning as a minimization problem in a pseudoEuclidean space in [21]. In [22], a maxmin optimization problem is further proposed so as to find a proxy kernel for the indefinite kernel. Based on confidence function, a simple generalization of SVMs is suggested by Guo and Schuurmans [23]. Kernel principal component analysis is developed as a kernel transformation method to deal with indefinite kernels [24].
In this paper, we develop a superior and effective method, i.e., projection method, to convert an indefinite kernel into a PSD one. Compared with the existing methods, our proposed one is much more flexible and comprehensive. One can easily obtain different type of methods such as flipping or denoising method by varying its parameters. Furthermore, our suggested λ under Logdet Divergence and perturbed vonNeumann Divergence can always yield near optimal performance, which can be regarded as a good choice for dealing with indefinite kernels. Besides, our suggested projection matrix also has certain special mathematical properties. Furthermore, the connection between spectrum method and projection method can be investigated through analysis on eigenvalues.
The rest of the paper is organized as follows. Firstly, we present the projection method and also the associated theorem. Then we propose the optimal λ determination in the projection matrix under unconstrained optimization framework. After that, we apply two indefinite kernels on some real world data sets which range from cancer prediction to glycan classification. And we also validate the suggested optimal λ with the experimental data. Discussions of the experimental validation on the suggested optimal λ under Logdet Divergence and perturbed vonNeumann Divergence are followed. Finally, in the last section, we give the concluding remarks with possible future work.
Methods
And according to Mercer’s theorem, a valid kernel should be positive semidefinite. Thus, to deal with invalid kernels, kernel transformation strategy is increasingly popular. In the case of nonpositive semidefinite kernel K, we may decompose it into this form K=P·D·P ^{′}. Where D is a diagonal matrix and not all the diagonal entries are nonnegative, P is orthonormal matrix with the jth column corresponding to the eigenvector for jth eigenvalue in D and P ^{′} represents the transpose of matrix P. Eigenvalue transform is the representative method in kernel transformation [17–20].
In the following, we present our suggested projection method for transforming an indefinite kernel to a PSD one.
Lemma 1
There exists an n×m (m<n) matrix B satisfying \(B^{\prime }B=I_{m}\) such that \((I_{n}\lambda BB^{\prime })\) has 1−λ and 1 as its eigenvalues, the multiplicities for whom are m and n−m respectively. Besides, it shares the same set of eigenvectors with K.
Proof
Consider that K is a real and symmetric matrix, we decompose it as K=P·D·P ^{′} where \(P=[\vec {p}_{1},\vec {p}_{2},\ldots,\vec {p}_{n}]\) and D=diag[d _{1},d _{2},…,d _{ n }] is a diagonal matrix with the diagonal elements d _{ i },i=1,2,…,n. W.L.O.G, we may assume all the eigenvalues are sorted in ascending order. We further assume the positive inertia index is l and the negative inertia index is m.
Thus, it has 1 and (1−λ) as its eigenvalues, and the multiplicity for (1−λ) is m and for 1 the multiplicity is n−m. Furthermore, the eigenvectors of (I _{ n }−λ B B ^{′}) are exactly the same as the kernel K. □
Theorem 1
Let K be an n×n real symmetric matrix which is indefinite. Then there exists an n×m (m<n) matrix B satisfying B ^{′} B=I _{ m } such that (I _{ n }−λ B B ^{′})K is a positive semidefinite kernel where λ≥1 is a regularization parameter.
Proof
Since λ≥1, we have (1−λ)d _{ i }≥0 for 1≤i≤m. This will guarantee the kernel matrix (I _{ n }−λ B B ^{′})K is positive semidefinite. □
In particular, we get denoising method by letting λ=1 according to Eq. (3). And flipping method is the particular case of Projection method when λ=2.
Optimal λ determination
Considering that λ is a embedded parameter in the projection method, it is necessary to study optimal λ determination which can demonstrate excellent prediction power for λ>0. To this end, we begin with the definition of Bregman matrix divergence [25].
Definition 1
Here ϕ(K)is a strictly convex differentiable function of K and tr(K) means the trace of matrix K.
 1.Mahalanobis Divergence(p=2):$$ D_{\phi}(K,K_{0})=\text{tr}\left(K^{2}2{KK}_{0}+K_{0}^{2}\right). $$(4)
 2.Frobenius Divergence\(\left (\phi (K) = \K\_{F}^{2}\right)\):$$ D_{\phi}(K,K_{0})=\KK_{0}\_{F}^{2}. $$(5)
 3.vonNeumann Divergence(ϕ(K)=tr(K log(K)−K)):$$ D_{\phi}(K,K_{0})=\text{tr}(K\log K K \log K_{0} K + K_{0}). $$(6)
 4.LogDet Divergence(ϕ(K)=− log det(K)):$$ {}D_{\phi}(K,K_{0})=\text{tr}\left({KK}_{0}^{1}\right)\log \det\left({KK}_{0}^{1}\right)n. $$(7)
The optimal λ can be quickly obtained as 0.
It is easy to see that the optimal λ is 0.
Applying differentiation to Eq. (10), we obtain the optimal value of λ=0.
Results
Materials
Data set information
Data set  Number of instances  Number of attributes 

Sonar  208  60 
Live disorder  345  6 
Breast cancer  680  10 
Cystic fibrosis  177  Depends on q 
Leukemia  355  Depends on q 
Lung cancer  91  54675 
Experiments
We perform the experiments in 5fold crossvalidation setting and measure the performance of models with the Area Under Curve (AUC). AUC (calculated as the area under the ROC curve) is commonly used for model evaluation. We measure the averaged AUC values for the considered methods through 10 times 5fold crossvalidations. Here we introduce two kernels: the Generalized Histogram Intersection (GHI) kernel [12] and the cosine kernel for illustration purpose. These two kernels in most cases are indefinite (shown in Additional file 1: Table SI), both of which have not been used in biological applications like glycan classification or cancer prediction.
When α=β, the kernel can be proved to be a positive semidefinite matrix. Experimental results in Table SI (in Additional file 1) also show consistence with the statement, as the minimal eigenvalue in GHI kernel is 0 when α=β. We in experimental settings use different values of α,β∈{1,2,3},α≠β to evaluate the performance of our proposed projection method.
Experiments on GHI kernel
Averaged AUC values (%) of projection method and GHI kernel using sonar data, live disorder data, breast cancer data and NSCLC data
Data sets  Parameters  Projection method  GHI kernel 

Sonar  α=1,β=1  82.87 ± 0.99  82.87 ± 0.99 
α=1,β=2  81.47 ± 0.99  53.42 ± 4.94  
α=1,β=3  84.02 ± 1.19  54.10 ± 4.92  
α=2,β=2  84.29 ± 1.54  84.29 ± 1.54  
α=2,β=3  84.31 ± 1.56  83.06 ± 2.04  
α=3,β=3  83.62 ± 1.17  83.62 ± 1.17  
Live  α=1,β=1  82.87 ± 0.99  82.87 ± 0.99 
α=1,β=2  81.47 ± 0.99  53.42 ± 4.94  
α=1,β=3  84.02 ± 1.19  54.10 ± 4.92  
α=2,β=2  84.29 ± 1.54  84.29 ± 1.54  
α=2,β=3  84.31 ± 1.56  83.06 ± 2.04  
α=3,β=3  83.62 ± 1.17  83.62 ± 1.17  
Breast  α=1,β=1  96.73 ± 0.11  96.73 ± 0.11 
α=1,β=2  97.06 ± 0.01  90.12 ± 4.78  
α=1,β=3  97.01 ± 0.01  75.61 ± 7.44  
α=2,β=2  96.71 ± 0.11  96.71 ± 0.11  
α=2,β=3  96.92 ± 0.01  96.96 ± 0.01  
α=3,β=3  96.63 ± 0.10  96.63 ± 0.10  
NSCLC  α=1,β=1  100 ± 0  100 ± 0 
α=1,β=2  99.72 ± 0.01  64.07 ± 7.42  
α=1,β=3  61.46 ± 1.57  51.47 ± 5.53  
α=2,β=2  100 ± 0  100 ± 0  
α=2,β=3  99.99 ± 0  73.07 ± 8.17  
α=3,β=3  100 ± 0  100 ± 0 
Averaged AUC values (%) of projection method and GHI kernel using cystic fibrosis data
Parameters  Projection method(q=1)  GHI(q=1)  Projection method(q=2)  GHI(q=2) 
α=1,β=1  78.57 ± 1.75  78.57 ± 1.75  81.32 ± 1.25  81.32 ± 1.25 
α=1,β=2  78.94 ± 1.86  78.94 ± 1.86  81.74 ± 1.60  81.74 ± 1.60 
α=1,β=3  78.64 ± 1.01  78.63 ± 1.01  80.82 ± 1.30  80.82 ± 1.29 
α=2,β=2  79.33 ± 1.42  79.33 ± 1.41  80.53 ± 1.72  80.53 ± 1.72 
α=2,β=3  79.32 ± 1.19  79.32 ± 1.19  81.06 ± 1.37  81.06 ± 1.36 
α=3,β=3  78.14 ± 1.11  78.13 ± 1.11  80.79 ± 1.12  80.78 ± 1.12 
Parameters  Projection method(q=3)  GHI(q=3)  Projection method(q=4)  GHI(q=4) 
α=1,β=1  80.77 ± 1.44  80.76 ± 1.44  83.10 ± 2.10  83.09 ± 2.10 
α=1,β=2  80.98 ± 1.81  80.97 ± 1.81  82.11 ± 1.77  82.13 ± 1.77 
α=1,β=3  81.20 ± 1.95  81.19 ± 1.94  83.54 ± 1.46  83.51 ± 1.48 
α=2,β=2  81.32 ± 1.26  81.30 ± 1.27  82.75 ± 2.14  82.79 ± 2.15 
α=2,β=3  81.10 ± 1.10  81.09 ± 1.11  83.62 ± 1.61  83.65 ± 1.65 
α=3,β=3  81.06 ± 1.39  81.04 ± 1.39  83.49 ± 0.77  83.56 ± 0.82 
Parameters  Projection method(q=5)  GHI(q=5)  Projection method(q=6)  GHI(q=6) 
α=1,β=1  74.03 ± 2.18  74.00 ± 2.18  72.30 ± 1.93  72.50 ± 1.87 
α=1,β=2  71.67 ± 2.52  71.62 ± 2.58  73.62 ± 2.69  73.80 ± 2.70 
α=1,β=3  74.77 ± 2.27  74.73 ± 2.28  71.94 ± 1.77  72.11 ± 1.65 
α=2,β=2  73.73 ± 1.36  73.73 ± 1.38  71.49 ± 2.78  71.60 ± 2.84 
α=2,β=3  72.62 ± 2.97  72.61 ± 2.92  72.81 ± 1.91  73.01 ± 1.92 
α=3,β=3  75.23 ± 2.64  75.20 ± 2.55  73.53 ± 2.62  73.80 ± 2.67 
Parameters  Projection method(q=7)  GHI(q=7)  Projection method(q=8)  GHI(q=8) 
α=1,β=1  67.99 ± 2.78  67.60 ± 2.87  60.65 ± 4.20  60.90 ± 4.36 
α=1,β=2  68.28 ± 3.51  67.89 ± 3.60  58.19 ± 3.72  58.33 ± 3.77 
α=1,β=3  67.75 ± 2.20  67.25 ± 2.19  58.98 ± 3.67  59.28 ± 3.69 
α=2,β=2  67.90 ± 3.11  67.23 ± 3.04  58.28 ± 4.20  58.34 ± 4.13 
α=2,β=3  67.58 ± 2.91  66.96 ± 2.88  58.66 ± 2.40  58.86 ± 2.37 
α=3,β=3  68.85 ± 2.28  68.44 ± 2.13  59.62 ± 3.34  59.77 ± 3.37 
Parameters  Projection method(q=9)  GHI(q=9)  
α=1,β=1  53.25 ± 3.99  53.25 ± 3.99  
α=1,β=2  52.12 ± 4.28  52.12 ± 4.28  
α=1,β=3  52.54 ± 3.22  52.54 ± 3.22  
α=2,β=2  51.16 ± 2.37  51.16 ± 2.37  
α=2,β=3  51.62 ± 4.18  51.62 ± 4.18  
α=3,β=3  51.96 ± 5.01  51.96 ± 5.01 
Averaged AUC values (%) of projection method and GHI kernel using leukemia data
Parameters  Projection method(q=1)  GHI(q=1)  Projection method(q=2)  GHI(q=2) 
α=1,β=1  93.68 ± 0.62  93.68 ± 0.62  95.90 ± 0.84  95.90 ± 0.84 
α=1,β=2  93.75 ± 0.59  87.00 ± 4.10  95.85 ± 0.41  94.93 ± 0.81 
α=1,β=3  93.34 ± 0.91  86.94 ± 3.37  95.41 ± 0.64  94.53 ± 0.89 
α=2,β=2  93.33 ± 0.74  93.32 ± 0.74  95.61 ± 0.46  95.61 ± 0.46 
α=2,β=3  93.31 ± 0.47  93.51 ± 0.46  95.17 ± 0.69  95.66 ± 0.85 
α=3,β=3  93.54 ± 0.66  93.54 ± 0.66  95.77 ± 0.40  95.77 ± 0.40 
Parameters  Projection method(q=3)  GHI(q=3)  Projection method(q=4)  GHI(q=4) 
α=1,β=1  95.07 ± 0.64  95.08 ± 0.65  93.51 ± 0.54  93.54 ± 0.54 
α=1,β=2  95.13 ± 0.46  95.10 ± 0.471  93.86 ± 0.77  93.88 ± 0.77 
α=1,β=3  94.83 ± 0.41  94.77 ± 0.42  94.13 ± 0.52  94.15 ± 0.52 
α=2,β=2  95.13 ± 0.53  95.13 ± 0.53  94.05 ± 0.49  94.06 ± 0.49 
α=2,β=3  94.84 ± 0.67  94.85 ± 0.67  93.81 ± 0.69  93.82 ± 0.69 
α=3,β=3  94.77 ± 0.61  94.77 ± 0.61  93.98 ± 0.36  93.99 ± 0.36 
Parameters  Projection method(q=5)  GHI(q=5)  Projection method(q=6)  GHI(q=6) 
α=1,β=1  93.40 ± 0.58  93.44 ± 0.58  93.23 ± 0.26  93.38 ± 0.26 
α=1,β=2  93.12 ± 0.70  93.16 ± 0.70  93.07 ± 0.75  93.21 ± 0.74 
α=1,β=3  93.20 ± 0.27  93.25 ± 0.28  93.05 ± 0.63  93.18 ± 0.63 
α=2,β=2  93.61 ± 0.73  93.64 ± 0.74  93.21 ± 0.48  93.35 ± 0.48 
α=2,β=3  93.78 ± 0.56  93.83 ± 0.56  93.26 ± 0.70  93.41 ± 0.72 
α=3,β=3  93.71 ± 0.72  93.75 ± 0.73  93.38 ± 0.65  93.51 ± 0.67 
Parameters  Projection method(q=7)  GHI(q=7)  Projection method(q=8)  GHI(q=8) 
α=1,β=1  92.15 ± 0.68  92.37 ± 0.67  90.10 ± 0.71  90.36 ± 0.70 
α=1,β=2  92.33 ± 0.57  92.53 ± 0.59  90.68 ± 1.14  90.92 ± 1.13 
α=1,β=3  92.11 ± 0.86  92.31 ± 0.86  90.72 ± 0.73  90.96 ± 0.73 
α=2,β=2  92.01 ± 0.50  92.23 ± 0.50  90.67 ± 1.06  90.93 ± 1.04 
α=2,β=3  92.06 ± 0.45  92.27 ± 0.43  90.31 ± 0.90  90.53 ± 0.89 
α=3,β=3  92.28 ± 0.71  92.48 ± 0.73  90.66 ± 0.65  90.92 ± 0.67 
Parameters  Projection method(q=9)  GHI(q=9)  
α=1,β=1  88.92 ± 0.59  89.20 ± 0.62  
α=1,β=2  89.61 ± 0.62  89.86 ± 0.63  
α=1,β=3  89.33 ± 0.68  89.60 ± 0.67  
α=2,β=2  89.54 ± 0.96  89.80 ± 0.96  
α=2,β=3  88.57 ± 0.67  88.82 ± 0.68  
α=3,β=3  88.56 ± 0.63  88.84 ± 0.63 
The performance for sonar data set is reported in Table 2. For example, when α=1,β=2, Projection Method shows the averaged AUC value 81.47% with standard deviation 0.99% while in GHI kernel method the averaged AUC value is 53.42% with standard deviation 4.94%. When (α,β)=(1,3), Projection Method shows 84.02% in the averaged AUC value, with standard deviation 1.19%. However, the averaged AUC value for GHI kernel method is only 54.10% with standard deviation 4.92%. When (α,β)=(2,3), the averaged AUC value for Projection Method is 84.31%, larger than the averaged AUC value for GHI Method 83.06%. The standard deviation in Projection Method is 1.56%, while in GHI kernel method standard deviation is 2.04%. This implies that Projection method is more powerful and stable compared to original GHI kernel method.
For live disorder data set, we can see from Table 2 that the Projection method is significantly better performance than the GHI kernel method when α≠β. The best performance of GHI kernel when indefinite achieves around 60% in AUC value which is not satisfying. When α=β, both methods show comparable performance.
For breast cancer data set, results in Table 2 indicate that when α=1,β=2 and α=1,β=3, the Projection method is clearly superior to the GHI kernel method except for α=2,β=3 where comparable performance is detected in both methods. This illustrates the fact that indefinite kernels sometimes can also perform well. However, the superiority of projection method over the original GHI kernel method is clearly shown in this data set.
In cystic fibrosis data set, we get 9 different comparison results when values of q vary from 1 to 9 as shown in Table 3. There is no clear difference between Projection method and GHI kernel, as GHI kernel is positive semidefinite for almost all considered pairs of α and β (see Additional file 1: Table SI for reference). The only 2 cases when GHI kernel indefinite are α=1,β=2 and α=1,β=3 for q = 1, and the minimal eigenvalue for the generated GHI kernel in these 2 cases is only 0.08, quite close to 0, demonstrating that the generated kernel is almost positive semidefinite.
Results for Leukemia Data are summarized in Table 4. Similar to the results in cystic fibrosis data, projection method and GHI kernel method show similar performance in most of the cases for q from 1 to 9. From Additional file 1: Table SI we can see that, GHI kernel is indefinite when α≠β for q=1,2,3. When q=1,2, Projection method is better than GHI kernel method for α=1,β=2 and α=1,β=3; However, GHI kernel method is comparable to Projection method for α=2,β=3.
Some interesting results can be found for NSCLC data as shown in Table 2. Projection method and GHI kernel method show exact performance when α=β, yielding 100% in AUC values. Note that Projection method does not make any perturbation to the original kernel when positive semidefinite(GHI kernel when α=β), we can get conclusion that GHI kernel is a preferred kernel for tumor differentiation with NSCLC data. When α differs from β, different results are shown. When α=1,β=2, Projection method shows 99.72% in averaged AUC values with 0.01% standard deviation, while GHI kernel method only can get 64.07% in Averaged AUC values with a large standard deviation 7.42%. When α=2,β=3, Projection method shows 99.99% in averaged AUC values with 0 standard deviation, while GHI kernel method can get 73.07% in averaged AUC values with a large standard deviation 8.17%. Exceptions happen when α=1,β=3 where Projection method can only get 61.46% in averaged AUC values and GHI kernel method is even worse, achieving only 51.47% in averaged AUC values.
We can conclude that the performance of projection method is not always similar for different pairs of (α,β). There exists best (α,β) for inducing best projection method, but different data sets may be suitable to different pairs. GHI kernel method sometimes when kernel is indefinite can also perform well. But in general, projection method is clearly better than the GHI kernel for the above considered data sets.
Experiments on Cosine kernel
Averaged AUC values (%) of projection method and Cosine kernel for the considered datasets
Dataset  Projection method  Cosine kernel 

Live disorder data  73.71 ± 1.21  65.63 ± 2.75 
Sonar data  89.57 ± 1.37  67.46 ± 4.32 
Breast data  99.37 ± 0.06  97.99 ± 3.09 
Cystic (q=1)  79.25 ± 1.80  76.89 ± 3.24 
Cystic (q=2)  80.55 ± 1.38  79.80 ± 1.84 
Cystic (q=3)  78.27 ± 1.59  70.10 ± 4.01 
Cystic (q=4)  73.24 ± 2.15  58.52 ± 4.95 
Cystic (q=5)  64.38 ± 3.85  52.13 ± 4.30 
Cystic (q=6)  69.26 ± 2.11  60.72 ± 5.36 
Cystic (q=7)  64.6 ± 2.38  58.54 ± 3.80 
Cystic (q=8)  63.17 ± 2.89  63.66 ± 3.21 
Cystic (q=9)  54.21 ± 2.30  43.05 ± 2.38 
Leukemia (q=1)  94.36 ± 0.43  90.73 ± 1.94 
Leukemia (q=2)  94.38 ± 0.79  69.45 ± 4.81 
Leukemia (q=3)  95.20 ± 0.49  69.97 ± 6.58 
Leukemia (q=4)  94.73 ± 0.45  73.33 ± 5.99 
Leukemia (q=5)  91.23 ± 0.44  71.81 ± 9.62 
Leukemia (q=6)  93.19 ± 0.66  79.08 ± 6.96 
Leukemia (q=7)  90.56 ± 1.25  65.26 ± 6.90 
Leukemia (q=8)  87.81 ± 0.98  58.31 ± 2.87 
Leukemia (q=9)  87.52 ± 1.20  55.88 ± 3.82 
NSCLC  52.91 ± 4.45  48.64 ± 5.30 
Discussion
Experimental results show that the Projection method is better or comparable with the compared kernel methods: GHI kernel and Cosine kernel. Despite the fact that GHI kernel and Cosine kernel when indefinite sometimes can yield good performance, Projection method still demonstrate comparable performance. The necessity of Projection transformation for the considered indefinite kernels is clearly demonstrated. Projection method when λ≥1 can transform an indefinite kernel into a PSD one. The optimal λ determination for Projection Method focusing on four different divergences is also considered. From the deduced optimal λ, we focus on the one with LogDet Divergence as it is more realistic.
In the following, we will conduct experiments on the considered data sets, to confirm if suggested optimal λ of the Projection Method can show optimal performance in various values of λ>0.
Optimal λ in the projection method for sonar data
The suggested optimal λ in Fig. 2 is 2.0 in Projected GHI kernel for all pairs of (α,β),α≠β. The performance of Projection Method shows a steady decrement when λ>2, implying that λ=2 is a good choice for projection method. When λ<1, the performance of projection method is quite unstable because the PSD property cannot be guaranteed.
It is very interesting to see that the suggested optimal λ is uniformly the same in the two considered kernels. Take projected GHI kernel with different (α,β) pairs for comparison, we can see that projected method with α=1,β=3 shows best performance, 0.8733, where the experimental best performance is shown to be 0.8735 achieving at λ=1.9. When α=1,β=2, the projection method with suggested optimal λ achieves 0.8246 in averaged AUC value, and the experimental best result is 0.8276. When α=2,β=3, the projection method with suggested optimal λ achieves 0.8540 in averaged AUC value, and the experimental best result is 0.8557. Considering the projected Cosine Kernel, the experimental best AUC value for Projected Cosine Kernel 0.9126 is achieved at λ=3.8, while our suggested optimal λ=2 yielding AUC value 0.9051, the difference between the two values is little: 0.0075. We can conclude that the suggested optimal λ can guarantee at least an near optimal performance.
Optimal λ in the projection method for live disorder data
Optimal λ in the projection method for breast cancer data
Optimal λ in the projection method for cystic fibrosis data
Optimal λ in the projection method for leukemia data
Optimal λ in the projection method with NSCLC data
Optimal λ suggested in projection method with considered kernels
Methods  GHI Kernel  Cosine kernel  

Dataset  α=1,β=2  α=1,β=3  α=2,β=3  
Live disorder data  2.38  2.37  2.45  2.17 
Sonar data  2  2  2  2 
Breast data  4.6  6.57  4.06  4.29 
Cystic (q=1)  71  71  100  5.8 
Cystic (q=2)  100  100  1  2.5 
Cystic (q=3)  100  100  1  2.8 
Cystic (q=4)  100  100  1  3.67 
Cystic (q=5)  100  100  100  6.2 
Cystic (q=6)  1  1  1  14 
Cystic (q=7)  1  1  1  21 
Cystic (q=8)  1  1  1  37 
Cystic (q=9)  1  1  1  85 
Leukemia (q=1)  47.33  47.33  46.67  10 
Leukemia (q=2)  28.25  28.25  22.80  7.4 
Leukemia (q=3)  46.5  46.5  47  5.42 
Leukemia (q=4)  1  1  1  3.06 
Leukemia (q=5)  1  1  1  2.33 
Leukemia (q=6)  1  1  1  2.39 
Leukemia (q=7)  100  100  100  2.56 
Leukemia (q=8)  100  100  100  2.67 
Leukemia (q=9)  1  1  1  2.98 
NSCLC  2.0  46  2  2 
Any better optimal λ for projection method?

Lambda Comparison with Projection Method in Sonar Data, Live Disorder Data, Breast Cancer Data and NSCLC Data
As shown in the table (Table 7), we can see that the newly determined optimal λ through perturbed Von Neumann Divergence shows similar performance with the optimal λ generated by Logdet divergence. The only clear difference can be detected for Sonar data in GHI kernel when α=2,β=3 and Cosine Kernel. For GHI kernel α=2,β=3 we can see that λ _{opt} is superior to λ _{opt1}, while for Cosine kernel, λ _{opt1} is superior to λ _{opt}. Regarding the determined optimal λ under different divergences, we can see that λ _{opt} differs from λ _{opt1}. For GHI kernel case, the determined optimal λ under Logdet Divergence and perturbed vonNeumann Divergence is similar to each other in Live data set but quite different in other data sets. For cosine kernel case, λ _{opt} and λ _{opt1} are quite different from each other. We can see that though the determined optimal λ under Logdet Divergence and perturbed vonNeumann Divergence is different, the performance is comparable. When we compare both kernels, we can see that Cosine kernel with λ _{opt1} is a preferred option.Table 7Optimal λ comparison in projection method with considered kernels in sonar data, live disorder data, breast cancer data and NSCLC data
α=1,β=2
α=1,β=3
α=2,β=3
Cosine
Sonar
(λ _{opt},AUC_{opt})
(2.00,0.8266)
(2.00,0.8787)
(2.00,0.8585)
(2.00,0.9034)
(λ _{opt1},AUC_{opt1})
(2.59,0.8284)
(2.16,0.8784)
(4.32,0.8486)
(8.30,0.9118)
Live
(λ _{opt},AUC_{opt})
(2.38,0.7559)
(2.37,0.7397)
(2.45,0.7543)
(2.17,0.7292)
(λ _{opt1},AUC_{opt1})
(2.08,0.7571)
(2.04,0.7415)
(2.09,0.7542)
(6.70,0.7249)
Breast
(λ _{opt},AUC_{opt})
(4.60,0.9689)
(6.57,0.9659)
(4.06,0.9684)
(4.29,0.9937)
(λ _{opt1},AUC_{opt1})
(2.03,0.9702)
(2.02,0.9675)
(2.20,0.9686)
(13.04,0.9936)
NSCLC
(λ _{opt},AUC_{opt})
(2.00,0.9996)
(2.00,0.9959)
(2.00,0.9903)
(2.00,0.4059)
(λ _{opt1},AUC_{opt1})
(4.96,0.9990)
(3.58,0.9978)
(2.69,0.9910)
(2.30,0.4010)

Lambda Comparison with Projection Method in Cystic Fibrosis Data
From Table 8, we can get some conclusions. For GHI kernel, it is obvious that projection method shows almost identical performance with λ _{opt} and λ _{opt1}. It is interesting to see that in GHI kernel case, λ _{opt} and λ _{opt1} are equal to each other expect when q=1 for α=1,β=2 and α=1,β=3. From Table SI we know that GHI kernel in these 2 cases is indefinite. Although the values of λ _{opt} and λ _{opt1} are quite different from each other when q=1 for α=1,β=2 and α=1,β=3, the performances are similar to each other. When it comes to Cosine kernel, we can see that projection method with λ _{opt1} tends to perform better for q∈{1,2,3,4,5,6,9}. Clear differences can be detected when q=3,4,5 that are marked in bold face. Besides, λ _{opt1} in Cosine kernel is larger than λ _{opt} for most cases (q∈{1,2,…,7}), meaning that projection method with Cosine kernel tends to show better performance for relatively large λ. When we compare GHI kernel and Cosine kernel, we find that GHI kernel in general tends to show better performance for small q, and Cosine kernel shows better performance when q is large.Table 8Optimal λ comparison in projection method with considered kernels in cystic fibrosis data
α=1,β=2
α=1,β=3
α=2,β=3
Cosine
q=1
(λ _{opt},AUC_{opt})
(71,0.7771)
(71,0.7711)
(100,0.7829)
(5.8,0.7889)
(λ _{opt1},AUC_{opt1})
(36.5,0.7775)
(36.5,0.7713)
(100,0.7829)
(28.3,0.7912)
q=2
(λ _{opt},AUC_{opt})
(100,0.8031)
(100,0.8114)
(1,0.8209)
(2.5,0.7951)
(λ _{opt1},AUC_{opt1})
(100,0.8031)
(100,0.8114)
(1,0.8209)
(43.38,0.7959)
q=3
(λ _{opt},AUC_{opt})
(100,0.8103)
(100,0.8140))
(1,0.8033)
(2.8,0.7978)
(λ _{opt1},AUC_{opt1})
(100,0.8103)
(100,0.8140)
(1,0.8033)
(34.3,0.8111)
q=4
(λ _{opt},AUC_{opt})
(100,0.8296)
(100,0.8356)
(1,0.8286)
(3.67,0.7825)
(λ _{opt1},AUC_{opt1})
(100,0.8296)
(100,0.8356)
(1,0.8286)
(26.58,0.7979)
q=5
(λ _{opt},AUC_{opt})
(100,0.7400)
(100,0.7272)
(100,0.7405)
(6.2,0.6973)
(λ _{opt1},AUC_{opt1})
(100,0.7400)
(100,0.7272)
(100,0.7405)
(27,0.7137)
q=6
(λ _{opt},AUC_{opt})
(1,0.7173)
(1,0.7164)
(1,0.7224)
(14,0.7144)
(λ _{opt1},AUC_{opt1})
(1,0.7173)
(1,0.7164)
(1,0.7224)
(34.16,0.7156)
q=7
(λ _{opt},AUC_{opt})
(1,0.6702)
(1,0.6721)
(1,0.6713)
(21,0.6620)
(λ _{opt1},AUC_{opt1})
(1,0.6702)
(1,0.6721)
(1,0.6713)
(22.17,0.6616)
q=8
(λ _{opt},AUC_{opt})
(1,0.5928)
(1,0.5791)
(1,0.5935)
(37,0.6388)
(λ _{opt1},AUC_{opt1})
(1,0.5928)
(1,0.5791)
(1,0.5935)
(19.25,0.6387)
q=9
(λ _{opt},AUC_{opt})
(1,0.5146)
(1,0.5107)
(1,0.5254)
(85,0.5637)
(λ _{opt1},AUC_{opt1})
(1,0.5146)
(1,0.5107)
(1,0.5254)
(17.5,0.5688)

Lambda Comparison with Projection Method in Leukemia Data
We can get similar conclusions for Leukemia data. As shown in the table (Table 9), projection method shows almost identical performance with λ _{opt} and λ _{opt1} though different optimal λ values are obtained (Please check q=1,2,3 respectively). When q=1, λ _{opt1} is smaller than λ _{opt}. When q=2,3 respectively, λ _{opt1} is larger than λ _{opt}. Though values of optimal λ differ from each other, the performances are quite similar, meaning that projection method with GHI kernel for Leukemia data is less sensitive in the optimal λ. When q∈{4,5,6,7,8,9}, λ _{opt} and λ _{opt1} are identical, we can see from Table SI that GHI kernel in these cases are PSD already. For Cosine Kernel, optimal λ determined by Logdet Divergence and perturbed VonNeumann Divergence differs. Projection method with λ _{opt1} performs slightly better than projection method with λ _{opt}. Besides, λ _{opt1} in Cosine kernel is larger than λ _{opt}, implying that projection method tends to show better performance for large λ. When we focus on the performance of projection method with λ _{opt1}, we can find that different from Cystic Fibrosis data set, the performance of projected cosine kernel with λ _{opt1} tends to show better performance for small q while projected GHI kernel with λ _{opt} tends to show better performance for large q.Table 9Optimal λ comparison in projection method with considered kernels in leukemia data
α=1,β=2
α=1,β=3
α=2,β=3
Cosine
q=1
(λ _{opt},AUC_{opt})
(47.3,0.9418)
(47.3,0.9377)
(46.7,0.9365)
(10,0.9469)
(λ _{opt1},AUC_{opt1})
(26.8,0.9419)
(26.8,0.9377)
(27.8,0.9367)
(16.6,0.9472)
q=2
(λ _{opt},AUC_{opt})
(28.3,0.9551)
((28.3,0.9551)
(22.8,0.9582)
(7.4,0.9541)
(λ _{opt1},AUC_{opt1})
(35.9,0.9550)
(35.9,0.9551)
(29.3,0.9582)
(24.7,0.9555)
q=3
(λ _{opt},AUC_{opt})
(46.5,0.9512)
(46.5,0.9540)
(47,0.9500)
(5.42,0.9573)
(λ _{opt1},AUC_{opt1})
(87.8,0.9512)
(87.8,0.9551)
(88.8,0.9500)
(23.1,0.9593)
q=4
(λ _{opt},AUC_{opt})
(1,0.9427)
(1,0.9416)
(1,0.9405)
(3.06,0.9485)
(λ _{opt1},AUC_{opt1})
(1,0.9427)
(1,0.9416)
(1,0.9405)
(18.6,0.9522)
q=5
(λ _{opt},AUC_{opt})
(1,0.9352)
(1,0.9362)
(1,0.9363)
(2.33,0.9175)
(λ _{opt1},AUC_{opt1})
(1,0.9352)
(1,0.9362)
(1,0.9363)
(11.08,0.9259)
q=6
(λ _{opt},AUC_{opt})
(1,0.9310)
(1,0.9319)
(1,0.9311)
(2.39,0.9333)
(λ _{opt1},AUC_{opt1})
(1,0.9310)
(1,0.9319)
(1,0.9311)
(7.74,0.9337)
q=7
(λ _{opt},AUC_{opt})
(100,0.9201)
(100,0.9236)
(1,0.9212)
(2.56,0.8993)
(λ _{opt1},AUC_{opt1})
(100,0.9201)
(100,0.9236)
(1,0.9212)
(5.03,0.8921)
q=8
(λ _{opt},AUC_{opt})
(100,0.9035)
(100,0.9096)
(100,0.9059)
(2.67,0.8795)
(λ _{opt1},AUC_{opt1})
(100,0.9035)
(100,0.9096)
(100,0.9059)
(3.56,0.8845)
q=9
(λ _{opt},AUC_{opt})
(1,0.8936)
(1,0.8915)
(1,0.8899)
(2.98,0.8734)
(λ _{opt1},AUC_{opt1})
(1,0.8936)
(1,0.8915)
(1,0.8899)
(3.72,0.8735)
In summary, when λ∈(0,1), the positive semidefiniteness of the projected kernel matrix cannot be assured, and the performance tends to be extremely unstable. The suggested optimal λ in Projection method is related to the eigenvalues in original kernel matrix, and thus varies in different data sets. Besides, the suggested optimal λ under Logdet Divergence and perturbed VonNeumann Divergence differs from each other in the same data sets in most cases. Even in that case, projection method under the two different cases can still guarantee near optimal performance. It can be seen that when optimal λ under Logdet Divergence and optimal λ under perturbed VonNeumann Divergence is very different, the performance of projection method in both cases is still similar, showing that in this case projection method is relatively insensitive to the values of suggested optimal λ (projection method with a large range of λ values can suggest near optimal performance). Our suggested theoretical λ under Logdet Divergence and perturbed VonNeumann Divergence sometimes cannot guarantee the best performance. There are two possible reasons. One possible reason is that the optimal λ determination by unconstrained optimization in framework of kernel learning hypothesized the positive definiteness of the kernels, but we use indefinite kernels in this case. Another possible reason is that the inverse of kernel was substituted by pseudo inverse.
Conclusions
In this paper, we propose projection method for addressing indefinite kernel learning problems. The projection method is construed from an eigenspace perspective. It is very flexible by varying the parameter λ, to change from the denoising method to the flipping method. These two spectrum based methods are wellknown techniques in dealing with indefinite kernels. Two kernels that are not generally PSD are introduced for comparison: GHI kernel method and the Cosine kernel method. We show better performance for projection method in terms of AUC values under 5fold crossvalidations. The optimal λ embedded in the Projection Method can be determined by solving an unconstrained optimization problem. Experimental studies show consistence with theoretical analysis as projection method with our suggested λ can always guarantee at least near optimal performance for λ>0. In the pursuit of precise optimal λ determination method, we also compared optimal λ determination with Logdet Divergence and perturbed VonNeumann Divergence, aiming at finding better λ in projection method. The determined optimal λ differs from each other for different kernels and data sets involved, and the results obtained are in general similar. Our proposed projection method may be regarded as a good choice for dealing with indefinite kernels. Future work may contribute to the development of more precise optimal λ determination method and the development of more variants of projection method for indefinite kernels, hoping to be applied in other areas.
Declarations
Acknowledgements
The authors would like to thank Prof.Kiyoko AokiKinoshita for providing cystic fibrosis data and Samuel Emersion Harvey for helping to polish the manuscript. The Preliminary version of the paper has been published in proceedings of the ISORA2015 [33].
Funding
This research is supported in part by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China, HKU Strategic Theme in Computation and Information, National Natural Science Foundation of China Grant Nos. 11626229, 11271144, 11671158, and 61472428 and Natural Science Foundation of SZU (No. 2017058). The publication costs are funded by Natural Science Foundation of SZU (No. 2017058).
Availability of data and materials
All the data sets are publicly available and can be accessed from the databases: LIBSVM Data, CFG (Consortium for Functional Glycomics) and NCBI (National Center for Biotechnology Information). The following links are: (http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets. http://www.functionalglycomics.org/. https://www.ncbi.nlm.nih.gov/.)
About this supplement
This article has been published as part of BMC Systems Biology Volume 11 Supplement 6, 2017: Selected articles from the IEEE BIBM International Conference on Bioinformatics & Biomedicine (BIBM) 2016: systems biology. The full contents of the supplement are available online at https://bmcsystbiol.biomedcentral.com/articles/supplements/volume11supplement6.
Authors’ contributions
JH designed the research. JH and WKC proposed the methods and did theoretical analysis. JH and QYS collected the data. JH, QYS and CXQ conducted the experiments and analyze the results. JH, QYS, WKC and CXQ wrote the manuscript. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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