### Mutual Kernel Matrix Completion

With the huge influx of various data nowadays, extracting knowledge from them has become an interesting but tedious task among data scientists, particularly when the data come in heterogeneous form and have missing information. Many data completion techniques had been introduced, especially in the advent of kernel methods. However, among the many data completion techniques available in the literature, studies about mutually completing several incomplete kernel matrices have not been given much attention yet. In this paper, we present a new method, called Mutual Kernel Matrix Completion (MKMC) algorithm, that tackles this problem of mutually inferring the missing entries of multiple kernel matrices by combining the notions of data fusion and kernel matrix completion, applied on biological data sets to be used for classification task. We first introduced an objective function that will be minimized by exploiting the EM algorithm, which in turn results to an estimate of the missing entries of the kernel matrices involved. The completed kernel matrices are then combined to produce a model matrix that can be used to further improve the obtained estimates. An interesting result of our study is that the E-step and the M-step are given in closed form, which makes our algorithm efficient in terms of time and memory. After completion, the (completed) kernel matrices are then used to train an SVM classifier to test how well the relationships among the entries are preserved. Our empirical results show that the proposed algorithm bested the traditional completion techniques in preserving the relationships among the data points, and in accurately recovering the missing kernel matrix entries. By far, MKMC offers a promising solution to the problem of mutual estimation of a number of relevant incomplete kernel matrices.

### On the Maximum Entropy Property of the First-Order Stable Spline Kernel and its Implications

A new nonparametric approach for system identification has been recently proposed where the impulse response is seen as the realization of a zero--mean Gaussian process whose covariance, the so--called stable spline kernel, guarantees that the impulse response is almost surely stable. Maximum entropy properties of the stable spline kernel have been pointed out in the literature. In this paper we provide an independent proof that relies on the theory of matrix extension problems in the graphical model literature and leads to a closed form expression for the inverse of the first order stable spline kernel as well as to a new factorization in the form $UWU^\top$ with $U$ upper triangular and $W$ diagonal. Interestingly, all first--order stable spline kernels share the same factor $U$ and $W$ admits a closed form representation in terms of the kernel hyperparameter, making the factorization computationally inexpensive. Maximum likelihood properties of the stable spline kernel are also highlighted. These results can be applied both to improve the stability and to reduce the computational complexity associated with the computation of stable spline estimators.

### Integrating Features and Similarities: Flexible Models for Heterogeneous Multiview Data

We present a probabilistic framework for learning with heterogeneous multiview data where some views are given as ordinal, binary, or real-valued feature matrices, and some views as similarity matrices. Our framework has the following distinguishing aspects: (i) a unified latent factor model for integrating information from diverse feature (ordinal, binary, real) and similarity based views, and predicting the missing data in each view, leveraging view correlations; (ii) seamless adaptation to binary/multiclass classification where data consists of multiple feature and/or similarity-based views; and (iii) an efficient, variational inference algorithm which is especially flexible in modeling the views with ordinal-valued data (by learning the cutpoints for the ordinal data), and extends naturally to streaming data settings. Our framework subsumes methods such as multiview learning and multiple kernel learning as special cases. We demonstrate the effectiveness of our framework on several real-world and benchmarks datasets.

### Kernel transfer over multiple views for missing data completion

We consider the kernel completion problem with the presence of multiple views in the data. In this context the data samples can be fully missing in some views, creating missing columns and rows to the kernel matrices that are calculated individually for each view. We propose to solve the problem of completing the kernel matrices by transferring the features of the other views to represent the view under consideration. We align the known part of the kernel matrix with a new kernel built from the features of the other views. We are thus able to find generalizable structures in the kernel under completion, and represent it accurately. Its missing values can be predicted with the data available in other views. We illustrate the benefits of our approach with simulated data and multivariate digits dataset, as well as with real biological datasets from studies of pattern formation in early \textit{Drosophila melanogaster} embryogenesis.

### Multi-view Kernel Completion

In this paper, we introduce the first method that (1) can complete kernel matrices with completely missing rows and columns as opposed to individual missing kernel values, (2) does not require any of the kernels to be complete a priori, and (3) can tackle non-linear kernels. These aspects are necessary in practical applications such as integrating legacy data sets, learning under sensor failures and learning when measurements are costly for some of the views. The proposed approach predicts missing rows by modelling both within-view and between-view relationships among kernel values. We show, both on simulated data and real world data, that the proposed method outperforms existing techniques in the restricted settings where they are available, and extends applicability to new settings.