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.
Graduate School of Science and Technology, Gunma University（群马大学科学技术研究生院）
Recent studies utilize multiple kernel learning to deal with incomplete-data problem. In this study, we introduce new methods that do not only complete multiple incomplete kernel matrices simultaneously, but also allow control of the flexibility of the model by parameterizing the model matrix. By imposing restrictions on the model covariance, overfitting of the data is avoided. A limitation of kernel matrix estimations done via optimization of an objective function is that the positive definiteness of the result is not guaranteed. In view of this limitation, our proposed methods employ the Log…
This work studies low-rank approximation of a positive semidefinite matrix from partial entries via nonconvex optimization. We characterized how well local-minimum based low-rank factorization approximates a fixed positive semidefinite matrix without any assumptions on the rank-matching, the condition number or eigenspace incoherence parameter. Furthermore, under certain assumptions on rank-matching and well-boundedness of condition numbers and eigenspace incoherence parameters, a corollary of our main theorem improves the state-of-the-art sampling rate results for nonconvex matrix completion …
The ubiquitous proliferation of online social networks has led to the widescale emergence of relational graphs expressing unique patterns in link formation and descriptive user node features. Matrix Factorization and Completion have become popular methods for Link Prediction due to the low rank nature of mutual node friendship information, and the availability of parallel computer architectures for rapid matrix processing. Current Link Prediction literature has demonstrated vast performance improvement through the utilization of sparsity in addition to the low rank matrix assumption. However, …
Recent advances in matrix completion enable data imputation in full-rank matrices by exploiting low dimensional (nonlinear) latent structure. In this paper, we develop a new model for high rank matrix completion (HRMC), together with batch and online methods to fit the model and out-of-sample extension to complete new data. The method works by (implicitly) mapping the data into a high dimensional polynomial feature space using the kernel trick; importantly, the data occupies a low dimensional subspace in this feature space, even when the original data matrix is of full-rank. We introduce an ex…
We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. In addition, varieties can be used to model a richer class of nonlinear quadratic and higher degree curves and surfaces. We…
Universitat Politècnica de Catalunya-Barcelona Tech（加泰罗尼亚理工大学-巴塞罗那理工）；University of Minnesota（明尼苏达大学）
Prior information can be incorporated in matrix completion to improve estimation accuracy and extrapolate the missing entries. Reproducing kernel Hilbert spaces provide tools to leverage the said prior information, and derive more reliable algorithms. This paper analyzes the generalization error of such approaches, and presents numerical tests confirming the theoretical results.
This paper develops new methods to recover the missing entries of a high-rank or even full-rank matrix when the intrinsic dimension of the data is low compared to the ambient dimension. Specifically, we assume that the columns of a matrix are generated by polynomials acting on a low-dimensional intrinsic variable, and wish to recover the missing entries under this assumption. We show that we can identify the complete matrix of minimum intrinsic dimension by minimizing the rank of the matrix in a high dimensional feature space. We develop a new formulation of the resulting problem using the ker…
University of California, Davis（加利福尼亚大学戴维斯分校）；Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）
Matrix completion has attracted attention in many fields, including statistics, applied mathematics, and electrical engineering. Most of the works focus on the independent sampling models under which the observed entries are sampled independently. Motivated by applications in the integration of knowledge graphs derived from multi-source biomedical data such as those from Electronic Health Records (EHR) and biomedical text, we propose the {\bf B}lock-wise {\bf O}verlapping {\bf N}oisy {\bf M}atrix {\bf I}ntegration (BONMI) to treat blockwise missingness of symmetric matrices representing relate…
Universitat Politècnica de Catalunya-Barcelona Tech（加泰罗尼亚理工大学-巴塞罗那理工大学）；University of Minnesota（明尼苏达大学）
Matrix completion and extrapolation (MCEX) are dealt with here over reproducing kernel Hilbert spaces (RKHSs) in order to account for prior information present in the available data. Aiming at a faster and low-complexity solver, the task is formulated as a kernel ridge regression. The resultant MCEX algorithm can also afford online implementation, while the class of kernel functions also encompasses several existing approaches to MC with prior information. Numerical tests on synthetic and real datasets show that the novel approach performs faster than widespread methods such as alternating lea…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；Aalto University School of Science and Technology（阿尔托大学科学与技术学院）
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, b…
We propose Bayesian extensions of two nonparametric regression methods which are kernel and mutual $k$-nearest neighbor regression methods. Derived based on Gaussian process models for regression, the extensions provide distributions for target value estimates and the framework to select the hyperparameters. It is shown that both the proposed methods asymptotically converge to kernel and mutual $k$-nearest neighbor regression methods, respectively. The simulation results show that the proposed methods can select proper hyperparameters and are better than or comparable to the former methods for…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；Aalto University（阿尔托大学）；Aix-Marseille University（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；Turing Center for Living Systems (CENTURI)（图灵生命系统中心（CENTURI））
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 with Cross-View Kernel Transfer (CVKT) procedure, in which the features of the other views are transformed to represent the view under consideration. The transformations are learned with kernel alignment to the known part of the kernel matrix, allowing for finding gene…
The abundance of training data is not guaranteed in various supervised learning applications. One of these situations is the post-earthquake regional damage assessment of buildings. Querying the damage label of each building requires a thorough inspection by experts, and thus, is an expensive task. A practical approach is to sample the most informative buildings in a sequential learning scheme. Active learning methods recommend the most informative cases that are able to maximally reduce the generalization error. The information theoretic measure of mutual information (MI) is one of the most e…
Massachusetts Institute of Technology（麻省理工学院）；University of California, San Diego（加州大学圣地亚哥分校）；Broad Institute of MIT and Harvard（布罗德研究所（麻省理工学院与哈佛大学））
Matrix completion problems arise in many applications including recommendation systems, computer vision, and genomics. Increasingly larger neural networks have been successful in many of these applications, but at considerable computational costs. Remarkably, taking the width of a neural network to infinity allows for improved computational performance. In this work, we develop an infinite width neural network framework for matrix completion that is simple, fast, and flexible. Simplicity and speed come from the connection between the infinite width limit of neural networks and kernels known as…
Recently, a method called the Mutual Information Neural Estimator (MINE) that uses neural networks has been proposed to estimate mutual information and more generally the Kullback-Leibler (KL) divergence between two distributions. The method uses the Donsker-Varadhan representation to arrive at the estimate of the KL divergence and is better than the existing estimators in terms of scalability and flexibility. The output of MINE algorithm is not guaranteed to be a consistent estimator. We propose a new estimator that instead of searching among functions characterized by neural networks searche…
We develop a new collaborative filtering (CF) method that combines both previously known users' preferences, i.e. standard CF, as well as product/user attributes, i.e. classical function approximation, to predict a given user's interest in a particular product. Our method is a generalized low rank matrix completion problem, where we learn a function whose inputs are pairs of vectors -- the standard low rank matrix completion problem being a special case where the inputs to the function are the row and column indices of the matrix. We solve this generalized matrix completion problem using tenso…
Transformers Provably Learn Directed Acyclic Graphs via Kernel-Guided Mutual Information
Department of Mathematics, National University of Singapore, Singapore（新加坡国立大学数学系）；Department of Statistics and Data Science, University of Pennsylvania, USA（宾夕法尼亚大学统计与数据科学系）；Department of Electrical and Computer Engineering, The Ohio State University, USA（俄亥俄州立大学电子与计算机工程系）
We consider the problem of learning Stochastic Differential Equations of the form $dX_t = f(X_t)dt+σ(X_t)dW_t $ from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions $f$, $σ$, and stochastic process $dW_t$ representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion and data adapted kernels learned via a new variant of cross validation. Our approach can be decompo…
Laboratoire Hubert Curien, CNRS (UMR 5516) Universit\'e Jean Monnet F-42000, Saint-Etienne, France Email；Laboratoire d'Informatique de Paris-Nord, CNRS (UMR 7030) Universit\'e Paris 13, Sorbonne Paris Cit\'e F-93430, Villetaneuse, France Email
The ability of a human being to extrapolate previously gained knowledge to other domains inspired a new family of methods in machine learning called transfer learning. Transfer learning is often based on the assumption that objects in both target and source domains share some common feature and/or data space. In this paper, we propose a simple and intuitive approach that minimizes iteratively the distance between source and target task distributions by optimizing the kernel target alignment (KTA). We show that this procedure is suitable for transfer learning by relating it to Hilbert-Schmidt I…
Centre for Robotics Research（机器人研究中心）；Department of Engineering, King’s College London（伦敦国王学院工程系）
Human beings are able to understand objectives and learn by simply observing others perform a task. Imitation learning methods aim to replicate such capabilities, however, they generally depend on access to a full set of optimal states and actions taken with the agent's actuators and from the agent's point of view. In this paper, we introduce a new algorithm - called Disentangling Generative Adversarial Imitation Learning (DisentanGAIL) - with the purpose of bypassing such constraints. Our algorithm enables autonomous agents to learn directly from high dimensional observations of an expert per…
