Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. A key ingredient in solving the upper-level problem is the exact solution of the lower-level problem which is practically infeasible. In this work we propose to solve these problems using inexact derivative-free optimization algorithms which never require to solve the lower-level problem exactly. We provide global convergence and worst-case complexity analysis of our approach, and test our proposed framework on ROF-denoising and learning MRI sampling patterns. Dynamically adjusting the lower-level accuracy yields learned parameters with similar reconstruction quality as high-accuracy evaluations but with dramatic reductions in computational work (up to 100 times faster in some cases).

Many modern data analyses benefit from explicitly modeling dependence structure in data - such as measurements across time or space, ordered words in a sentence, or genes in a genome. A gold standard evaluation technique is structured cross-validation (CV), which leaves out some data subset (such as data within a time interval or data in a geographic region) in each fold. But CV here can be prohibitively slow due to the need to rerun already-expensive learning algorithms many times. Previous work has shown approximate cross-validation (ACV) methods provide a fast and provably accurate alternative in the setting of empirical risk minimization. But this existing ACV work is restricted to simpler models by the assumptions that (i) data across CV folds are independent and (ii) an exact initial model fit is available. In structured data analyses, both these assumptions are often untrue. In the present work, we address (i) by extending ACV to CV schemes with dependence structure between the folds. To address (ii), we verify - both theoretically and empirically - that ACV quality deteriorates smoothly with noise in the initial fit. We demonstrate the accuracy and computational benefits of our proposed methods on a diverse set of real-world applications.

The spectral evolution model aims to characterize the growth of large networks (i.e., how they evolve as new edges are established) in terms of the eigenvalue decomposition of the adjacency matrices. It assumes that, while eigenvectors remain constant, eigenvalues evolve in a predictable manner over time. This paper extends the original formulation of the model twofold. First, it presents a method to compute an approximation of the spectral evolution of eigenvalues based on the Rayleigh quotient. Second, it proposes an algorithm to estimate the evolution of eigenvalues by extrapolating only a fraction of their approximated values. The proposed model is used to characterize mention networks of users who posted tweets that include the most popular political hashtags in Colombia from August 2017 to August 2018 (the period which concludes the disarmament of the Revolutionary Armed Forces of Colombia). To evaluate the extent to which the spectral evolution model resembles these networks, link prediction methods based on learning algorithms (i.e., extrapolation and regression) and graph kernels are implemented. Experimental results show that the learning algorithms deployed on the approximated trajectories outperform the usual kernel and extrapolation methods at predicting the formation of new edges.

Within the machine learning community, the widely-used uniform convergence framework seeks to answer the question of how complex models such as modern neural networks can generalize well to new data. This approach bounds the test error of the \emph{worst-case} model one could have fit to the data, which presents fundamental limitations. In this paper, we revisit the statistical mechanics approach to learning, which instead attempts to understand the behavior of the \emph{typical} model. To quantify this typicality in the setting of over-parameterized linear classification, we develop a methodology to compute the full distribution of test errors among interpolating classifiers. We apply our method to compute this distribution for several real and synthetic datasets. We find that in many regimes of interest, an overwhelming proportion of interpolating classifiers have good test performance, even though---as we demonstrate---classifiers with very high test error do exist. This shows that the behavior of the worst-case model can deviate substantially from that of the usual model. Furthermore, we observe that for a given training set and testing distribution, there is a critical value $\varepsilon^* > 0$ which is \emph{typical}, in the sense that nearly all test errors eventually concentrate around it. Based on these empirical results, we study this phenomenon theoretically under simplifying assumptions on the data, and we derive simple asymptotic expressions for both the distribution of test errors as well as the critical value $\varepsilon^*$. Both of these results qualitatively reproduce our empirical findings. Our results show that the usual style of analysis in statistical learning theory may not be fine-grained enough to capture the good generalization performance observed in practice, and that approaches based on the statistical mechanics of learning offer a promising alternative.

In this paper we study Meta learning of Gaussian graphical models. In our setup, each task has a different true precision matrix, each with a possibly different support (i.e., set of edges in the graph). We assume that the union of the supports of all the true precision matrices (i.e., the true support union) is small in size, which relates to sparse graphs. We propose to pool all the samples from different tasks, and estimate a single precision matrix by $\ell_1$-regularized maximum likelihood estimation. We show that with high probability, the support of the estimated single precision matrix is equal to the true support union, provided a sufficient number of samples per task $n \in O((\log N)/K)$, for $N$ nodes and $K$ tasks. That is, one requires less samples per task when more tasks are available. We prove a matching information-theoretic lower bound for the necessary number of samples, which is $n \in \Omega((\log N)/K)$, and thus, our algorithm is minimax optimal. Synthetic experiments validate our theory.

Cluster analysis organizes data into sensible groupings and is one of fundamental modes of understanding and learning. The widely used K-means and hierarchical clustering methods can be dramatically suboptimal due to local minima. Recently introduced convex clustering approach formulates clustering as a convex optimization problem and ensures a globally optimal solution. However, the state-of-the-art convex clustering algorithms, based on the alternating direction method of multipliers (ADMM) or the alternating minimization algorithm (AMA), require large computation and memory space, which limits their applications. In this paper, we develop a very efficient smoothing proximal gradient algorithm (Sproga) for convex clustering. Our Sproga is faster than ADMM- or AMA-based convex clustering algorithms by one to two orders of magnitude. The memory space required by Sproga is less than that required by ADMM and AMA by at least one order of magnitude. Computer simulations and real data analysis show that Sproga outperforms several well known clustering algorithms including K-means and hierarchical clustering. The efficiency and superior performance of our algorithm will help convex clustering to find its wide application.

Identifying causal relationships for a treatment intervention is a fundamental problem in health sciences. Randomized controlled trials (RCTs) are considered the gold standard for identifying causal relationships. However, recent advancements in the theory of causal inference based on the foundations of structural causal models (SCMs) have allowed the identification of causal relationships from observational data, under certain assumptions. Survival analysis provides standard measures, such as the hazard ratio, to quantify the effects of an intervention. While hazard ratios are widely used in clinical and epidemiological studies for RCTs, a principled approach does not exist to compute hazard ratios for observational studies with SCMs. In this work, we review existing approaches to compute hazard ratios as well as their causal interpretation, if it exists. We also propose a novel approach to compute hazard ratios from observational studies using backdoor adjustment through SCMs and do-calculus. Finally, we evaluate the approach using experimental data for Ewing's sarcoma.

The information contained in hierarchical topology, intrinsic to many networks, is currently underutilised. A novel architecture is explored which exploits this information through a multiscale decomposition. A dendrogram is produced by a Girvan-Newman hierarchical clustering algorithm. It is segmented and fed through graph convolutional layers, allowing the architecture to learn multiple scale latent space representations of the network, from fine to coarse grained. The architecture is tested on a benchmark citation network, demonstrating competitive performance. Given the abundance of hierarchical networks, possible applications include quantum molecular property prediction, protein interface prediction and multiscale computational substrates for partial differential equations.

Background: The expanding research in the field of long non-coding RNAs(lncRNAs) showed abnormal expression of lncRNAs in many complex diseases. Accurately identifying lncRNA-disease association is essential in understanding lncRNA functionality and disease mechanism. There are many machine learning techniques involved in the prediction of lncRNA-disease association which use different biological interaction networks and associated features. Feature learning from the network structured data is one of the limiting factors of machine learning-based methods. Graph neural network based techniques solve this limitation by unsupervised feature learning. Deep belief networks (DBN) are recently used in biological network analysis to learn the latent representations of network features. Method: In this paper, we propose a DBN based lncRNA-disease association prediction model (DBNLDA) from lncRNA, disease and miRNA interactions. The architecture contains three major modules-network construction, DBN based feature learning and neural network-based prediction. First, we constructed three heterogeneous networks such as lncRNA-miRNA similarity (LMS), disease-miRNA similarity (DMS) and lncRNA-disease association (LDA) network. From the node embedding matrices of similarity networks, lncRNA-disease representations were learned separately by two DBN based subnetworks. The joint representation of lncRNA-disease was learned by a third DBN from outputs of the two subnetworks mentioned. This joint feature representation was used to predict the association score by an ANN classifier. Result: The proposed method obtained AUC of 0.96 and AUPR of 0.967 when tested against standard dataset used by the state-of-the-art methods. Analysis on breast, lung and stomach cancer cases also affirmed the effectiveness of DBNLDA in predicting significant lncRNA-disease associations.

We study the problem of PAC learning one-hidden-layer ReLU networks with $k$ hidden units on $\mathbb{R}^d$ under Gaussian marginals in the presence of additive label noise. For the case of positive coefficients, we give the first polynomial-time algorithm for this learning problem for $k$ up to $\tilde{O}(\sqrt{\log d})$. Previously, no polynomial time algorithm was known, even for $k=3$. This answers an open question posed by~\cite{Kliv17}. Importantly, our algorithm does not require any assumptions about the rank of the weight matrix and its complexity is independent of its condition number. On the negative side, for the more general task of PAC learning one-hidden-layer ReLU networks with arbitrary real coefficients, we prove a Statistical Query lower bound of $d^{\Omega(k)}$. Thus, we provide a separation between the two classes in terms of efficient learnability. Our upper and lower bounds are general, extending to broader families of activation functions.