Replica theory for learning curves for Gaussian processes on random graphs

Statistical physics approaches can be used to derive accurate predictions for the performance of inference methods learning from potentially noisy data, as quantified by the learning curve defined as the average error versus number of training examples. We analyse a challenging problem in the area of non-parametric inference where an effectively infinite number of parameters has to be learned, specifically Gaussian process regression. When the inputs are vertices on a random graph and the outputs noisy function values, we show that replica techniques can be used to obtain exact performance predictions in the limit of large graphs. The covariance of the Gaussian process prior is defined by a random walk kernel, the discrete analogue of squared exponential kernels on continuous spaces. Conventionally this kernel is normalised only globally, so that the prior variance can differ between vertices; as a more principled alternative we consider local normalisation, where the prior variance is uniform

Prediction of Atomization Energy Using Graph Kernel and Active Learning

Data-driven prediction of molecular properties presents unique challenges to the design of machine learning methods concerning data structure/dimensionality, symmetry adaption, and confidence management. In this paper, we present a kernel-based pipeline that can learn and predict the atomization energy of molecules with high accuracy. The framework employs Gaussian process regression to perform predictions based on the similarity between molecules, which is computed using the marginalized graph kernel. To apply the marginalized graph kernel, a spatial adjacency rule is first employed to convert molecules into graphs whose vertices and edges are labeled by elements and interatomic distances, respectively. We then derive formulas for the efficient evaluation of the kernel. Specific functional components for the marginalized graph kernel are proposed, while the effect of the associated hyperparameters on accuracy and predictive confidence are examined. We show that the graph kernel is particularly suitable for predicting extensive properties because its convolutional structure coincides with that of the covariance formula between sums of random variables. Using an active learning procedure, we demonstrate that the proposed method can achieve a mean absolute error of 0.62 +- 0.01 kcal/mol using as few as 2000 training samples on the QM7 data set

Searching for rewards in graph-structured spaces

How do people generalize and explore structured spaces? We study human behavior on a multi-armed bandit task, where rewards are influenced by the connectivity structure of a graph. A detailed predictive model comparison shows that a Gaussian Process regression model using a diffusion kernel is able to best describe participant choices, and also predict judgments about expected reward and confidence. This model unifies psychological models of function learning with the Successor Representation used in reinforcement learning, thereby building a bridge between different models of generalization

Empirical stationary correlations for semi-supervised learning on graphs

In semi-supervised learning on graphs, response variables observed at one node are used to estimate missing values at other nodes. The methods exploit correlations between nearby nodes in the graph. In this paper we prove that many such proposals are equivalent to kriging predictors based on a fixed covariance matrix driven by the link structure of the graph. We then propose a data-driven estimator of the correlation structure that exploits patterns among the observed response values. By incorporating even a small fraction of observed covariation into the predictions, we are able to obtain much improved prediction on two graph data sets.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS293 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian Semi-supervised Learning with Graph Gaussian Processes

We propose a data-efficient Gaussian process-based Bayesian approach to the semi-supervised learning problem on graphs. The proposed model shows extremely competitive performance when compared to the state-of-the-art graph neural networks on semi-supervised learning benchmark experiments, and outperforms the neural networks in active learning experiments where labels are scarce. Furthermore, the model does not require a validation data set for early stopping to control over-fitting. Our model can be viewed as an instance of empirical distribution regression weighted locally by network connectivity. We further motivate the intuitive construction of the model with a Bayesian linear model interpretation where the node features are filtered by an operator related to the graph Laplacian. The method can be easily implemented by adapting off-the-shelf scalable variational inference algorithms for Gaussian processes.Comment: To appear in NIPS 2018 Fixed an error in Figure 2. The previous arxiv version contains two identical sub-figure

Graph Laplacians and their convergence on random neighborhood graphs

Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero. We show that for a uniform measure on the submanifold all graph Laplacians have the same limit up to constants. However in the case of a non-uniform measure on the submanifold only the so called random walk graph Laplacian converges to the weighted Laplace-Beltrami operator.Comment: Improved presentation, typos corrected, to appear in JML

Matérn Gaussian Processes on Graphs.

Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process models are readily available when the input space is Euclidean, the choice is much more limited for Gaussian processes whose input space is an undirected graph. In this work, we leverage the stochastic partial differential equation characterization of Mat´ern Gaussian processes—a widelyused model class in the Euclidean setting—to study their analog for undirected graphs. We show that the resulting Gaussian processes inherit various attractive properties of their Euclidean and Riemannian analogs and provide techniques that allow them to be trained using standard methods, such as inducing points. This enables graph Mat´ern Gaussian processes to be employed in mini-batch and non-conjugate settings, thereby making them more accessible to practitioners and easier to deploy within larger learning frameworks

Mat\'ern Gaussian Processes on Graphs

Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process models are readily available when the input space is Euclidean, the choice is much more limited for Gaussian processes whose input space is an undirected graph. In this work, we leverage the stochastic partial differential equation characterization of Mat\'ern Gaussian processes - a widely-used model class in the Euclidean setting - to study their analog for undirected graphs. We show that the resulting Gaussian processes inherit various attractive properties of their Euclidean and Riemannian analogs and provide techniques that allow them to be trained using standard methods, such as inducing points. This enables graph Mat\'ern Gaussian processes to be employed in mini-batch and non-conjugate settings, thereby making them more accessible to practitioners and easier to deploy within larger learning frameworks

Kern-basierte Lernverfahren für das virtuelle Screening

We investigate the utility of modern kernel-based machine learning methods for ligand-based virtual screening. In particular, we introduce a new graph kernel based on iterative graph similarity and optimal assignments, apply kernel principle component analysis to projection error-based novelty detection, and discover a new selective agonist of the peroxisome proliferator-activated receptor gamma using Gaussian process regression. Virtual screening, the computational ranking of compounds with respect to a predicted property, is a cheminformatics problem relevant to the hit generation phase of drug development. Its ligand-based variant relies on the similarity principle, which states that (structurally) similar compounds tend to have similar properties. We describe the kernel-based machine learning approach to ligand-based virtual screening; in this, we stress the role of molecular representations, including the (dis)similarity measures defined on them, investigate effects in high-dimensional chemical descriptor spaces and their consequences for similarity-based approaches, review literature recommendations on retrospective virtual screening, and present an example workflow. Graph kernels are formal similarity measures that are defined directly on graphs, such as the annotated molecular structure graph, and correspond to inner products. We review graph kernels, in particular those based on random walks, subgraphs, and optimal vertex assignments. Combining the latter with an iterative graph similarity scheme, we develop the iterative similarity optimal assignment graph kernel, give an iterative algorithm for its computation, prove convergence of the algorithm and the uniqueness of the solution, and provide an upper bound on the number of iterations necessary to achieve a desired precision. In a retrospective virtual screening study, our kernel consistently improved performance over chemical descriptors as well as other optimal assignment graph kernels. Chemical data sets often lie on manifolds of lower dimensionality than the embedding chemical descriptor space. Dimensionality reduction methods try to identify these manifolds, effectively providing descriptive models of the data. For spectral methods based on kernel principle component analysis, the projection error is a quantitative measure of how well new samples are described by such models. This can be used for the identification of compounds structurally dissimilar to the training samples, leading to projection error-based novelty detection for virtual screening using only positive samples. We provide proof of principle by using principle component analysis to learn the concept of fatty acids. The peroxisome proliferator-activated receptor (PPAR) is a nuclear transcription factor that regulates lipid and glucose metabolism, playing a crucial role in the development of type 2 diabetes and dyslipidemia. We establish a Gaussian process regression model for PPAR gamma agonists using a combination of chemical descriptors and the iterative similarity optimal assignment kernel via multiple kernel learning. Screening of a vendor library and subsequent testing of 15 selected compounds in a cell-based transactivation assay resulted in 4 active compounds. One compound, a natural product with cyclobutane scaffold, is a full selective PPAR gamma agonist (EC50 = 10 +/- 0.2 muM, inactive on PPAR alpha and PPAR beta/delta at 10 muM). The study delivered a novel PPAR gamma agonist, de-orphanized a natural bioactive product, and, hints at the natural product origins of pharmacophore patterns in synthetic ligands.Wir untersuchen moderne Kern-basierte maschinelle Lernverfahren für das Liganden-basierte virtuelle Screening. Insbesondere entwickeln wir einen neuen Graphkern auf Basis iterativer Graphähnlichkeit und optimaler Knotenzuordnungen, setzen die Kernhauptkomponentenanalyse für Projektionsfehler-basiertes Novelty Detection ein, und beschreiben die Entdeckung eines neuen selektiven Agonisten des Peroxisom-Proliferator-aktivierten Rezeptors gamma mit Hilfe von Gauß-Prozess-Regression. Virtuelles Screening ist die rechnergestützte Priorisierung von Molekülen bezüglich einer vorhergesagten Eigenschaft. Es handelt sich um ein Problem der Chemieinformatik, das in der Trefferfindungsphase der Medikamentenentwicklung auftritt. Seine Liganden-basierte Variante beruht auf dem Ähnlichkeitsprinzip, nach dem (strukturell) ähnliche Moleküle tendenziell ähnliche Eigenschaften haben. In unserer Beschreibung des Lösungsansatzes mit Kern-basierten Lernverfahren betonen wir die Bedeutung molekularer Repräsentationen, einschließlich der auf ihnen definierten (Un)ähnlichkeitsmaße. Wir untersuchen Effekte in hochdimensionalen chemischen Deskriptorräumen, ihre Auswirkungen auf Ähnlichkeits-basierte Verfahren und geben einen Literaturüberblick zu Empfehlungen zur retrospektiven Validierung, einschließlich eines Beispiel-Workflows. Graphkerne sind formale Ähnlichkeitsmaße, die inneren Produkten entsprechen und direkt auf Graphen, z.B. annotierten molekularen Strukturgraphen, definiert werden. Wir geben einen Literaturüberblick über Graphkerne, insbesondere solche, die auf zufälligen Irrfahrten, Subgraphen und optimalen Knotenzuordnungen beruhen. Indem wir letztere mit einem Ansatz zur iterativen Graphähnlichkeit kombinieren, entwickeln wir den iterative similarity optimal assignment Graphkern. Wir beschreiben einen iterativen Algorithmus, zeigen dessen Konvergenz sowie die Eindeutigkeit der Lösung, und geben eine obere Schranke für die Anzahl der benötigten Iterationen an. In einer retrospektiven Studie zeigte unser Graphkern konsistent bessere Ergebnisse als chemische Deskriptoren und andere, auf optimalen Knotenzuordnungen basierende Graphkerne. Chemische Datensätze liegen oft auf Mannigfaltigkeiten niedrigerer Dimensionalität als der umgebende chemische Deskriptorraum. Dimensionsreduktionsmethoden erlauben die Identifikation dieser Mannigfaltigkeiten und stellen dadurch deskriptive Modelle der Daten zur Verfügung. Für spektrale Methoden auf Basis der Kern-Hauptkomponentenanalyse ist der Projektionsfehler ein quantitatives Maß dafür, wie gut neue Daten von solchen Modellen beschrieben werden. Dies kann zur Identifikation von Molekülen verwendet werden, die strukturell unähnlich zu den Trainingsdaten sind, und erlaubt so Projektionsfehler-basiertes Novelty Detection für virtuelles Screening mit ausschließlich positiven Beispielen. Wir führen eine Machbarkeitsstudie zur Lernbarkeit des Konzepts von Fettsäuren durch die Hauptkomponentenanalyse durch. Der Peroxisom-Proliferator-aktivierte Rezeptor (PPAR) ist ein im Zellkern vorkommender Rezeptor, der den Fett- und Zuckerstoffwechsel reguliert. Er spielt eine wichtige Rolle in der Entwicklung von Krankheiten wie Typ-2-Diabetes und Dyslipidämie. Wir etablieren ein Gauß-Prozess-Regressionsmodell für PPAR gamma-Agonisten mit chemischen Deskriptoren und unserem Graphkern durch gleichzeitiges Lernen mehrerer Kerne. Das Screening einer kommerziellen Substanzbibliothek und die anschließende Testung 15 ausgewählter Substanzen in einem Zell-basierten Transaktivierungsassay ergab vier aktive Substanzen. Eine davon, ein Naturstoff mit Cyclobutan-Grundgerüst, ist ein voller selektiver PPAR gamma-Agonist (EC50 = 10 +/- 0,2 muM, inaktiv auf PPAR alpha und PPAR beta/delta bei 10 muM). Unsere Studie liefert einen neuen PPAR gamma-Agonisten, legt den Wirkmechanismus eines bioaktiven Naturstoffs offen, und erlaubt Rückschlüsse auf die Naturstoffursprünge von Pharmakophormustern in synthetischen Liganden