Learning inducing points and uncertainty on molecular data

Uncertainty control and scalability to large datasets are the two main issues for the deployment of Gaussian process models into the autonomous material and chemical space exploration pipelines. One way to address both of these issues is by introducing the latent inducing variables and choosing the right approximation for the marginal log-likelihood objective. Here, we show that variational learning of the inducing points in the high-dimensional molecular descriptor space significantly improves both the prediction quality and uncertainty estimates on test configurations from a sample molecular dynamics dataset. Additionally, we show that inducing points can learn to represent the configurations of the molecules of different types that were not present within the initialization set of inducing points. Among several evaluated approximate marginal log-likelihood objectives, we show that the predictive log-likelihood provides both the predictive quality comparable to the exact Gaussian process model and excellent uncertainty control. Finally, we comment on whether a machine learning model makes predictions by interpolating the molecular configurations in high-dimensional descriptor space. We show that despite our intuition, and even for densely sampled molecular dynamics datasets, most of the predictions are done in the extrapolation regime.Comment: 8 pages, 5 figure

Signals on Graphs: Uncertainty Principle and Sampling

In many applications, the observations can be represented as a signal defined over the vertices of a graph. The analysis of such signals requires the extension of standard signal processing tools. In this work, first, we provide a class of graph signals that are maximally concentrated on the graph domain and on its dual. Then, building on this framework, we derive an uncertainty principle for graph signals and illustrate the conditions for the recovery of band-limited signals from a subset of samples. We show an interesting link between uncertainty principle and sampling and propose alternative signal recovery algorithms, including a generalization to frame-based reconstruction methods. After showing that the performance of signal recovery algorithms is significantly affected by the location of samples, we suggest and compare a few alternative sampling strategies. Finally, we provide the conditions for perfect recovery of a useful signal corrupted by sparse noise, showing that this problem is also intrinsically related to vertex-frequency localization properties.Comment: This article is the revised version submitted to the IEEE Transactions on Signal Processing on May, 2016; first revision was submitted on January, 2016; original manuscript was submitted on July, 2015. The work includes 16 pages, 8 figure

An introduction to hypergraph signal processing

Developing tools to analyze signals defined over a graph is a research area that is attracting a significant amount of contributions because of its many applications. However, a graph representation does not capture the overall information about the data, as it implicitly takes into account only pairwise relations. The goal of this paper is to extend signal processing tools to signals defined over hypergraphs, which represent a formal framework to describe multi-way relations among the data. First, we suggest alternative ways to introduce a Fourier Transform (FT) for signals defined over hypergraphs and, in particular, for simplicial complexes. Then, building on the notion of Fourier Transform, we derive a sampling theorem aimed at identifying the minimum number of samples necessary to encode all information about band-limited hypergraph signals

On the degrees of freedom of signals on graphs

Continuous-time signals are well known for not being perfectly localized in both time and frequency domains. Conversely, a signal defined over the vertices of a graph can be perfectly localized in both vertex and frequency domains. We derive the conditions ensuring the validity of this property and then, building on this theory, we provide the conditions for perfect reconstruction of a graph signal from its samples. Next, we provide a finite step algorithm for the reconstruction of a band-limited signal from its samples and then we show the effect of sampling a non perfectly band-limited signal and show how to select the bandwidth that minimizes the mean square reconstruction error

Analytic signal in many dimensions

Submitted to Applied and Computational Harmonic AnalysisIn this paper we extend analytic signal to the multidimensional case. First it is shown how to obtain separate phase-shifted components and how to combine them into instantaneous amplitude and phase. Secondly we define the proper hypercomplex analytic signal as a holo-morphic hypercomplex function on the boundary of polydisk in the hypercomplex space. Next it is shown that the correct phase-shifted components can be obtained by positive frequency restriction of the Scheffers-Fourier transform based on the commutative and associative algebra generated by the set of elliptic hypercomplex numbers. Moreover we demonstrate that for d > 2 there is no corresponding Clifford-Fourier transform that allows to recover phase-shifted components correctly. Finally the euclidean-domain construction of instantaneous amplitude is extended to manifold and manifold-like graphs and point clouds

Analytic signal in many dimensions

Submitted to Applied and Computational Harmonic AnalysisIn this paper we extend analytic signal to the multidimensional case. First it is shown how to obtain separate phase-shifted components and how to combine them into instantaneous amplitude and phase. Secondly we define the proper hypercomplex analytic signal as a holo-morphic hypercomplex function on the boundary of polydisk in the hypercomplex space. Next it is shown that the correct phase-shifted components can be obtained by positive frequency restriction of the Scheffers-Fourier transform based on the commutative and associative algebra generated by the set of elliptic hypercomplex numbers. Moreover we demonstrate that for d > 2 there is no corresponding Clifford-Fourier transform that allows to recover phase-shifted components correctly. Finally the euclidean-domain construction of instantaneous amplitude is extended to manifold and manifold-like graphs and point clouds

Multidimensional analytic signal with application on graphs

International audienceIn this work we provide an extension to analytic signal method for multidimensional signals. First, expressions for separate phase-shifted components are given. Second, we show that phase-shifted components could be obtained from hypercomplex Fourier transform in the algebra of commuta-tive hypercomplex numbers. Finally we apply multidimen-sional analytic signal method to signals defined on graphs