Dual-to-kernel learning with ideals

In this paper, we propose a theory which unifies kernel learning and symbolic algebraic methods. We show that both worlds are inherently dual to each other, and we use this duality to combine the structure-awareness of algebraic methods with the efficiency and generality of kernels. The main idea lies in relating polynomial rings to feature space, and ideals to manifolds, then exploiting this generative-discriminative duality on kernel matrices. We illustrate this by proposing two algorithms, IPCA and AVICA, for simultaneous manifold and feature learning, and test their accuracy on synthetic and real world data.Comment: 15 pages, 1 figur

A Computational Algebra Approach to the Reverse Engineering of Gene Regulatory Networks

This paper proposes a new method to reverse engineer gene regulatory networks from experimental data. The modeling framework used is time-discrete deterministic dynamical systems, with a finite set of states for each of the variables. The simplest examples of such models are Boolean networks, in which variables have only two possible states. The use of a larger number of possible states allows a finer discretization of experimental data and more than one possible mode of action for the variables, depending on threshold values. Furthermore, with a suitable choice of state set, one can employ powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. To perform well, the algorithm requires wildtype together with perturbation time courses. This makes it suitable for small to meso-scale networks rather than networks on a genome-wide scale. The complexity of the algorithm is quadratic in the number of variables and cubic in the number of time points. The algorithm is validated on a recently published Boolean network model of segment polarity development in Drosophila melanogaster.Comment: 28 pages, 5 EPS figures, uses elsart.cl

A Multiscale Pyramid Transform for Graph Signals

Multiscale transforms designed to process analog and discrete-time signals and images cannot be directly applied to analyze high-dimensional data residing on the vertices of a weighted graph, as they do not capture the intrinsic geometric structure of the underlying graph data domain. In this paper, we adapt the Laplacian pyramid transform for signals on Euclidean domains so that it can be used to analyze high-dimensional data residing on the vertices of a weighted graph. Our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling, graph reduction, and filtering and interpolation of signals on graphs. Equipped with appropriate notions of these operations, we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure

Neural networks and support vector machines based bio-activity classification

Classification of various compounds into their respective biological activity classes is important in drug discovery applications from an early phase virtual compound filtering and screening point of view. In this work two types of neural networks, multi layer perceptron (MLP) and radial basis functions (RBF), and support vector machines (SVM) were employed for the classification of three types of biologically active enzyme inhibitors. Both of the networks were trained with back propagation learning method with chemical compounds whose active inhibition properties were previously known. A group of topological indices, selected with the help of principle component analysis (PCA) were used as descriptors. The results of all the three classification methods show that the performance of both the neural networks is better than the SVM

Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI

With the objective of employing graphs toward a more generalized theory of signal processing, we present a novel sampling framework for (wavelet-)sparse signals defined on circulant graphs which extends basic properties of Finite Rate of Innovation (FRI) theory to the graph domain, and can be applied to arbitrary graphs via suitable approximation schemes. At its core, the introduced Graph-FRI-framework states that any K-sparse signal on the vertices of a circulant graph can be perfectly reconstructed from its dimensionality-reduced representation in the graph spectral domain, the Graph Fourier Transform (GFT), of minimum size 2K. By leveraging the recently developed theory of e-splines and e-spline wavelets on graphs, one can decompose this graph spectral transformation into the multiresolution low-pass filtering operation with a graph e-spline filter, and subsequent transformation to the spectral graph domain; this allows to infer a distinct sampling pattern, and, ultimately, the structure of an associated coarsened graph, which preserves essential properties of the original, including circularity and, where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017

Entropy of Overcomplete Kernel Dictionaries

In signal analysis and synthesis, linear approximation theory considers a linear decomposition of any given signal in a set of atoms, collected into a so-called dictionary. Relevant sparse representations are obtained by relaxing the orthogonality condition of the atoms, yielding overcomplete dictionaries with an extended number of atoms. More generally than the linear decomposition, overcomplete kernel dictionaries provide an elegant nonlinear extension by defining the atoms through a mapping kernel function (e.g., the gaussian kernel). Models based on such kernel dictionaries are used in neural networks, gaussian processes and online learning with kernels. The quality of an overcomplete dictionary is evaluated with a diversity measure the distance, the approximation, the coherence and the Babel measures. In this paper, we develop a framework to examine overcomplete kernel dictionaries with the entropy from information theory. Indeed, a higher value of the entropy is associated to a further uniform spread of the atoms over the space. For each of the aforementioned diversity measures, we derive lower bounds on the entropy. Several definitions of the entropy are examined, with an extensive analysis in both the input space and the mapped feature space.Comment: 10 page

Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201