Efficient Semidefinite Spectral Clustering via Lagrange Duality

We propose an efficient approach to semidefinite spectral clustering (SSC), which addresses the Frobenius normalization with the positive semidefinite (p.s.d.) constraint for spectral clustering. Compared with the original Frobenius norm approximation based algorithm, the proposed algorithm can more accurately find the closest doubly stochastic approximation to the affinity matrix by considering the p.s.d. constraint. In this paper, SSC is formulated as a semidefinite programming (SDP) problem. In order to solve the high computational complexity of SDP, we present a dual algorithm based on the Lagrange dual formalization. Two versions of the proposed algorithm are proffered: one with less memory usage and the other with faster convergence rate. The proposed algorithm has much lower time complexity than that of the standard interior-point based SDP solvers. Experimental results on both UCI data sets and real-world image data sets demonstrate that 1) compared with the state-of-the-art spectral clustering methods, the proposed algorithm achieves better clustering performance; and 2) our algorithm is much more efficient and can solve larger-scale SSC problems than those standard interior-point SDP solvers.Comment: 13 page

G-Protein Coupled Receptor Signaling Architecture of Mammalian Immune Cells

A series of recent studies on large-scale networks of signaling and metabolic systems revealed that a certain network structure often called “bow-tie network” are observed. In signaling systems, bow-tie network takes a form with diverse and redundant inputs and outputs connected via a small numbers of core molecules. While arguments have been made that such network architecture enhances robustness and evolvability of biological systems, its functional role at a cellular level remains obscure. A hypothesis was proposed that such a network function as a stimuli-reaction classifier where dynamics of core molecules dictate downstream transcriptional activities, hence physiological responses against stimuli. In this study, we examined whether such hypothesis can be verified using experimental data from Alliance for Cellular Signaling (AfCS) that comprehensively measured GPCR related ligands response for B-cell and macrophage. In a GPCR signaling system, cAMP and Ca2+ act as core molecules. Stimuli-response for 32 ligands to B-Cells and 23 ligands to macrophages has been measured. We found that ligands with correlated changes of cAMP and Ca2+ tend to cluster closely together within the hyperspaces of both cell types and they induced genes involved in the same cellular processes. It was found that ligands inducing cAMP synthesis activate genes involved in cell growth and proliferation; cAMP and Ca2+ molecules that increased together form a feedback loop and induce immune cells to migrate and adhere together. In contrast, ligands without a core molecules response are scattered throughout the hyperspace and do not share clusters. G-protein coupling receptors together with immune response specific receptors were found in cAMP and Ca2+ activated clusters. Analyses have been done on the original software applicable for discovering ‘bow-tie’ network architectures within the complex network of intracellular signaling where ab initio clustering has been implemented as well. Groups of potential transcription factors for each specific group of genes were found to be partly conserved across B-Cell and macrophage. A series of findings support the hypothesis

Doubly Stochastic Neighbor Embedding on Spheres

Stochastic Neighbor Embedding (SNE) methods minimize the divergence between the similarity matrix of a high-dimensional data set and its counterpart from a low-dimensional embedding, leading to widely applied tools for data visualization. Despite their popularity, the current SNE methods experience a crowding problem when the data include highly imbalanced similarities. This implies that the data points with higher total similarity tend to get crowded around the display center. To solve this problem, we introduce a fast normalization method and normalize the similarity matrix to be doubly stochastic such that all the data points have equal total similarities. Furthermore, we show empirically and theoretically that the doubly stochasticity constraint often leads to embeddings which are approximately spherical. This suggests replacing a flat space with spheres as the embedding space. The spherical embedding eliminates the discrepancy between the center and the periphery in visualization, which efficiently resolves the crowding problem. We compared the proposed method (DOSNES) with the state-of-the-art SNE method on three real-world datasets and the results clearly indicate that our method is more favorable in terms of visualization quality. DOSNES is freely available at http://yaolubrain.github.io/dosnes/. (C) 2019 The Authors. Published by Elsevier B.V.Peer reviewe

Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry

Convex optimization is a well-established research area with applications in almost all fields. Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a more general set of convex programs known as semi-definite programs and second-order cone programs. However, it has been established that these methods are excessively slow for high dimensions, i.e., they suffer from the curse of dimensionality. On the other hand, optimization algorithms on manifold have shown great ability in finding solutions to nonconvex problems in reasonable time. This paper is interested in solving a subset of convex optimization using a different approach. The main idea behind Riemannian optimization is to view the constrained optimization problem as an unconstrained one over a restricted search space. The paper introduces three manifolds to solve convex programs under particular box constraints. The manifolds, called the doubly stochastic, symmetric and the definite multinomial manifolds, generalize the simplex also known as the multinomial manifold. The proposed manifolds and algorithms are well-adapted to solving convex programs in which the variable of interest is a multidimensional probability distribution function. Theoretical analysis and simulation results testify the efficiency of the proposed method over state of the art methods. In particular, they reveal that the proposed framework outperforms conventional generic and specialized solvers, especially in high dimensions

An Efficient Approach to Solve the Large-Scale Semidefinite Programming Problems

Solving the large-scale problems with semidefinite programming (SDP) constraints is of great importance in modeling and model reduction of complex system, dynamical system, optimal control, computer vision, and machine learning. However, existing SDP solvers are of large complexities and thus unavailable to deal with large-scale problems. In this paper, we solve SDP using matrix generation, which is an extension of the classical column generation. The exponentiated gradient algorithm is also used to solve the special structure subproblem of matrix generation. The numerical experiments show that our approach is efficient and scales very well with the problem dimension. Furthermore, the proposed algorithm is applied for a clustering problem. The experimental results on real datasets imply that the proposed approach outperforms the traditional interior-point SDP solvers in terms of efficiency and scalability