A Spectral Approach to Network Design and Experimental Design

Over the last decade, the spectral sparsification technique has become a powerful tool in designing fast graph algorithms for various problems with numerous applications. In this thesis, we extend this spectral approach, and show that it is also very powerful in designing approximation algorithms for classical network design and experimental design problems. The central piece in this thesis is a problem called spectral rounding, which is inspired by spectral sparsification and studied in an earlier work on experimental design. In this problem, we are given vectors \vv_1, \ldots, \vv_m each with a non-negative cost, and a fractional solution \vx \in [0,1]^m. The task is to find an integral solution \vz \in \{0,1\}^m such that the spectrum of the integral solution is similar to the one of the fractional solution, i.e.~\sum_i \vz(i) \cdot \vv_i \vv_i^\top \approx \sum_i \vx(i) \cdot \vv_i \vv_i^\top, and the integral cost is approximately equal to the fractional cost. We observe that the spectral rounding problem underlies a large family of network design and experimental design problems. With this perspective, we bring new insights into these well-studied problems. For network design, we show that the spectral rounding technique provides a novel and general approach to significantly extend the scope of problems that can be solved efficiently. For experimental design, we show that the spectral rounding technique provides a unified and elegant framework that matches and improves all known existing algorithmic results. There are two key techniques that we will use in this thesis. The first one is regret minimization, which is well-known to the online optimization community and has been used for spectral sparsification. We use it to control the spectrum of the integral solution in the spectral rounding problem. The second key technique is concentration inequalities for analyzing adaptive random sampling processes, which enable us to satisfy spectral and linear constraints simultaneously with high probability

A stochastic spectral analysis of transcriptional regulatory cascades

The past decade has seen great advances in our understanding of the role of noise in gene regulation and the physical limits to signaling in biological networks. Here we introduce the spectral method for computation of the joint probability distribution over all species in a biological network. The spectral method exploits the natural eigenfunctions of the master equation of birth-death processes to solve for the joint distribution of modules within the network, which then inform each other and facilitate calculation of the entire joint distribution. We illustrate the method on a ubiquitous case in nature: linear regulatory cascades. The efficiency of the method makes possible numerical optimization of the input and regulatory parameters, revealing design properties of, e.g., the most informative cascades. We find, for threshold regulation, that a cascade of strong regulations converts a unimodal input to a bimodal output, that multimodal inputs are no more informative than bimodal inputs, and that a chain of up-regulations outperforms a chain of down-regulations. We anticipate that this numerical approach may be useful for modeling noise in a variety of small network topologies in biology

Functional control of network dynamics using designed Laplacian spectra

Complex real-world phenomena across a wide range of scales, from aviation and internet traffic to signal propagation in electronic and gene regulatory circuits, can be efficiently described through dynamic network models. In many such systems, the spectrum of the underlying graph Laplacian plays a key role in controlling the matter or information flow. Spectral graph theory has traditionally prioritized unweighted networks. Here, we introduce a complementary framework, providing a mathematically rigorous weighted graph construction that exactly realizes any desired spectrum. We illustrate the broad applicability of this approach by showing how designer spectra can be used to control the dynamics of various archetypal physical systems. Specifically, we demonstrate that a strategically placed gap induces chimera states in Kuramoto-type oscillator networks, completely suppresses pattern formation in a generic Swift-Hohenberg model, and leads to persistent localization in a discrete Gross-Pitaevskii quantum network. Our approach can be generalized to design continuous band gaps through periodic extensions of finite networks.Comment: 9 pages, 5 figure

BASiS: Batch Aligned Spectral Embedding Space

Graph is a highly generic and diverse representation, suitable for almost any data processing problem. Spectral graph theory has been shown to provide powerful algorithms, backed by solid linear algebra theory. It thus can be extremely instrumental to design deep network building blocks with spectral graph characteristics. For instance, such a network allows the design of optimal graphs for certain tasks or obtaining a canonical orthogonal low-dimensional embedding of the data. Recent attempts to solve this problem were based on minimizing Rayleigh-quotient type losses. We propose a different approach of directly learning the eigensapce. A severe problem of the direct approach, applied in batch-learning, is the inconsistent mapping of features to eigenspace coordinates in different batches. We analyze the degrees of freedom of learning this task using batches and propose a stable alignment mechanism that can work both with batch changes and with graph-metric changes. We show that our learnt spectral embedding is better in terms of NMI, ACC, Grassman distance, orthogonality and classification accuracy, compared to SOTA. In addition, the learning is more stable.Comment: 14 pages, 10 figure