Benjamin A. Miller

Toward signal processing theory for graphs and non-Euclidean data

Graphs are canonical examples of high-dimensional non-Euclidean data sets, and are emerging as a common data structure in many fields. While there are many algorithms to analyze such data, a signal processing theory for evaluating these techniques akin to detection and estimation in the classical Euclidean setting remains to be developed. In this paper we show the conceptual advantages gained by formulating graph analysis problems in a signal processing framework by way of a practical example: detection of a subgraph embedded in a background graph. We describe an approach based on detection theory and provide empirical results indicating that the test statistic proposed has reasonable power to detect dense subgraphs in large random graphs.

Graphulo: Linear Algebra Graph Kernels for NoSQL Databases

Big data and the Internet of Things era continue to challenge computational systems. Several technology solutions such as NoSQL databases have been developed to deal with this challenge. In order to generate meaningful results from large datasets, analysts often use a graph representation which provides an intuitive way to work with the data. Graph vertices can represent users and events, and edges can represent the relationship between vertices. Graph algorithms are used to extract meaningful information from these very large graphs. At MIT, the Graphulo initiative is an effort to perform graph algorithms directly in NoSQL databases such as Apache Accumulo or SciDB, which have an inherently sparse data storage scheme. Sparse matrix operations have a history of efficient implementations and the Graph Basic Linear Algebra Subprogram (Graph BLAS) community has developed a set of key kernels that can be used to develop efficient linear algebra operations. However, in order to use the Graph BLAS kernels, it is important that common graph algorithms be recast using the linear algebra building blocks. In this article, we look at common classes of graph algorithms and recast them into linear algebra operations using the Graph BLAS building blocks.

A scalable signal processing architecture for massive graph analysis

In many applications, it is convenient to represent data as a graph, and often these datasets will be quite large. This paper presents an architecture for analyzing massive graphs, with a focus on signal processing applications such as modeling, filtering, and signal detection. We describe the architecture, which covers the entire processing chain, from data storage to graph construction to graph analysis and subgraph detection. The data are stored in a new format that allows easy extraction of graphs representing any relationship existing in the data. The principal analysis algorithm is the partial eigendecomposition of the modularity matrix, whose running time is discussed. A large document dataset is analyzed, and we present subgraphs that stand out in the principal eigenspace of the time-varying graphs, including behavior we regard as clutter as well as small, tightly-connected clusters that emerge over time.

Matched filtering for subgraph detection in dynamic networks

Graphs are high-dimensional, non-Euclidean data, whose utility spans a wide variety of disciplines. While their non-Euclidean nature complicates the application of traditional signal processing paradigms, it is desirable to seek an analogous detection framework. In this paper we present a matched filtering method for graph sequences, extending to a dynamic setting a previous method for the detection of anomalously dense subgraphs in a large background. In simulation, we show that this temporal integration technique enables the detection of weak subgraph anomalies than are not detectable in the static case. We also demonstrate background/foreground separation using a real background graph based on a computer network.

The cube coefficient subspace architecture for nonlinear digital predistortion

In this paper, we present the cube coefficient subspace (CCS) architecture for linearizing power amplifiers (PAs), which divides the overparametrized Volterra kernel into small, computationally efficient subkernels spanning only the portions of the full multidimensional coefficient space with the greatest impact on linearization. Using measured results from a Q-band solid state PA, we demonstrate that the CCS predistorter architecture achieves better linearization performance than state-of-the-art memory polynomials and generalized memory polynomials.

A Spectral Framework for Anomalous Subgraph Detection

A wide variety of application domains is concerned with data consisting of entities and their relationships or connections, formally represented as graphs. Within these diverse application areas, a common problem of interest is the detection of a subset of entities whose connectivity is anomalous with respect to the rest of the data. While the detection of such anomalous subgraphs has received a substantial amount of attention, no application-agnostic framework exists for analysis of signal detectability in graph-based data. In this paper, we describe a framework that enables such analysis using the principal eigenspace of a graphs residuals matrix, commonly called the modularity matrix in community detection. Leveraging this analytical tool, we show that the framework has a natural power metric in the spectral norm of the anomalous subgraphs adjacency matrix (signal power) and of the background graphs residuals matrix (noise power). We propose several algorithms based on spectral properties of the residuals matrix, with more computationally expensive techniques providing greater detection power. Detection and identification performance are presented for a number of signal and noise models, including clusters and bipartite foregrounds embedded into simple random backgrounds, as well as graphs with community structure and realistic degree distributions. The trends observed verify intuition gleaned from other signal processing areas, such as greater detection power when the signal is embedded within a less active portion of the background. We demonstrate the utility of the proposed techniques in detecting small, highly anomalous subgraphs in real graphs derived from Internet traffic and product co-purchases.

Efficient anomaly detection in dynamic, attributed graphs: Emerging phenomena and big data

When working with large-scale network data, the interconnected entities often have additional descriptive information. This additional metadata may provide insight that can be exploited for detection of anomalous events. In this paper, we use a generalized linear model for random attributed graphs to model connection probabilities using vertex metadata. For a class of such models, we show that an approximation to the exact model yields an exploitable structure in the edge probabilities, allowing for efficient scaling of a spectral framework for anomaly detection through analysis of graph residuals, and a fast and simple procedure for estimating the model parameters. In simulation, we demonstrate that taking into account both attributes and dynamics in this analysis has a much more significant impact on the detection of an emerging anomaly than accounting for either dynamics or attributes alone. We also present an analysis of a large, dynamic citation graph, demonstrating that taking additional document metadata into account emphasizes parts of the graph that would not be considered significant otherwise.

A Log-Frequency Approach to the Identification of the Wiener–Hammerstein Model

In this paper we present a simple closed-form solution to the Wiener-Hammerstein (W-H) identification problem. The identification process occurs in the log-frequency domain where magnitudes and phases are separable. We show that the theoretically optimal W-H identification is unique up to an amplitude, phase and delay ambiguity, and that the nonlinearity enables the separate identification of the individual linear time invariant (LTI) components in a W-H architecture.

Anomalous subgraph detection via Sparse Principal Component Analysis

Network datasets have become ubiquitous in many fields of study in recent years. In this paper we investigate a problem with applicability to a wide variety of domains — detecting small, anomalous subgraphs in a background graph. We characterize the anomaly in a subgraph via the well-known notion of network modularity, and we show that the optimization problem formulation resulting from our setup is very similar to a recently introduced technique in statistics called Sparse Principal Component Analysis (Sparse PCA), which is an extension of the classical PCA algorithm. The exact version of our problem formulation is a hard combinatorial optimization problem, so we consider a recently introduced semidefinite programming relaxation of the Sparse PCA problem. We show via results on simulated data that the technique is very promising.

Efficient reconstruction of block-sparse signals

In many sparse reconstruction problems, M observations are used to estimate K components in an N dimensional basis, where N > M ≫ K. The exact basis vectors, however, are not known a priori and must be chosen from an M × N matrix. Such under-determined problems can be solved using an ℓ<inf>2</inf> optimization with an ℓ<inf>1</inf> penalty on the sparsity of the solution. There are practical applications in which multiple measurements can be grouped together, so that K × P data must be estimated from M × P observations, where the ℓ<inf>1</inf> sparsity penalty is taken with respect to the vector formed using the ℓ<inf>2</inf> norms of the rows of the data matrix. In this paper we develop a computationally efficient block partitioned ho-motopy method for reconstructing K × P data from M × P observations using a grouped sparsity constraint, and compare its performance to other block reconstruction algorithms.