Aliaksei Sandryhaila

Discrete Signal Processing on Graphs

In social settings, individuals interact through webs of relationships. Each individual is a node in a complex network (or graph) of interdependencies and generates data, lots of data. We label the data by its source, or formally stated, we index the data by the nodes of the graph. The resulting signals (data indexed by the nodes) are far removed from time or image signals indexed by well ordered time samples or pixels. DSP, discrete signal processing, provides a comprehensive, elegant, and efficient methodology to describe, represent, transform, analyze, process, or synthesize these well ordered time or image signals. This paper extends to signals on graphs DSP and its basic tenets, including filters, convolution, z-transform, impulse response, spectral representation, Fourier transform, frequency response, and illustrates DSP on graphs by classifying blogs, linear predicting and compressing data from irregularly located weather stations, or predicting behavior of customers of a mobile service provider.

Big Data Analysis with Signal Processing on Graphs: Representation and processing of massive data sets with irregular structure

Analysis and processing of very large data sets, or big data, poses a significant challenge. Massive data sets are collected and studied in numerous domains, from engineering sciences to social networks, biomolecular research, commerce, and security. Extracting valuable information from big data requires innovative approaches that efficiently process large amounts of data as well as handle and, moreover, utilize their structure. This article discusses a paradigm for large-scale data analysis based on the discrete signal processing (DSP) on graphs (DSPG). DSPG extends signal processing concepts and methodologies from the classical signal processing theory to data indexed by general graphs. Big data analysis presents several challenges to DSPG, in particular, in filtering and frequency analysis of very large data sets. We review fundamental concepts of DSPG, including graph signals and graph filters, graph Fourier transform, graph frequency, and spectrum ordering, and compare them with their counterparts from the classical signal processing theory. We then consider product graphs as a graph model that helps extend the application of DSPG methods to large data sets through efficient implementation based on parallelization and vectorization. We relate the presented framework to existing methods for large-scale data processing and illustrate it with an application to data compression.

Discrete Signal Processing on Graphs: Frequency Analysis

Signals and datasets that arise in physical and engineering applications, as well as social, genetics, biomolecular, and many other domains, are becoming increasingly larger and more complex. In contrast to traditional time and image signals, data in these domains are supported by arbitrary graphs. Signal processing on graphs extends concepts and techniques from traditional signal processing to data indexed by generic graphs. This paper studies the concepts of low and high frequencies on graphs, and low-, high- and band-pass graph signals and graph filters. In traditional signal processing, these concepts are easily defined because of a natural frequency ordering that has a physical interpretation. For signals residing on graphs, in general, there is no obvious frequency ordering. We propose a definition of total variation for graph signals that naturally leads to a frequency ordering on graphs and defines low-, high-, and band-pass graph signals and filters. We study the design of graph filters with specified frequency response, and illustrate our approach with applications to sensor malfunction detection and data classification.

Discrete Signal Processing on Graphs: Sampling Theory

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform. The sampled signal coefficients form a new graph signal, whose corresponding graph structure preserves the first-order difference of the original graph signal. For general graphs, an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise; for graphs whose graph Fourier transforms are frames with maximal robustness to erasures as well as for Erdös-Rényi graphs, random sampling leads to perfect recovery with high probability. We further establish the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs. To handle full-band graph signals, we propose a graph filter bank based on sampling theory on graphs. Finally, we apply the proposed sampling theory to semi-supervised classification of online blogs and digit images, where we achieve similar or better performance with fewer labeled samples compared to previous work.

Signal Recovery on Graphs: Variation Minimization

We consider the problem of signal recovery on graphs. Graphs model data with complex structure as signals on a graph. Graph signal recovery recovers one or multiple smooth graph signals from noisy, corrupted, or incomplete measurements. We formulate graph signal recovery as an optimization problem, for which we provide a general solution through the alternating direction methods of multipliers. We show how signal inpainting, matrix completion, robust principal component analysis, and anomaly detection all relate to graph signal recovery and provide corresponding specific solutions and theoretical analysis. We validate the proposed methods on real-world recovery problems, including online blog classification, bridge condition identification, temperature estimation, recommender system for jokes, and expert opinion combination of online blog classification.

Efficient Compression of QRS Complexes Using Hermite Expansion

We propose a novel algorithm for the compression of ECG signals, in particular QRS complexes. The algorithm is based on the expansion of signals with compact support into a basis of discrete Hermite functions. These functions can be constructed by sampling continuous Hermite functions at specific sampling points. They form an orthogonal basis in the underlying signal space. The proposed algorithm relies on the theory of signal models based on orthogonal polynomials. We demonstrate that the constructed discrete Hermite functions have important ad- vantages compared to continuous Hermite functions, which have previously been suggested for the compression of QRS complexes. Our algorithm achieves higher compression ratios compared with previously reported algorithms based on continuous Hermite functions, discrete Fourier, cosine, or wavelet transforms.

Generating FPGA-Accelerated DFT Libraries

We present a domain-specific approach to generate high-performance hardware-software partitioned implementations of the discrete Fourier transform (DFT) in fixed point precision. The partitioning strategy is a heuristic based on the DFTs divide-and-conquer algorithmic structure and fine tuned by the feedback-driven exploration of candidate designs. We have integrated this approach in the Spiral linear-transform code-generation framework to support push-button automatic implementation. We present evaluations of hardware-software DFT implementations running on the embedded PowerPC processor and the reconfigurable fabric of the Xilinx Virtex-II Pro FPGA. In our experiments, the 1D and 2D DFTs FPGA-accelerated libraries exhibit between 2 and 7.5 times higher performance (operations per second) and up to 2.5 times better energy efficiency (operations per Joule) than the software-only version.

Signal denoising on graphs via graph filtering

Signal recovery from noisy measurements is an important task that arises in many areas of signal processing. In this paper, we consider this problem for signals represented with graphs using a recently developed framework of discrete signal processing on graphs. We formulate graph signal denoising as an optimization problem and derive an exact closed-form solution expressed by an inverse graph filter, as well as an approximate iterative solution expressed by a standard graph filter. We evaluate the obtained algorithms by applying them to measurement denoising for temperature sensors and opinion combination for multiple experts.

Finite-time distributed consensus through graph filters

We propose a new framework for distributed computation of average consensus. The presented framework leads to a systematic design of iterative algorithms that compute the consensus exactly, are guaranteed to converge in finite time, are computationally efficient, and require no online memory. We demonstrate that our approach is applicable to a broad class of networks. For remaining networks, our framework leads to the construction of approximating algorithms for consensus that are also guaranteed to compute in finite time. Our approach is inspired by graph filters introduced by the theoretical framework of signal processing on graphs.

Discrete signal processing on graphs: Graph fourier transform

We propose a novel discrete signal processing framework for the representation and analysis of datasets with complex structure. Such datasets arise in many social, economic, biological, and physical networks. Our framework extends traditional discrete signal processing theory to structured datasets by viewing them as signals represented by graphs, so that signal coefficients are indexed by graph nodes and relations between them are represented by weighted graph edges. We discuss the notions of signals and filters on graphs, and define the concepts of the spectrum and Fourier transform for graph signals. We demonstrate their relation to the generalized eigenvector basis of the graph adjacency matrix and study their properties. As a potential application of the graph Fourier transform, we consider the efficient representation of structured data that utilizes the sparseness of graph signals in the frequency domain.