Robust Subspace Tracking Algorithms in Signal Processing: A Brief Survey

Principal component analysis (PCA) and subspace estimation (SE) are popular data analysis tools and used in a wide range of applications. The main interest in PCA/SE is for dimensionality reduction and low-rank approximation purposes. The emergence of big data streams have led to several essential issues for performing PCA/SE. Among them are (i) the size of such data streams increases over time, (ii) the underlying models may be time-dependent, and (iii) problem of dealing with the uncertainty and incompleteness in data. A robust variant of PCA/SE for such data streams, namely robust online PCA or robust subspace tracking (RST), has been introduced as a good alternative. The main goal of this paper is to provide a brief survey on recent RST algorithms in signal processing. Particularly, we begin this survey by introducing the basic ideas of the RST problem. Then, different aspects of RST are reviewed with respect to different kinds of non-Gaussian noises and sparse constraints. Our own contributions on this topic are also highlighted.

Rastreador recursivo de subespaços e ordem do modelo aplicado à estimação adaptativa do número de componentes harmônicos e inter-harmônicos em sinais elétricos

Em muitas aplicações se faz necessário conhecer previamente o número exato de componentes senoidais presentes em um sinal, como é o caso dos métodos paramétricos de alta resolução de frequência para estimação de parâmetros de componentes em sinais elétricos na presença de ruído. Este trabalho propõe a utilização de um método para estimação do número de componentes harmônicos e inter-harmônicos em sinais elétricos baseado em técnicas de subespaços e em teoria da informação. Trata-se de um método recursivo que se utiliza do rastreador de subespaços PASTd (Projection Approximation Subspace Tracking with deflation), que apresenta complexidade computacional baixa de ordem O(mr), onde m denota o tamanho do vetor de entrada e r a dimensão do subespaço de sinal. Além disso, o método utiliza um detector de ordem baseado nos critérios advindos da teoria da informação AIC (Akaike Information Criterion) e MDL (Minimum Description Length). Resultados de diversos testes de simulação serão gerados e discutidos com o intuito de caracterizar o método e identificar possíveis pontos fortes e fracos.

Seismic noise attenuation using an online subspace tracking algorithm

SUMMARY We propose a new low-rank based noise attenuation method using an efficient algorithm for tracking subspaces from highly corrupted seismic observations. The subspace tracking algorithm requires only basic linear algebraic manipulations. The algorithm is derived by analysing incremental gradient descent on the Grassmannian manifold of subspaces. When the multidimensional seismic data are mapped to a low-rank space, the subspace tracking algorithm can be directly applied to the input low-rank matrix to estimate the useful signals. Since the subspace tracking algorithm is an online algorithm, it is more robust to random noise than traditional truncated singular value decomposition (TSVD) based subspace tracking algorithm. Compared with the state-of-the-art algorithms, the proposed denoising method can obtain better performance. More specifically, the proposed method outperforms the TSVD-based singular spectrum analysis method in causing less residual noise and also in saving half of the computational cost. Several synthetic and field data examples with different levels of complexities demonstrate the effectiveness and robustness of the presented algorithm in rejecting different types of noise including random noise, spiky noise, blending noise, and coherent noise.