What is robust PCA for anomaly detection?

What is robust PCA for anomaly detection?


Robust PCA (Principal Component Analysis) is a technique for decomposing a data matrix into low-rank and sparse components.

Moreover, a general-purpose anomaly detection algorithm deployable in tiny iot devices!

An IoT (Internet of Things) device is a physical object or device that is connected to the internet and has the ability to collect and exchange data with other devices and systems over the internet. These devices are often small, embedded systems that are designed to perform specific functions, such as sensing or actuating, and are connected to the internet via wireless or wired networks.

IoT devices can range from simple sensors that collect and transmit data. Such as temperature or humidity sensors. To complex devices that can control other systems, such as smart home devices, industrial control systems, or autonomous vehicles. IoT devices can also include wearables, smart appliances, security cameras, and more.

The data collected by IoT devices can be analyzed and used to improve processes, optimize performance, and make more informed decisions. For example, data collected from smart home devices can be used to optimize energy usage and improve home security, while data collected from industrial sensors can be used to improve manufacturing processes and reduce downtime.

Emmanuel Candes and others in a series of papers published in the early 2010 first developed the technique.

The scientific background of Robust PCA lies in the field of compressive sensing.

Compressive sensing (also known as compressive sampling or sparse signal sampling) is a relatively new field of signal processing that has emerged over the last two decades. It is based on the idea that many signals, particularly in the natural world, are sparse or compressible in some domain.

The scientific background of compressive sensing lies in the field of information theory and signal processing, particularly in the development of the Shannon-Nyquist sampling theorem. The Shannon-Nyquist theorem states that in order to perfectly reconstruct a signal, it must be sampled at a rate of at least twice its highest frequency component.

Example of magnitude of the Fourier transform of a bandlimited function

While this theorem is fundamental to digital signal processing, it can be quite inefficient in situations where the signal has a sparse representation.

X(f) (top blue) and XA(f) (bottom blue) are continuous Fourier transforms of two different functions, and (not shown). When the functions are sampled at rate , the images (green) are added to the original transforms (blue) when one examines the discrete-time Fourier transforms (DTFT) of the sequences. In this hypothetical example, the DTFTs are identical, which means the sampled sequences are identical, even though the original continuous pre-sampled functions are not. If these were audio signals, and might not sound the same. But their samples (taken at rate fs) are identical and would lead to identical reproduced sounds; thus xA(t) is an alias of x(t) at this sample rate.

Compressive sensing seeks to overcome this inefficiency by exploiting the sparsity or compressibility of the signal in some domain.

Spectrum, Xs(f), of a properly sampled bandlimited signal (blue). And the adjacent DTFT images (green) that do not overlap. A brick-wall low-pass filter, H(f), removes the images, leaves the original spectrum, X(f), additionally recovers the original signal from its samples.

The basic idea is to acquire a small number of linear measurements of the signal. However, that are still sufficient to recover the underlying sparse representation.

A family of sinusoids at the critical frequency, all having the same sample sequences of alternating +1 and –1. That is, thus they all are aliases of each other, even though their frequency is not above half the sample rate.

This can become accomplished by designing a sensing matrix.

Example of the retrieval of an unknown signal (gray line) from few measurements (black dots) using the knowledge that the signal is sparse in the Hermite polynomials basis (purple dots show the retrieved coefficients). Jacopo Bertolotti – https://twitter.com/j_bertolotti/status/1214918749838594048

One that maps the high-dimensional signal to a low-dimensional space. While preserving enough information to allow for accurate recovery of the signal.

In the context of compressive sensing, a sensing matrix is a mathematical matrix used to map a high-dimensional signal or data onto a lower-dimensional space.

Typically designed to collect only a small number of linear measurements of the signal or data. While still retaining enough information to allow for accurate recovery of the original signal.

The sensing matrix typically becomes represented as an m x n matrix. Where m is the number of measurements and n is the dimensionality of the signal or data.

Furthermore, the elements of the sensing matrix often become chosen randomly from a known distribution. Such as a Gaussian or Bernoulli distribution, although other structured designs are also possible.

The design of the sensing matrix is critical to the success of compressive sensing.

As it determines the number of measurements required for accurate recovery of the signal or data. A well-designed sensing matrix should capture the essential features of the signal or data. While reducing the number of measurements needed for reconstruction.

The choice of sensing matrix depends on the specific application and the nature of the signal or data under measurement. Commonly used sensing matrices include random Gaussian matrices, Bernoulli matrices, and Fourier matrices.

As well as structured matrices such as wavelet or curvelet dictionaries. Furthermore, the choice of sensing matrix significantly impact the accuracy and efficiency of your compressive sensing algorithm.

Robust PCA (Principal Component Analysis) works well with data that has outliers!

Supernova remnant SNR E0519-69.0 in the Large Magellanic Cloud

Because it is designed to separate the data into low-rank and sparse components, where the sparse component contains the outliers.

Robust PCA builds on the idea of low-rank matrix recovery. And furthermore extends it to handle outliers or other forms of noise. Traditional PCA assumes that the data becomes Gaussian distributed and contains no outliers, which can cause problems when outliers are present.

Robust PCA, on the other hand, uses a technique called Principal Component Pursuit (PCP) to decompose the data matrix into a low-rank matrix and a sparse matrix. The low-rank matrix captures the underlying structure of the data, while the sparse matrix captures the outliers or noise. A robust technique to outliers as a result of becoming based on the assumption that the low-rank structure of the data remains preserved. However, even in the presence of outliers.

Robust PCA enjoys a wide range of applications!

Such as in computer vision, image processing, and data analysis, where often becomes used for denoising, background subtraction, and feature extraction. Moreover, proven an effective tool for dealing with data containing outliers or other forms of noise. And thus, making it a valuable addition to the data scientist’s toolbox.

In conclusion, Robust PCA distinguishes between the underlying structure of the data and the outliers, and separates them accordingly. As a result, making it a powerful tool for dealing with data that has outliers or other forms of noise.

Deep Learning God Yann LeCun – Facebook / Meta’s Director of Artificial Intelligence & Courant Prof.


What is robust PCA for anomaly detection?