Last week I looked at Singular Value Decomposition unsupervised machine learning technique as part of a four-part series on data science concepts for beginners. Remember that unsupervised machine learning is data driven rather than task driven (supervised machine learning). Today we'll be staying in the dimension reduction part of unsupervised machine learning as shown in the Cheat-sheet below and will talk about principal component analysis or PCA. In a similar manner to SVD, PCA is trying to reduce the number of dimensions for data exploration. The PCA method is trying to maximize variance of the data to make a predictive model and converts a set of possibly correlated variables into a set of linearly uncorrelated variables.

What if our data have way more than 3-dimensions? In the table is the average consumption of 17 types of food in grams per person per week for every country in the UK. The table shows some interesting variations across different food types, but overall differences aren't so notable. Let's see if PCA can eliminate dimensions to emphasize how countries differ. Already we can see something is different about Northern Ireland.

We consider an online version of the robust Principle Component Analysis (PCA), which arises naturally in time-varying source separations such as video foreground-background separation. This paper proposes a compressive online robust PCA with prior information for recursively separating a sequences of frames into sparse and low-rank components from a small set of measurements. In contrast to conventional batch-based PCA, which processes all the frames directly, the proposed method processes measurements taken from each frame. Moreover, this method can efficiently incorporate multiple prior information, namely previous reconstructed frames, to improve the separation and thereafter, update the prior information for the next frame. We utilize multiple prior information by solving $n\text{-}\ell_{1}$ minimization for incorporating the previous sparse components and using incremental singular value decomposition ($\mathrm{SVD}$) for exploiting the previous low-rank components. We also establish theoretical bounds on the number of measurements required to guarantee successful separation under assumptions of static or slowly-changing low-rank components. Using numerical experiments, we evaluate our bounds and the performance of the proposed algorithm. In addition, we apply the proposed algorithm to online video foreground and background separation from compressive measurements. Experimental results show that the proposed method outperforms the existing methods.

Various problems in data analysis and statistical genetics call for recovery of a column-sparse, low-rank matrix from noisy observations. We propose ReFACTor, a simple variation of the classical Truncated Singular Value Decomposition (TSVD) algorithm. In contrast to previous sparse principal component analysis (PCA) algorithms, our algorithm can provably reveal a low-rank signal matrix better, and often significantly better, than the widely used TSVD, making it the algorithm of choice whenever column-sparsity is suspected. Empirically, we observe that ReFACTor consistently outperforms TSVD even when the underlying signal is not sparse, suggesting that it is generally safe to use ReFACTor instead of TSVD and PCA. The algorithm is extremely simple to implement and its running time is dominated by the runtime of PCA, making it as practical as standard principal component analysis.

This formula-free summary provides a short overview about how PCA (principal component analysis) works for dimension reduction, that is, to select k features (also called variables) among a larger set of n features, with k much smaller than n. This smaller set of k features built with PCA is the best subset of k features, in the sense that it minimizes the variance of the residual noise when fitting data to a linear model. Note that PCA transforms the initial features into new ones, that are linear combinations of the original features.

Here is a short overview about how PCA (principal component analysis) works for dimension reduction, that is, to select k features (also called variables) among a larger set of n features, with k much smaller than n. This smaller set of k features built with PCA is the best subset of k features, in the sense that it minimizes the variance of the residual noise when fitting data to a linear model. Note that PCA transforms the initial features into new ones, that are linear combinations of the original features.

The sheer size of data in the modern age is not only a challenge for computer hardware but also the main bottleneck for the performance of many machine learning algorithms. The main goal of a PCA analysis is to identify patterns in data. PCA aims to detect the correlation between variables. If a strong correlation between variables exists, the attempt to reduce the dimensionality only makes sense. It is a statistical method used to reduce the number of variables in a data-set.

In a previous post I summarized the tasks and procedures available in SAS Viya Data Mining and Machine Learning. In this post, I'll dive into the unsupervised learning category which currently hosts several tasks: Kmeans, Kmodes, and Kprototypes Clustering, Outlier Detection, and a few variants of Principal Component Analysis. In unsupervised learning there are no known labels (outcomes), only attributes (inputs). Examples include clustering, association, and segmentation. Machine learning finds high density areas (in multidimensional space) that are more or less similar to each other, and identifies structures in the data that separate these areas.

Inferring macroscopic properties of physical systems from their microscopic description is an ongoing work in many disciplines of physics, like condensed matter, ultra cold atoms or quantum chromo dynamics. The most drastic changes in the macroscopic properties of a physical system occur at phase transitions, which often involve a symmetry breaking process. The theory of such phase transitions was formulated by Landau as a phenomenological model [1] and later devised from microscopic principles using the renormalization group [2, 3]. One can identify phases by knowledge of an order parameter which is zero in the disordered phase and nonzero in the ordered phase. Whereas in many known models the order parameter can be determined by symmetry considerations of the underlying Hamiltonian, there are states of matter where such a parameter can only be defined in a complicated non-local way [4]. These systems include topological states like topological insulators, quantum spin hall states [5] or quantum spin liquids [6].

Soheil Feizi and David Tse Stanford University Abstract In the era of big data, reducing data dimensionality is critical in many areas of science. Widely used Principal Component Analysis (PCA) addresses this problem by computing a low dimensional data embedding that maximally explain variance of the data. However, PCA has two major weaknesses. Firstly, it only considers linear correlations among variables (features), and secondly it is not suitable for categorical data. We resolve these issues by proposing Maximally Correlated Principal Component Analysis (MCPCA). MCPCA computes transformations of variables whose covariance matrix has the largest Ky Fan norm. Variable transformations are unknown, can be nonlinear and are computed in an optimization. MCPCA can also be viewed as a multivariate extension of Maximal Correlation. For jointly Gaussian variables we show that the covariance matrix corresponding to the identity (or the negative of the identity) transformations majorizes covariance matrices of non-identity functions. Using this result we characterize global MCPCA optimizers for nonlinear functions of jointly Gaussian variables for every rank constraint. For categorical variables we characterize global MCPCA optimizers for the rank one constraint based on the leading eigenvector of a matrix computed using pairwise joint distributions. For a general rank constraint we propose a block coordinate descend algorithm and show its convergence to stationary points of the MCPCA optimization. We compare MCPCA with PCA and other state-of-the-art dimensionality reduction methods including Isomap, LLE, multilayer autoencoders (neural networks), kernel PCA, probabilistic PCA and diffusion maps on several synthetic and real datasets. We show that MCPCA consistently provides improved performance compared to other methods. 1 Introduction Let X 1 and X 2 be two mean zero and unit variance random variables. Pearson's correlation [1] defined as ρ Pearson(X 1,X 2) E [X 1X 2 ] (1.1) is a basic statistical parameter and plays a central role in many statistical and machine learning methods such as linear regression [2], principal component analysis [3], and support vector machines [4], partially owing to its simplicity and computational efficiency. Pearson's correlation however has two main weaknesses: firstly it only captures linear dependency between variables, and secondly for discrete (categorical) variables the value of Pearson's correlation depends somewhat arbitrarily on the labels. To overcome these weaknesses, Maximal Correlation (MC) has been proposed and 1 arXiv:1702.05471v2 MC tackles the two main drawbacks of the Pearson's correlation: it models a family of nonlinear relationships between the two variables.