**Sequential Subspace Changepoint Detection**

We consider the sequential changepoint detection problem of detecting changes that are characterized by a subspace structure which is manifested in the covariance matrix. In particular, the covariance structure changes from an identity matrix to an unknown spiked covariance model. We consider three sequential changepoint detection procedures: The exact cumulative sum (CUSUM) that assumes knowledge of all parameters, the largest eigenvalue procedure and a novel Subspace-CUSUM algorithm with the last two being used for the case when unknown parameters are present. By leveraging the extreme eigenvalue distribution from random matrix theory and modeling the non-negligible temporal correlation in the sequence of detection statistics due to the sliding window approach, we provide theoretical approximations to the average run length (ARL) and the expected detection delay (EDD) for the largest eigenvalue procedure. The three methods are compared to each other using simulations. … **Scale Adaptive Dictionary Learning (SADL)**

Dictionary learning has been widely used in many image processing tasks. In most of these methods, the number of basis vectors is either set by experience or coarsely evaluated empirically. In this paper we propose a new Scale Adaptive Dictionary Learning (SADL) framework, which jointly estimates suitable scales and corresponding atoms in an adaptive fashion according to the training data, without the need of prior information. We design an atom counting function and develop a reliable numerical scheme to solve the challenging optimization problem. Extensive experiments on texture and video datasets demonstrate quantitatively and visually that our method can estimate the scale, without damaging the sparse reconstruction ability. … **Ranking Relative Principal Component Attributes Network Model (REL-PCANet)**

In 2018, at the World Economic Forum in Davos it was presented a new countries’ economic performance metric named the Inclusive Development Index (IDI) composed of 12 indicators. The new metric implies that countries might need to realize structural reforms for improving both economic expansion and social inclusion performance. That is why, it is vital for the IDI calculation method to have strong statistical and mathematical basis, so that results are accurate and transparent for public purposes. In the current work, we propose a novel approach for the IDI estimation – the Ranking Relative Principal Component Attributes Network Model (REL-PCANet). The model is based on RELARM and RankNet principles and combines elements of PCA, techniques applied in image recognition and learning to rank mechanisms. Also, we define a new approach for estimation of target probabilities matrix to reflect dynamic changes in countries’ inclusive development. Empirical study proved that REL-PCANet ensures reliable and robust scores and rankings, thus is recommended for practical implementation. … **Relaxed Online Maximum Margin Algorithm (ROMMA)**

An incremental algorithm for training linear threshold functions: the Relaxed Online Maximum Margin Algorithm, or ROMMA. ROMMA can be viewed as an approximation to the algorithm that repeatedly chooses the hyperplane that classifies previously seen examples correctly with the maximum margin. It is known that such a maximum-margin hypothesis can be computed by minimizing the length of the weight vector subject to a number of linear constraints. ROMMA works by maintaining a relatively simple relaxation of these constraints that can be efficiently updated. We prove a mistake bound for ROMMA that is the same as that proved for the perceptron algorithm. Our analysis implies that the maximum-margin algorithm also satisfies this mistake bound; this is the first worst-case performance guarantee for this algorithm. We describe some experiments using ROMMA and a variant that updates its hypothesis more aggressively as batch algorithms to recognize handwritten digits. The computational complexity and simplicity of these algorithms is similar to that of perceptron algorithm, but their generalization is much better. We show that a batch algorithm based on aggressive ROMMA converges to the fixed threshold SVM hypothesis. …

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