This work concentrates on optimization on Riemannian manifolds. The Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm is a commonly used quasi-Newton method for numerical optimization in Euclidean spaces. Riemannian LBFGS (RLBFGS) is an extension of this method to Riemannian manifolds. RLBFGS involves computationally expensive vector transports as well as unfolding recursions using adjoint vector transports. In this article, we propose two mappings in the tangent space using the inverse second root and Cholesky decomposition. These mappings make both vector transport and adjoint vector transport identity and therefore isometric. Identity vector transport makes RLBFGS less computationally expensive and its isometry is also very useful in convergence analysis of RLBFGS. Moreover, under the proposed mappings, the Riemannian metric reduces to Euclidean inner product, which is much less computationally expensive. We focus on the Symmetric Positive Definite (SPD) manifolds which are beneficial in various fields such as data science and statistics. This work opens a research opportunity for extension of the proposed mappings to other well-known manifolds.

Variants of Triplet networks are robust entities for learning a discriminative embedding subspace. There exist different triplet mining approaches for selecting the most suitable training triplets. Some of these mining methods rely on the extreme distances between instances, and some others make use of sampling. However, sampling from stochastic distributions of data rather than sampling merely from the existing embedding instances can provide more discriminative information. In this work, we sample triplets from distributions of data rather than from existing instances. We consider a multivariate normal distribution for the embedding of each class. Using Bayesian updating and conjugate priors, we update the distributions of classes dynamically by receiving the new mini-batches of training data. The proposed triplet mining with Bayesian updating can be used with any triplet-based loss function, e.g., triplet-loss or Neighborhood Component Analysis (NCA) loss. Accordingly, Our triplet mining approaches are called Bayesian Updating Triplet (BUT) and Bayesian Updating NCA (BUNCA), depending on which loss function is being used. Experimental results on two public datasets, namely MNIST and histopathology colorectal cancer (CRC), substantiate the effectiveness of the proposed triplet mining method.

Siamese neural network is a very powerful architecture for both feature extraction and metric learning. It usually consists of several networks that share weights. The Siamese concept is topology-agnostic and can use any neural network as its backbone. The two most popular loss functions for training these networks are the triplet and contrastive loss functions. In this paper, we propose two novel loss functions, named Fisher Discriminant Triplet (FDT) and Fisher Discriminant Contrastive (FDC). The former uses anchor-neighbor-distant triplets while the latter utilizes pairs of anchor-neighbor and anchor-distant samples. The FDT and FDC loss functions are designed based on the statistical formulation of the Fisher Discriminant Analysis (FDA), which is a linear subspace learning method. Our experiments on the MNIST and two challenging and publicly available histopathology datasets show the effectiveness of the proposed loss functions.

We propose a new method, named isolation Mondrian forest (iMondrian forest), for batch and online anomaly detection. The proposed method is a novel hybrid of isolation forest and Mondrian forest which are existing methods for batch anomaly detection and online random forest, respectively. iMondrian forest takes the idea of isolation, using the depth of a node in a tree, and implements it in the Mondrian forest structure. The result is a new data structure which can accept streaming data in an online manner while being used for anomaly detection. Our experiments show that iMondrian forest mostly performs better than isolation forest in batch settings and has better or comparable performance against other batch and online anomaly detection methods.

This paper proposes a novel trading system which plays the role of an artificial counselor for stock investment. In this paper, the stock future prices (technical features) are predicted using Support Vector Regression. Thereafter, the predicted prices are used to recommend which portions of the budget an investor should invest in different existing stocks to have an optimum expected profit considering their level of risk tolerance. Two different methods are used for suggesting best portions, which are Markowitz portfolio theory and fuzzy investment counselor. The first approach is an optimization-based method which considers merely technical features, while the second approach is based on Fuzzy Logic taking into account both technical and fundamental features of the stock market. The experimental results on New York Stock Exchange (NYSE) show the effectiveness of the proposed system.