Analysis of histopathological images is critical in cancer diagnosis and treatment. Due to the huge size of histopathological images and the varied number of imaging records per patient, many existing works analyze the Whole Slide Image (WSI) as a bag in which its patches are instances. However, these approaches are limited to analyzing the patches in a fixed shape, while the malignant lesions can form varied shapes. To address this challenge, in this article we propose a Multi-Instance Multi-Shape Support Vector Machine (MIMSSVM) to analyze the multiple images (instances) jointly where each instance consists of multiple patches in various shapes. In our approach, we can identify the different morphologic abnormalities of nuclei shapes from the multiple images. In addition to the multi-instance multi-shape learning capability, we derive an efficient solution algorithm to optimize the proposed model that scales well to a large number of features. Our experimental results show our new method outperforms the existing SVMs and deep learning models in histopathological classification. The proposed model also identifies the tissue segments in an image exhibiting an indication of an abnormality which provides utility in the early detection of malignant tumors. All these promising experimental results have demonstrated the effectiveness of our new method. We anticipate that our new method is of interest to biomedical engineering communities beyond WSI research and have open sourced the code of our method online. The implementation of our proposed MIMSSVM model is publicly available at https://github.com/hoonseo0409/MIMSSVM .

Multi-instance learning (MIL) is an area of machine learning that handles data that is organized into sets of instances known as bags. Traditionally, MIL is used in the supervised-learning setting and is able to classify bags which can contain any number of instances. This property allows MIL to be naturally applied to solve the problems in a wide variety of real-world applications from computer vision to healthcare. However, many traditional MIL algorithms do not scale efficiently to large datasets. In this paper we present a novel Primal-Dual Multi-Instance Support Vector Machine (pdMISVM) derivation and implementation that can operate efficiently on large scale data. Our method relies on an algorithm derived using a multi-block variation of the alternating direction method of multipliers (ADMM). The approach presented in this work is able to scale to large-scale data since it avoids iteratively solving quadratic programming problems which are generally used to optimize MIL algorithms based on SVMs. In addition, we modify our derivation to include an additional optimization designed to avoid solving a least-squares problem during our algorithm; this optimization increases the utility of our approach to handle a large number of features as well as bags. Finally, we apply our approach to synthetic and real-world multi-instance datasets to illustrate the scalability, promising predictive performance, and interpretability of our proposed method. We end our discussion with an extension of our approach to handle non-linear decision boundaries. Code and data for our methods are available online at: https://github.com/minds-mines/pdMISVM.jl.

Nonnegative Matrix Factorization (NMF) is broadly used to determine class membership in a variety of clustering applications. From movie recommendations and image clustering to visual feature extractions, NMF has applications to solve a large number of knowledge discovery and data mining problems. Traditional optimization methods, such as the Multiplicative Updating Algorithm (MUA), solves the NMF problem by utilizing an auxiliary function to ensure that the objective monotonically decreases. Although the objective in MUA converges, there exists no proof to show that the learned matrix factors converge as well. Without this rigorous analysis, the clustering performance and stability of the NMF algorithms cannot be guaranteed. To address this knowledge gap, in this article, we study the factor-bounded NMF problem and provide a solution algorithm with proven convergence by rigorous mathematical analysis, which ensures that both the objective and matrix factors converge. In addition, we show the relationship between MUA and our solution followed by an analysis of the convergence of MUA. Experiments on both toy data and real-world datasets validate the correctness of our proposed method and its utility as an effective clustering algorithm.