Low-Rank Matrices on Graphs: Generalized Recovery & Applications

Many real world datasets subsume a linear or non-linear low-rank structure in a very low-dimensional space. Unfortunately, one often has very little or no information about the geometry of the space, resulting in a highly under-determined recovery problem. Under certain circumstances, state-of-the-art algorithms provide an exact recovery for linear low-rank structures but at the expense of highly inscalable algorithms which use nuclear norm. However, the case of non-linear structures remains unresolved. We revisit the problem of low-rank recovery from a totally different perspective, involving graphs which encode pairwise similarity between the data samples and features. Surprisingly, our analysis confirms that it is possible to recover many approximate linear and non-linear low-rank structures with recovery guarantees with a set of highly scalable and efficient algorithms. We call such data matrices as \textit{Low-Rank matrices on graphs} and show that many real world datasets satisfy this assumption approximately due to underlying stationarity. Our detailed theoretical and experimental analysis unveils the power of the simple, yet very novel recovery framework \textit{Fast Robust PCA on Graphs

A Cross-Disciplinary Review of Blockchain Research Trends and Methodologies: Topic Modeling Approach

Given the increasing interest in blockchain technology, we present a large-scale cross-disciplinary literature analysis of research on the blockchain using topic modelling with the goal of identifying the major research trends, research methodologies, and fruitful areas for further research. In particular, the analysis focuses on abstracting out research trends from relevant terms and topics related to the research disciplines of Business, Computer Science, Economics, Social Sciences, Engineering, Healthcare, and Law. A total of 2,125 articles published between 2008 to up until early 2019 in academic journals and conferences were analyzed. Results of our analysis reveal that research is bipartite between practical and research domains, with academic research on blockchain not clearly aligning with organizational and social benefits. Also, we found – 1) few inter-disciplinary publications, and 2) a small number of studies that use surveys, experiments, and case studies as their research method. Our findings also reveal that research on Blockchain in the social sciences and law is still in the embryonic stage, thus making it essential to develop more direct research efforts for Blockchain to thrive in all research disciplines

Compressive PCA for Low-Rank Matrices on Graphs

We introduce a novel framework for an approxi- mate recovery of data matrices which are low-rank on graphs, from sampled measurements. The rows and columns of such matrices belong to the span of the first few eigenvectors of the graphs constructed between their rows and columns. We leverage this property to recover the non-linear low-rank structures efficiently from sampled data measurements, with a low cost (linear in n). First, a Resrtricted Isometry Property (RIP) condition is introduced for efficient uniform sampling of the rows and columns of such matrices based on the cumulative coherence of graph eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is suggested for the sampled data. Finally, several efficient, parallel and parameter-free decoders are presented along with their theoretical analysis for decoding the low-rank and cluster indicators for the full data matrix. Thus, we overcome the computational limitations of the standard linear low-rank recovery methods for big datasets. Our method can also be seen as a major step towards efficient recovery of non- linear low-rank structures. For a matrix of size n X p, on a single core machine, our method gains a speed up of $p^2/k$ over Robust Principal Component Analysis (RPCA), where k << p is the subspace dimension. Numerically, we can recover a low-rank matrix of size 10304 X 1000, 100 times faster than Robust PCA

Understanding the Influence of Cultural Dimensions on the Interpretative Ability of People to Infer Personality from the Avatars: Evidence from Cultural Dimensions of Greece, Pakistan, Russia, and Singapore

Avatar is a customized cartoon representation of the self and many people develop inferences about individuals’ online representations through their avatar’s facial appearance. Research has shown that avatars can signal information about the personality and social desires of a person [1]. Nonetheless, customizing an avatar enables control of self-representation that could potentially moderate the true personality traits of an individual. The customized facial appearance of the avatar affects people’s ability to draw expressions [2], whereas, several cultural dimensions affect the interpretative ability of the people to construct personality inferences from the facial appearance of avatars. We found a significant relationship between neuroticism to uncertainty avoidance and masculinity, whereas, negative relationships were found between extraversion and masculinity, and agreeableness to uncertainty avoidance. The study uses three-dimensional avatars to capture detailed features and expressions on avatar faces

Robust Principal Component Analysis on Graphs

Principal Component Analysis (PCA) is the most widely used tool for linear dimensionality reduction and clustering. Still it is highly sensitive to outliers and does not scale well with respect to the number of data samples. Robust PCA solves the first issue with a sparse penalty term. The second issue can be handled with the matrix factorization model, which is however non-convex. Besides, PCA based clustering can also be enhanced by using a graph of data similarity. In this article, we introduce a new model called "Robust PCA on Graphs" which incorporates spectral graph regularization into the Robust PCA framework. Our proposed model benefits from 1) the robustness of principal components to occlusions and missing values, 2) enhanced low-rank recovery, 3) improved clustering property due to the graph smoothness assumption on the low-rank matrix, and 4) convexity of the resulting optimization problem. Extensive experiments on 8 benchmark, 3 video and 2 artificial datasets with corruptions clearly reveal that our model outperforms 10 other state-of-the-art models in its clustering and low-rank recovery tasks

Fast Robust PCA on Graphs

Mining useful clusters from high dimensional data has received significant attention of the computer vision and pattern recognition community in the recent years. Linear and non-linear dimensionality reduction has played an important role to overcome the curse of dimensionality. However, often such methods are accompanied with three different problems: high computational complexity (usually associated with the nuclear norm minimization), non-convexity (for matrix factorization methods) and susceptibility to gross corruptions in the data. In this paper we propose a principal component analysis (PCA) based solution that overcomes these three issues and approximates a low-rank recovery method for high dimensional datasets. We target the low-rank recovery by enforcing two types of graph smoothness assumptions, one on the data samples and the other on the features by designing a convex optimization problem. The resulting algorithm is fast, efficient and scalable for huge datasets with O(nlog(n)) computational complexity in the number of data samples. It is also robust to gross corruptions in the dataset as well as to the model parameters. Clustering experiments on 7 benchmark datasets with different types of corruptions and background separation experiments on 3 video datasets show that our proposed model outperforms 10 state-of-the-art dimensionality reduction models. Our theoretical analysis proves that the proposed model is able to recover approximate low-rank representations with a bounded error for clusterable data

Scalable Low-rank Matrix and Tensor Decomposition on Graphs

In many signal processing, machine learning and computer vision applications, one often has to deal with high dimensional and big datasets such as images, videos, web content, etc. The data can come in various forms, such as univariate or multivariate time series, matrices or high dimensional tensors. The goal of the data mining community is to reveal the hidden linear or non-linear structures in the datasets. Over the past couple of decades matrix factorization, owing to its intrinsic association with dimensionality reduction has been adopted as one of the key methods in this context. One can either use a single linear subspace to approximate the data (the standard Principal Component Analysis (PCA) approach) or a union of low dimensional subspaces where each data class belongs to a different subspace. In many cases, however, the low dimensional data follows some additional structure. Knowledge of such structure is beneficial, as we can use it to enhance the representativity of our models by adding structured priors. A nowadays standard way to represent pairwise affinity between objects is by using graphs. The introduction of graph-based priors to enhance matrix factorization models has recently brought them back to the highest attention of the data mining community. Representation of a signal on a graph is well motivated by the emerging field of signal processing on graphs, based on notions of spectral graph theory. The underlying assumption is that high-dimensional data samples lie on or close to a smooth low-dimensional manifold. Interestingly, the underlying manifold can be represented by its discrete proxy, i.e. a graph. A primary limitation of the state-of-the-art low-rank approximation methods is that they do not generalize for the case of non-linear low-rank structures. Furthermore, the standard low-rank extraction methods for many applications, such as low-rank and sparse decomposition, are computationally cumbersome. We argue, that for many machine learning and signal processing applications involving big data, an approximate low-rank recovery suffices. Thus, in this thesis, we present solutions to the above two limitations by presenting a new framework for scalable but approximate low-rank extraction which exploits the hidden structure in the data using the notion of graphs. First, we present a novel signal model, called `Multilinear low-rank tensors on graphs (MLRTG)' which states that a tensor can be encoded as a multilinear combination of the low-frequency graph eigenvectors, where the graphs are constructed along the various modes of the tensor. Since the graph eigenvectors have the interpretation of \textit{non-linear} embedding of a dataset on the low-dimensional manifold, we propose a method called `Graph Multilinear SVD (GMLSVD)' to recover PCA based linear subspaces from these eigenvectors. Finally, we propose a plethora of highly scalable matrix and tensor based problems for low-rank extraction which implicitly or explicitly make use of the GMLSVD framework. The core idea is to replace the expensive iterative SVD operations by updating the linear subspaces from the fixed non-linear ones via low-cost operations. We present applications in low-rank and sparse decomposition and clustering of the low-rank features to evaluate all the proposed methods. Our theoretical analysis shows that the approximation error of the proposed framework depends on the spectral properties of the graph Laplacian

Ewing\u27s sarcoma arising from the adrenal gland in a young male: a case report

Background: Ewing\u27s sarcoma uncommonly arises from extraosseous soft tissue or parenchymal organs. Primary adrenal Ewing\u27s Sarcoma, although very rare, is extremely aggressive and commonly fatal. CASE PRESENTATION: A 17 year old Pakistani male was referred to the outpatient oncology clinic at our center with a three month history of concomitant pain, swelling and dragging sensation in the right hypochondrium. Abdominal examination revealed a large, firm mass in the right hypochondrium extending into the right lumbar region and epigastrium. His genital exam was unremarkable and there were no stigmata of hepatic or adrenal disease.Computed tomography scans revealed a large peripherally enhancing mass in the hepatorenal area, biopsy of which showed a neoplastic lesion composed of small round blue cells which exhibited abundance of glycogen and stained diffusely positive for CD99 (MIC2 antigen). Fluorescence in situ hybridization demonstrated gene rearrangement at chromosome 22q12 which confirmed the diagnosis of Ewing\u27s sarcoma. Staging scans revealed pulmonary metastasis and hence he was commenced on systemic chemotherapy. CONCLUSION: This case report highlights the importance of keeping Ewing\u27s sarcoma in mind when a young patient presents with a large non-functional adrenal mass

Compressed Sensing and Adaptive Graph Total Variation for Tomographic Reconstructions

Compressed Sensing (CS) and Total Variation (TV)- based iterative image reconstruction algorithms have received increased attention recently. This is due to the ability of such methods to reconstruct from limited and noisy data. Local TV methods fail to preserve texture details and fine structures, which are tedious for the method to distinguish from noise. In many cases local methods also create additional artifacts due to over smoothing. Non-Local Total Variation (NLTV) has been increasingly used for medical imaging applications. However, it is not updated in every iteration of the algorithm, has a high computational complexity and depends on the scale of pairwise parameters. In this work we propose using Adaptive Graph- based TV in combination with CS (ACSGT). Similar to NLTV our proposed method goes beyond spatial similarity between different regions of an image being reconstructed by establishing a connection between similar regions in the image regardless of spatial distance. However, it is computationally much more efficient and scalable when compared to NLTV due to the use of approximate nearest neighbor search algorithm. Moreover, our method is adaptive, i.e, it involves updating the graph prior every iteration making the connection between similar regions stronger. Since TV is a special case of graph TV the proposed method can be seen as a generalization of CS and TV methods. We test our proposed algorithm by reconstructing a variety of different phantoms from limited and corrupted data and observe that we achieve a better result with ACSGT in every case

On the Soliton Solutions for the Stochastic Konno–Oono System in Magnetic Field with the Presence of Noise

In this study, we consider the stochastic Konno–Oono system to investigate the soliton solutions under the multiplicative sense. The multiplicative noise is considered firstly in the Stratonovich sense and secondly in the Itoˆ sense. Applications of the Konno–Oono system include current-fed strings interacting with an external magnetic field. The F-expansion method is used to find the different types of soliton solutions in the form of dark, singular, complex dark, combo, solitary, periodic, mixed periodic, and rational functions. These solutions are applicable in the magnetic field when we study it at the micro level. Additionally, the absolute, real, and imaginary physical representations in three dimensions and the corresponding contour plots of some solutions are drawn in the sense of noise by the different choices of parameters.This research was funded by Basque Government through Grants IT1555-22 and KK-2022/00090; and (MCIN/AEI 269.10.13039/501100011033/FEDER, UE) for Grants PID2021-1235430B-C21 and PID2021-1235430B-C22