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  1. Abstract Background

    Interpretation of high-throughput gene expression data continues to require mathematical tools in data analysis that recognizes the shape of the data in high dimensions. Topological data analysis (TDA) has recently been successful in extracting robust features in several applications dealing with high dimensional constructs. In this work, we utilize some recent developments in TDA to curate gene expression data. Our work differs from the predecessors in two aspects: (1) Traditional TDA pipelines use topological signatures called barcodes to enhance feature vectors which are used for classification. In contrast, this work involves curating relevant features to obtain somewhat better representatives with the help of TDA. This representatives of the entire data facilitates better comprehension of the phenotype labels. (2) Most of the earlier works employ barcodes obtained using topological summaries as fingerprints for the data. Even though they are stable signatures, there exists no direct mapping between the data and said barcodes.


    The topology relevant curated data that we obtain provides an improvement in shallow learning as well as deep learning based supervised classifications. We further show that the representative cycles we compute have an unsupervised inclination towards phenotype labels. This work thus shows that topological signatures are able to comprehend gene expression levels and classify cohorts accordingly.


    In this work, we engender representative persistent cycles to discern the gene expression data. These cycles allow us to directly procure genes entailed in similar processes.

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  2. Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in the purely topological persistence diagrams (also termed as barcodes). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with Z2 coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d-th persistence diagram of a weak (d+1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d+1)-pseudomanifold to be embedded in R^{d+1}Rd+1 so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data. 
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  3. With the rise in popularity of drones, their use in anti-social activities has also proliferated. Nationwide police increasingly report the appearance of drones in unauthorized settings such as public gatherings and also in the delivery of contraband to prisons. Detection and classification of drones in such environments is very challenging from both visual and acoustic perspective. Visual detection of drones is challenging due to their small size. There may be cases where views are obstructed, lighting conditions are poor, the field of view is narrow, etc. In contrast, acoustic-based detection methods are omnidirectional, however, they are prone to errors due to possible noise in the signal. This paper presents a method of predicting the presence (detection and classification) of a drone using a single microphone and other inexpensive computational devices. A Support Vector Machine classified the spectral and temporal features of pre-segments generated using a sliding window for the audio signal. Additionally, spectral subtraction was used to reconstruct the magnitude spectrum of drone sounds to reduce false alarms. To increase the accuracy of predictions, an added confidence script is proposed based on a queue-and-dump approach to make the system more robust. The proposed system was tested in real time in a realistic environment with various drone models and flight characteristics. Performance is satisfactory in a quiet setting but the system generates excessive false alarms when exposed to lawn equipment. 
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