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Data is said to follow the transform (or analysis) sparsity model if it becomes sparse when acted on by a linear operator called a sparsifying transform. Several algorithms have been designed to learn such a transform directly from data, and data-adaptive sparsifying transforms have demonstrated excellent performance in signal restoration tasks. Sparsifying transforms are typically learned using small sub-regions of data called patches, but these algorithms often ignore redundant information shared between neighboring patches. We show that many existing transform and analysis sparse representations can be viewed as filter banks, thus linking the local properties of patch-based model to the global properties of a convolutional model. We propose a new transform learning framework where the sparsifying transform is an undecimated perfect reconstruction filter bank. Unlike previous transform learning algorithms, the filter length can be chosen independently of the number of filter bank channels. Numerical results indicate filter bank sparsifying transforms outperform existing patch-based transform learning for image denoising while benefiting from additional flexibility in the design process.
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Automatic heart sound abnormality detection can play a vital role in the early diagnosis of heart diseases, particularly in low-resource settings. The state-of-the-art algorithms for this task utilize a set of Finite Impulse Response (FIR) band-pass filters as a front-end followed by a Convolutional Neural Network (CNN) model. In this work, we propound a novel CNN architecture that integrates the front-end bandpass filters within the network using time-convolution (tConv) layers, which enables the FIR filter-bank parameters to become learnable. Different initialization strategies for the learnable filters, including random parameters and a set of predefined FIR filter-bank coefficients, are examined. Using the proposed tConv layers, we add constraints to the learnable FIR filters to ensure linear and zero phase responses. Experimental evaluations are performed on a balanced 4-fold cross-validation task prepared using the PhysioNet/CinC 2016 dataset. Results demonstrate that the proposed models yield superior performance compared to the state-of-the-art system, while the linear phase FIR filterbank method provides an absolute improvement of 9.54% over the baseline in terms of an overall accuracy metric.
In this paper we characterize and construct novel oversampled filter banks implementing fusion frames. A fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way. When properly designed, fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures. However, up to this point, few implementable constructions of such frames were known; we show how to construct them using oversampled filter banks. In this work, we first provide polyphase domain characterizations of filter bank fusion frames. We then use these characterizations to construct filter bank fusion fra
We study the learnability of a class of compact operators known as Schatten--von Neumann operators. These operators between infinite-dimensional function spaces play a central role in a variety of applications in learning theory and inverse problems. We address the question of sample complexity of learning Schatten-von Neumann operators and provide an upper bound on the number of measurements required for the empirical risk minimizer to generalize with arbitrary precision and probability, as a function of class parameter $p$. Our results give generalization guarantees for regression of infinite-dimensional signals from infinite-dimensional data. Next, we adapt the representer theorem of Abernethy emph{et al.} to show that empirical risk minimization over an a priori infinite-dimensional, non-compact set, can be converted to a convex finite dimensional optimization problem over a compact set. In summary, the class of $p$-Schatten--von Neumann operators is probably approximately correct (PAC)-learnable via a practical convex program for any $p < infty$.
Connections between integration along hypersufaces, Radon transforms, and neural networks are exploited to highlight an integral geometric mathematical interpretation of neural networks. By analyzing the properties of neural networks as operators on probability distributions for observed data, we show that the distribution of outputs for any node in a neural network can be interpreted as a nonlinear projection along hypersurfaces defined by level surfaces over the input data space. We utilize these descriptions to provide new interpretation for phenomena such as nonlinearity, pooling, activation functions, and adversarial examples in neural network-based learning problems.
An electrocardiogram (ECG) is a time-series signal that is represented by one-dimensional (1-D) data. Higher dimensional representation contains more information that is accessible for feature extraction. Hidden variables such as frequency relation and morphology of segment is not directly accessible in the time domain. In this paper, 1-D time series data is converted into multi-dimensional representation in the form of multichannel 2-D images. Following that, deep learning was used to train a deep neural network based classifier to detect arrhythmias. The results of simulation on testing database demonstrate the effectiveness of the proposed methodology by showing an outstanding classification performance compared to other existing methods and hand-crafted annotations made by certified cardiologists.
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