Then this course is for you! This course has been designed by two professional Data Scientists so that we can share our knowledge and help you learn complex theory, algorithms, and coding libraries in a simple way. We will walk you step-by-step into the World of Machine Learning. With every tutorial, you will develop new skills and improve your understanding of this challenging yet lucrative sub-field of Data Science. This course is fun and exciting, but at the same time, we dive deep into Machine Learning.

There has been an intense recent activity in embedding of very high dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part we cover nonlinear methods such as principal curves, multidimensional scaling, local linear methods, ISOMAP, graph based methods and kernel based methods. The second part is concerned with topological embedding methods, in particular mapping topological properties into persistence diagrams. Another type of data sets with a tremendous growth is very high-dimensional network data. The task considered in part three is how to embed such data in a vector space of moderate dimension to make the data amenable to traditional techniques such as cluster and classification techniques. The final part of the survey deals with embedding in $\mathbb{R}^2$, which is visualization. Three methods are presented: $t$-SNE, UMAP and LargeVis based on methods in parts one, two and three, respectively. The methods are illustrated and compared on two simulated data sets; one consisting of a triple of noisy Ranunculoid curves, and one consisting of networks of increasing complexity and with two types of nodes.

This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [ACHA 45 (2018), no. 3, 616-641] to arbitrary (signed) signals on $\overline{\mathbb{R}}$. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demonstrate the application of the transform in classifying (detecting) signals under random displacements.

Approximate message passing (AMP) is a low-cost iterative parameter-estimation technique for certain high-dimensional linear systems with non-Gaussian distributions. However, AMP only applies to independent identically distributed (IID) transform matrices, but may become unreliable for other matrix ensembles, especially for ill-conditioned ones. To handle this difficulty, orthogonal/vector AMP (OAMP/VAMP) was proposed for general right-unitarily-invariant matrices. However, the Bayes-optimal OAMP/VAMP requires high-complexity linear minimum mean square error estimator. To solve the disadvantages of AMP and OAMP/VAMP, this paper proposes a memory AMP (MAMP), in which a long-memory matched filter is proposed for interference suppression. The complexity of MAMP is comparable to AMP. The asymptotic Gaussianity of estimation errors in MAMP is guaranteed by the orthogonality principle. A state evolution is derived to asymptotically characterize the performance of MAMP. Based on the state evolution, the relaxation parameters and damping vector in MAMP are optimized. For all right-unitarily-invariant matrices, the optimized MAMP converges to OAMP/VAMP, and thus is Bayes-optimal if it has a unique fixed point. Finally, simulations are provided to verify the validity and accuracy of the theoretical results.

In this paper, we describe an open source Python toolkit named Uncertainty Quantification 360 (UQ360) for the uncertainty quantification of AI models. The goal of this toolkit is twofold: first, to provide a broad range of capabilities to streamline as well as foster the common practices of quantifying, evaluating, improving, and communicating uncertainty in the AI application development lifecycle; second, to encourage further exploration of UQ's connections to other pillars of trustworthy AI such as fairness and transparency through the dissemination of latest research and education materials. Beyond the Python package (\url{https://github.com/IBM/UQ360}), we have developed an interactive experience (\url{http://uq360.mybluemix.net}) and guidance materials as educational tools to aid researchers and developers in producing and communicating high-quality uncertainties in an effective manner.

Recent works leveraging Graph Neural Networks to approach graph matching tasks have shown promising results. Recent progress in learning discrete distributions poses new opportunities for learning graph matching models. In this work, we propose a new model, Stochastic Iterative Graph MAtching (SIGMA), to address the graph matching problem. Our model defines a distribution of matchings for a graph pair so the model can explore a wide range of possible matchings. We further introduce a novel multi-step matching procedure, which learns how to refine a graph pair's matching results incrementally. The model also includes dummy nodes so that the model does not have to find matchings for nodes without correspondence. We fit this model to data via scalable stochastic optimization. We conduct extensive experiments across synthetic graph datasets as well as biochemistry and computer vision applications. Across all tasks, our results show that SIGMA can produce significantly improved graph matching results compared to state-of-the-art models. Ablation studies verify that each of our components (stochastic training, iterative matching, and dummy nodes) offers noticeable improvement.

Feature selection identifies subsets of informative features and reduces dimensions in the original feature space, helping provide insights into data generation or a variety of domain problems. Existing methods mainly depend on feature scoring functions or sparse regularizations; nonetheless, they have limited ability to reconcile the representativeness and inter-correlations of features. In this paper, we introduce a novel, simple yet effective regularization approach, named top-$k$ regularization, to supervised feature selection in regression and classification tasks. Structurally, the top-$k$ regularization induces a sub-architecture on the architecture of a learning model to boost its ability to select the most informative features and model complex nonlinear relationships simultaneously. Theoretically, we derive and mathematically prove a uniform approximation error bound for using this approach to approximate high-dimensional sparse functions. Extensive experiments on a wide variety of benchmarking datasets show that the top-$k$ regularization is effective and stable for supervised feature selection.

Cardiac auscultation is an essential point-of-care method used for the early diagnosis of heart diseases. Automatic analysis of heart sounds for abnormality detection is faced with the challenges of additive noise and sensor-dependent degradation. This paper aims to develop methods to address the cardiac abnormality detection problem when both types of distortions are present in the cardiac auscultation sound. We first mathematically analyze the effect of additive and convolutional noise on short-term filterbank-based features and a Convolutional Neural Network (CNN) layer. Based on the analysis, we propose a combination of linear and logarithmic spectrogram-image features. These 2D features are provided as input to a residual CNN network (ResNet) for heart sound abnormality detection. Experimental validation is performed on an open-access heart sound abnormality detection dataset involving noisy recordings obtained from multiple stethoscope sensors. The proposed method achieves significantly improved results compared to the conventional approaches, with an area under the ROC (receiver operating characteristics) curve (AUC) of 91.36%, F-1 score of 84.09%, and Macc (mean of sensitivity and specificity) of 85.08%. We also show that the proposed method shows the best mean accuracy across different source domains including stethoscope and noise variability, demonstrating its effectiveness in different recording conditions. The proposed combination of linear and logarithmic features along with the ResNet classifier effectively minimizes the impact of background noise and sensor variability for classifying phonocardiogram (PCG) signals. The proposed method paves the way towards developing computer-aided cardiac auscultation systems in noisy environments using low-cost stethoscopes.

Unlike the conventional wisdom in statistical learning theory, the test error of a deep neural network (DNN) often demonstrates double descent: as the model complexity increases, it first follows a classical U-shaped curve and then shows a second descent. Through bias-variance decomposition, recent studies revealed that the bell-shaped variance is the major cause of model-wise double descent (when the DNN is widened gradually). This paper investigates epoch-wise double descent, i.e., the test error of a DNN also shows double descent as the number of training epoches increases. By extending the bias-variance analysis to epoch-wise double descent of the zero-one loss, we surprisingly find that the variance itself, without the bias, varies consistently with the test error. Inspired by this result, we propose a novel metric, optimization variance (OV), to measure the diversity of model updates caused by the stochastic gradients of random training batches drawn in the same iteration. OV can be estimated using samples from the training set only but correlates well with the (unknown) \emph{test} error, and hence early stopping may be achieved without using a validation set.

Contrastive learning has shown superior performance in embedding global and spatial invariant features in computer vision (e.g., image classification). However, its overall success of embedding local and spatial variant features is still limited, especially for semantic segmentation. In a per-pixel prediction task, more than one label can exist in a single image for segmentation (e.g., an image contains both cat, dog, and grass), thereby it is difficult to define 'positive' or 'negative' pairs in a canonical contrastive learning setting. In this paper, we propose an attention-guided supervised contrastive learning approach to highlight a single semantic object every time as the target. With our design, the same image can be embedded to different semantic clusters with semantic attention (i.e., coerce semantic masks) as an additional input channel. To achieve such attention, a novel two-stage training strategy is presented. We evaluate the proposed method on multi-organ medical image segmentation task, as our major task, with both in-house data and BTCV 2015 datasets. Comparing with the supervised and semi-supervised training state-of-the-art in the backbone of ResNet-50, our proposed pipeline yields substantial improvement of 5.53% and 6.09% in Dice score for both medical image segmentation cohorts respectively. The performance of the proposed method on natural images is assessed via PASCAL VOC 2012 dataset, and achieves 2.75% substantial improvement.