Sparse subspace clustering methods with sparsity induced by $\ell^{0}$-norm, such as $\ell^{0}$-Sparse Subspace Clustering ($\ell^{0}$-SSC)~\citep{YangFJYH16-L0SSC-ijcv}, are demonstrated to be more effective than its $\ell^{1}$ counterpart such as Sparse Subspace Clustering (SSC)~\citep{ElhamifarV13}. However, the theoretical analysis of $\ell^{0}$-SSC is restricted to clean data that lie exactly in subspaces. Real data often suffer from noise and they may lie close to subspaces. In this paper, we show that an optimal solution to the optimization problem of noisy $\ell^{0}$-SSC achieves subspace detection property (SDP), a key element with which data from different subspaces are separated, under deterministic and semi-random model. Our results provide theoretical guarantee on the correctness of noisy $\ell^{0}$-SSC in terms of SDP on noisy data for the first time, which reveals the advantage of noisy $\ell^{0}$-SSC in terms of much less restrictive condition on subspace affinity. In order to improve the efficiency of noisy $\ell^{0}$-SSC, we propose Noisy-DR-$\ell^{0}$-SSC which provably recovers the subspaces on dimensionality reduced data. Noisy-DR-$\ell^{0}$-SSC first projects the data onto a lower dimensional space by random projection, then performs noisy $\ell^{0}$-SSC on the projected data for improved efficiency. Experimental results demonstrate the effectiveness of Noisy-DR-$\ell^{0}$-SSC.

We give the first polynomial time algorithm for \emph{list-decodable covariance estimation}. For any $\alpha > 0$, our algorithm takes input a sample $Y \subseteq \mathbb{R}^d$ of size $n\geq d^{\mathsf{poly}(1/\alpha)}$ obtained by adversarially corrupting an $(1-\alpha)n$ points in an i.i.d. sample $X$ of size $n$ from the Gaussian distribution with unknown mean $\mu_*$ and covariance $\Sigma_*$. In $n^{\mathsf{poly}(1/\alpha)}$ time, it outputs a constant-size list of $k = k(\alpha)= (1/\alpha)^{\mathsf{poly}(1/\alpha)}$ candidate parameters that, with high probability, contains a $(\hat{\mu},\hat{\Sigma})$ such that the total variation distance $TV(\mathcal{N}(\mu_*,\Sigma_*),\mathcal{N}(\hat{\mu},\hat{\Sigma}))<1-O_{\alpha}(1)$. This is the statistically strongest notion of distance and implies multiplicative spectral and relative Frobenius distance approximation for parameters with dimension independent error. Our algorithm works more generally for $(1-\alpha)$-corruptions of any distribution $D$ that possesses low-degree sum-of-squares certificates of two natural analytic properties: 1) anti-concentration of one-dimensional marginals and 2) hypercontractivity of degree 2 polynomials. Prior to our work, the only known results for estimating covariance in the list-decodable setting were for the special cases of list-decodable linear regression and subspace recovery due to Karmarkar, Klivans, and Kothari (2019), Raghavendra and Yau (2019 and 2020) and Bakshi and Kothari (2020). These results need superpolynomial time for obtaining any subconstant error in the underlying dimension. Our result implies the first polynomial-time \emph{exact} algorithm for list-decodable linear regression and subspace recovery that allows, in particular, to obtain $2^{-\mathsf{poly}(d)}$ error in polynomial-time. Our result also implies an improved algorithm for clustering non-spherical mixtures.

Recently, one critical issue looms large in the field of recommender systems -- there are no effective benchmarks for rigorous evaluation -- which consequently leads to unreproducible evaluation and unfair comparison. We, therefore, conduct studies from the perspectives of practical theory and experiments, aiming at benchmarking recommendation for rigorous evaluation. Regarding the theoretical study, a series of hyper-factors affecting recommendation performance throughout the whole evaluation chain are systematically summarized and analyzed via an exhaustive review on 141 papers published at eight top-tier conferences within 2017-2020. We then classify them into model-independent and model-dependent hyper-factors, and different modes of rigorous evaluation are defined and discussed in-depth accordingly. For the experimental study, we release DaisyRec 2.0 library by integrating these hyper-factors to perform rigorous evaluation, whereby a holistic empirical study is conducted to unveil the impacts of different hyper-factors on recommendation performance. Supported by the theoretical and experimental studies, we finally create benchmarks for rigorous evaluation by proposing standardized procedures and providing performance of ten state-of-the-arts across six evaluation metrics on six datasets as a reference for later study. Overall, our work sheds light on the issues in recommendation evaluation, provides potential solutions for rigorous evaluation, and lays foundation for further investigation.

ABSTRACT Coronal mass ejections (CMEs) are the most geoeffective space weather phenomena, being associated with large geomagnetic storms, having the potential to cause disturbances to telecommunication, satellite network disruptions, power grid damages and failures. Thus, considering these storms' potential effects on human activities, accurate forecasts of the geoeffectiveness of CMEs are paramount. This work focuses on experimenting with different machine learning methods trained on white-light coronagraph datasets of close to sun CMEs, to estimate whether such a newly erupting ejection has the potential to induce geomagnetic activity. We developed binary classification models using logistic regression, K-Nearest Neighbors, Support Vector Machines, feed forward artificial neural networks, as well as ensemble models. At this time, we limited our forecast to exclusively use solar onset parameters, to ensure extended warning times. We discuss the main challenges of this task, namely the extreme imbalance between the number of geoeffective and ineffective events in our dataset, along with their numerous similarities and the limited number of available variables. We show that even in such conditions, adequate hit rates can be achieved with these models. INTRODUCTION The purpose of this work is to develop a machine learning (ML) based model that can predict whether a coronal mass ejection (CME) will be geoeffective, using only numerical solar parameters as input. Coronal mass ejections are solar eruptive events whose magnetically charged particles can, directly or indirectly, under certain circumstances, reach Earth and cause geomagnetic storms (GSs), i.e., be geoeffective. These storms represent perturbations in the Earth's magnetic field, which have the potential to lead to electrical systems and grids failure and/or damage, power outages, navigation errors, radio signal perturbations, significant exposure to dangerous radiations for astronauts during space missions, etc. Given the potential negative impacts of such storms, predicting their occurrence is paramount for enabling safeguarding of human technology (Schwenn 2006; Pulkkinen 2007; Council 2013; Vourlidas et al. 2019; Temmer 2021). The intensity of the storms can be measured by various geomagnetic indices such as Ap, Kp, AE, PC or Dst (see Lockwood 2013, and references therein). Herein, we have chosen to use the values of the Dst index (Sugiura 1964) to establish whether the magnetic field perturbations do, in fact, manifest as storms. This is an index that is calculated using four geomagnetic stations situated at low latitudes. Depending on the value of this index, it can be established whether these perturbations are associated with geomagnetic storms or not. In terms of storm intensity, one of the most popular classifications that takes into consideration the minimum value of the Dst index is that of Gonzalez et al. (1994).

In this beginner-friendly program, you will learn the fundamentals of machine learning and how to use these techniques to build real-world AI applications. This Specialization is taught by Andrew Ng, an AI visionary who has led critical research at Stanford University and groundbreaking work at Google Brain, Baidu, and Landing.AI to advance the AI field. This 3-course Specialization is an updated and expanded version of Andrew's pioneering Machine Learning course, rated 4.9 out of 5 and taken by over 4.8 million learners since it launched in 2012. It provides a broad introduction to modern machine learning, including supervised learning (multiple linear regression, logistic regression, neural networks, and decision trees), unsupervised learning (clustering, dimensionality reduction, recommender systems), and some of the best practices used in Silicon Valley for artificial intelligence and machine learning innovation (evaluating and tuning models, taking a data-centric approach to improving performance, and more.) By the end of this Specialization, you will have mastered key concepts and gained the practical know-how to quickly and powerfully apply machine learning to challenging real-world problems.

K Nearest Neighbors (KNN) is one useful supervised ML algorithm, generally used for classification problems but can be used for regression problems too. It is non-parametric algorithm, meaning the algorithm does no make any particular assumption about the kind of mapping function or the underlying data. For now, let us assume we are working on a binary classification problem and each observation in the data set (2-D) belongs to either positive class or negative class The main purpose of classification is given a point'xq', we have to predict the output value'yq'. Please note: There are various types of methods to calculate the distance between two points. I will try to cover them in a separate article later.

In this paper, we study the problem of inferring spatially-varying Gaussian Markov random fields (SV-GMRF) where the goal is to learn a network of sparse, context-specific GMRFs representing network relationships between genes. An important application of SV-GMRFs is in inference of gene regulatory networks from spatially-resolved transcriptomics datasets. The current work on inference of SV-GMRFs are based on the regularized maximum likelihood estimation (MLE) and suffer from overwhelmingly high computational cost due to their highly nonlinear nature. To alleviate this challenge, we propose a simple and efficient optimization problem in lieu of MLE that comes equipped with strong statistical and computational guarantees. Our proposed optimization problem is extremely efficient in practice: we can solve instances of SV-GMRFs with more than 2 million variables in less than 2 minutes. We apply the developed framework to study how gene regulatory networks in Glioblastoma are spatially rewired within tissue, and identify prominent activity of the transcription factor HES4 and ribosomal proteins as characterizing the gene expression network in the tumor peri-vascular niche that is known to harbor treatment resistant stem cells.

Stochastic approximation (SA) with multiple coupled sequences has found broad applications in machine learning such as bilevel learning and reinforcement learning (RL). In this paper, we study the finite-time convergence of nonlinear SA with multiple coupled sequences. Different from existing multi-timescale analysis, we seek for scenarios where a fine-grained analysis can provide the tight performance guarantee for multi-sequence single-timescale SA (STSA). At the heart of our analysis is the smoothness property of the fixed points in multi-sequence SA that holds in many applications. When all sequences have strongly monotone increments, we establish the iteration complexity of $\mathcal{O}(\epsilon^{-1})$ to achieve $\epsilon$-accuracy, which improves the existing $\mathcal{O}(\epsilon^{-1.5})$ complexity for two coupled sequences. When all but the main sequence have strongly monotone increments, we establish the iteration complexity of $\mathcal{O}(\epsilon^{-2})$. The merit of our results lies in that applying them to stochastic bilevel and compositional optimization problems, as well as RL problems leads to either relaxed assumptions or improvements over their existing performance guarantees.

Knowing "what is happening" and "what will happen" of the mobility in a city is the building block of a data-driven smart city system. In recent years, mobility digital twin that makes a virtual replication of human mobility and predicting or simulating the fine-grained movements of the subjects in a virtual space at a metropolitan scale in near real-time has shown its great potential in modern urban intelligent systems. However, few studies have provided practical solutions. The main difficulties are four-folds. 1) The daily variation of human mobility is hard to model and predict; 2) the transportation network enforces a complex constraints on human mobility; 3) generating a rational fine-grained human trajectory is challenging for existing machine learning models; and 4) making a fine-grained prediction incurs high computational costs, which is challenging for an online system. Bearing these difficulties in mind, in this paper we propose a two-stage human mobility predictor that stratifies the coarse and fine-grained level predictions. In the first stage, to encode the daily variation of human mobility at a metropolitan level, we automatically extract citywide mobility trends as crowd contexts and predict long-term and long-distance movements at a coarse level. In the second stage, the coarse predictions are resolved to a fine-grained level via a probabilistic trajectory retrieval method, which offloads most of the heavy computations to the offline phase. We tested our method using a real-world mobile phone GPS dataset in the Kanto area in Japan, and achieved good prediction accuracy and a time efficiency of about 2 min in predicting future 1h movements of about 220K mobile phone users on a single machine to support more higher-level analysis of mobility prediction.

Python has become the most popular data science and machine learning programming language. But in order to obtain effective data and results, it's important that you have a basic understanding of how it works with machine learning. In this introductory tutorial, you'll learn the basics of Python for machine learning, including different model types and the steps to take to ensure you obtain quality data, using a sample machine learning problem. In addition, you'll get to know some of the most popular libraries and tools for machine learning. Machine learning (ML) is a form of artificial intelligence (AI) that teaches computers to make predictions and recommendations and solve problems based on data. Its problem-solving capabilities make it a useful tool in industries such as financial services, healthcare, marketing and sales, and education among others. There are three main types of machine learning: supervised, unsupervised, and reinforcement. In supervised learning, the computer is given a set of training data that includes both the input data (what we want to predict) and the output data (the prediction).