We propose a generic framework based on a new stochastic variance-reduced gradient descent algorithm for accelerating nonconvex low-rank matrix recovery. Starting from an appropriate initial estimator, our proposed algorithm performs projected gradient descent based on a novel semi-stochastic gradient specifically designed for low-rank matrix recovery. Based upon the mild restricted strong convexity and smoothness conditions, we derive a projected notion of the restricted Lipschitz continuous gradient property, and prove that our algorithm enjoys linear convergence rate to the unknown low-rank matrix with an improved computational complexity. Moreover, our algorithm can be employed to both noiseless and noisy observations, where the optimal sample complexity and the minimax optimal statistical rate can be attained respectively. We further illustrate the superiority of our generic framework through several specific examples, both theoretically and experimentally.

We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10-15 years, we present a Mixed Integer Linear Optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed in [8] and the continuous nonconvex quadratic optimization framework of [33]. We propose new discrete first-order methods, which when paired with state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular $\ell_{1}$-based approaches. We illustrate that our approach can handle problem instances with p = 10,000 features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling.

Imagine a case - if there is no straight line (or hyperplane) which can separate two classes? In the image shown below, there is a circle in 2D with red and blue data points all over it such that adjacent data points are of different colors. SVM handles the above case by using a kernel function to handle non-linear separable data. It is explained in the next section. In simple words, it is a method to make SVM run in case of non-linear separable data points.

I'll perform linear regression from scratch in Python using a method called'Gradient Descent' to determine the relationship between student test scores & amount of hours studied. This will be about 50 lines of code and I'll deep dive into the math behind this. That's what keeps me going.

Prerequisites: No prerequisites, knowledge of some undergraduate level mathematics would help but is not mandatory. Working knowledge of Python would be helpful if you want to run the source code that is provided. Taught by a Stanford-educated, ex-Googler and an IIT, IIM - educated ex-Flipkart lead analyst. This team has decades of practical experience in quant trading, analytics and e-commerce. The course is shy but confident: It is authoritative, drawn from decades of practical experience -but shies away from needlessly complicating stuff.

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Scikits are Python-based scientific toolboxes built around SciPy, the Python library for scientific computing. Scikit-learn is an open source project focused on machine learning: classification, regression, clustering, dimensionality reduction, model selection, and preprocessing. On the other hand, it has quite a nice selection of solid algorithms, and it uses Cython (the Python-to-C compiler) for functions that need to be fast, such as inner loops. Among the areas Scikit-learn does not cover are deep learning, reinforcement learning, graphical models, and sequence prediction. It is defined as being in and for Python, so it doesn't have APIs for other languages.

Clustering is a popular method essentially applied to data analysis, data mining, vector quantization and data compression. The most widely used clustering algorithm, which belongs to the group of partitioning algorithms, is the k-means. In this paper, we propose an extended version of k-means where the initial cluster centers are selected based on a heuristic data based formula, in contrast to random selection adopted by the traditional k-means algorithm. In particular, a new formula for selecting the initial cluster centers, before applying the k-means algorithm for clustering of a data set, is introduced. The new extended k-means algorithm is tested on clustering a set of 2-D data points.

Data scientists are professionals who use the most appropriate tools and methodologies to get their jobs done. The best data scientists avail themselves of the complete set of knowledge- and pattern-discovery approaches that involve statistical analysis. How should we refer to the sum total of data science techniques? Often, they are lumped under the term "advanced analytics." This phrase is deliberately vague in that it is intended as a catch-all for everything from statistical analysis and data mining to predictive modeling, natural language processing, support vector machines, and so on.