An important machine learning method for dimensionality reduction is called Principal Component Analysis. It is a method that uses simple matrix operations from linear algebra and statistics to calculate a projection of the original data into the same number or fewer dimensions. In this tutorial, you will discover the Principal Component Analysis machine learning method for dimensionality reduction and how to implement it from scratch in Python. How to Calculate the Principal Component Analysis from Scratch in Python Photo by mickey, some rights reserved. Take my free 7-day email crash course now (with sample code).

We show that Entropy-SGD (Chaudhari et al., 2017), when viewed as a learning algorithm, optimizes a PAC-Bayes bound on the risk of a Gibbs (posterior) classifier, i.e., a randomized classifier obtained by a risk-sensitive perturbation of the weights of a learned classifier. Entropy-SGD works by optimizing the bound's prior, violating the hypothesis of the PAC-Bayes theorem that the prior is chosen independently of the data. Indeed, available implementations of Entropy-SGD rapidly obtain zero training error on random labels and the same holds of the Gibbs posterior. In order to obtain a valid generalization bound, we rely on a result showing that data-dependent priors obtained by stochastic gradient Langevin dynamics (SGLD) yield valid PAC-Bayes bounds provided the target distribution of SGLD is $\epsilon$-differentially private. We observe that test error on MNIST and CIFAR10 falls within the (empirically nonvacuous) risk bounds computed under the assumption that SGLD reaches stationarity. In particular, Entropy-SGLD can be configured to yield relatively tight generalization bounds and still fit real labels, although these same settings do not obtain state-of-the-art performance.

Nonnegative matrix factorization (NMF) has become a workhorse for signal and data analytics, triggered by its model parsimony and interpretability. Perhaps a bit surprisingly, the understanding to its model identifiability---the major reason behind the interpretability in many applications such as topic mining and hyperspectral imaging---had been rather limited until recent years. Beginning from the 2010s, the identifiability research of NMF has progressed considerably: Many interesting and important results have been discovered by the signal processing (SP) and machine learning (ML) communities. NMF identifiability has a great impact on many aspects in practice, such as ill-posed formulation avoidance and performance-guaranteed algorithm design. On the other hand, there is no tutorial paper that introduces NMF from an identifiability viewpoint. In this paper, we aim at filling this gap by offering a comprehensive and deep tutorial on model identifiability of NMF as well as the connections to algorithms and applications. This tutorial will help researchers and graduate students grasp the essence and insights of NMF, thereby avoiding typical `pitfalls' that are often times due to unidentifiable NMF formulations. This paper will also help practitioners pick/design suitable factorization tools for their own problems.

Matrix completion method has been used in a wide range of applications such as collaborative filtering for recommendation (Koren et al., 2009), multi-label learning (Cabral et al., 2011) and clustering (Hsieh et al., 2012). In these applications, every entry is modeled as the inner product between factors corresponding to the row and column variables. For example, in movie recommendation, each row factor represents the latent representation of a user and each column factor represents the latent representation of a movie. In many applications of significant interest, besides the partially observed matrix, side information, in the form of features, is also available. These might correspond to demographic information (genders, occupation) for users or product information (genre, director) in a movie recommender system for example. With such features at hand, one can model an observation as a specific linear interaction between features to reduce the model complexity.

In many applications the process of generating label information is expensive and time consuming. We present a new method that combines active and semi-supervised deep learning to achieve high generalization performance from a deep convolutional neural network with as few known labels as possible. In a setting where a small amount of labeled data as well as a large amount of unlabeled data is available, our method first learns the labeled data set. This initialization is followed by an expectation maximization algorithm, where further training reduces classification entropy on the unlabeled data by targeting a low entropy fit which is consistent with the labeled data. In addition the algorithm asks at a specified frequency an oracle for labels of data with entropy above a certain entropy quantile. Using this active learning component we obtain an agile labeling process that achieves high accuracy, but requires only a small amount of known labels. For the MNIST dataset we report an error rate of 2.06% using only 300 labels and 1.06% for 1,000 labels. These results are obtained without employing any special network architecture or data augmentation.

Distributed Stochastic Gradient Descent (SGD) when run in a synchronous manner, suffers from delays in waiting for the slowest learners (stragglers). Asynchronous methods can alleviate stragglers, but cause gradient staleness that can adversely affect convergence. In this work we present the first theoretical characterization of the speed-up offered by asynchronous methods by analyzing the trade-off between the error in the trained model and the actual training runtime (wallclock time). The novelty in our work is that our runtime analysis considers random straggler delays, which helps us design and compare distributed SGD algorithms that strike a balance between stragglers and staleness. We also present a new convergence analysis of asynchronous SGD variants without bounded or exponential delay assumptions.

We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small step size regime and the presence of omnidirectional noise, our weak approximating diffusion process suggests the following dynamics for the SGD iteration starting from a local minimizer (resp.~saddle point): it escapes in a number of iterations exponentially (resp.~almost linearly) dependent on the inverse stepsize. The results are obtained using the theory for random perturbations of dynamical systems (theory of large deviations for local minimizers and theory of exiting for unstable stationary points). In addition, we discuss the effects of batch size for the deep neural networks, and we find that small batch size is helpful for SGD algorithms to escape unstable stationary points and sharp minimizers. Our theory indicates that one should increase the batch size at later stage for the SGD to be trapped in flat minimizers for better generalization.

Fundamental statistics are useful tools in applied machine learning for a better understanding your data. They are also the tools that provide the foundation for more advanced linear algebra operations and machine learning methods, such as the covariance matrix and principal component analysis respectively. As such, it is important to have a strong grip on fundamental statistics in the context of linear algebra notation. In this tutorial, you will discover how fundamental statistical operations work and how to implement them using NumPy with notation and terminology from linear algebra. A Gentle Introduction to Expected Value, Variance, and Covariance with NumPy Photo by Robyn Jay, some rights reserved.

Selection of the optimal parameters for machine learning tasks is challenging. Some results may be bad not because the data is noisy or the used learning algorithm is weak, but due to the bad selection of the parameters values. This article gives a brief introduction about evolutionary algorithms (EAs) and describes genetic algorithm (GA) which is one of the simplest random-based EAs. Suppose that a data scientist has an image dataset divided into a number of classes and an image classifier is to be created. After the data scientist investigated the dataset, the K-nearest neighbor (KNN) seems to be a good option.

An image pattern can be represented by a probability distribution whose density is concentrated on different low-dimensional subspaces in the high-dimensional image space. Such probability densities have an astronomical number of local modes corresponding to typical pattern appearances. Related groups of modes can join to form macroscopic image basins that represent pattern concepts. Recent works use neural networks that capture high-order image statistics to learn Gibbs models capable of synthesizing realistic images of many patterns. However, characterizing a learned probability density to uncover the Hopfield memories of the model, encoded by the structure of the local modes, remains an open challenge. In this work, we present novel computational experiments that map and visualize the local mode structure of Gibbs densities. Efficient mapping requires identifying the global basins without enumerating the countless modes. Inspired by Grenander's jump-diffusion method, we propose a new MCMC tool called Attraction-Diffusion (AD) that can capture the macroscopic structure of highly non-convex densities by measuring metastability of local modes. AD involves altering the target density with a magnetization potential penalizing distance from a known mode and running an MCMC sample of the altered density to measure the stability of the initial chain state. Using a low-dimensional generator network to facilitate exploration, we map image spaces with up to 12,288 dimensions (64 $\times$ 64 pixels in RGB). Our work shows: (1) AD can efficiently map highly non-convex probability densities, (2) metastable regions of pattern probability densities contain coherent groups of images, and (3) the perceptibility of differences between training images influences the metastability of image basins.