In pattern recognition, the K-Nearest Neighbor algorithm (KNN) is a method for classifying objects based on the closest training examples in the feature space. KNN is a type of instance-based learning, or lazy learning where the function is only approximated locally and all computation is deferred until classification. The KNN algorithm is amongst the simplest of all machine learning algorithms: an object is classified by a majority vote of its neighbors, with the object being assigned to the class most common amongst its k nearest neighbors (k is a positive integer, typically small). If k 1, then the object is simply assigned to the class of its nearest neighbor [Source: Wikipedia]. In today's post, we explore the application of KNN to an automobile manufacturer that has developed prototypes for two new vehicles, a car and a truck.

Gradient descent is one of the most popular algorithms to perform optimization and by far the most common way to optimize neural networks. At the same time, every state-of-the-art Deep Learning library contains implementations of various algorithms to optimize gradient descent (e.g. These algorithms, however, are often used as black-box optimizers, as practical explanations of their strengths and weaknesses are hard to come by. This blog post aims at providing you with intuitions towards the behaviour of different algorithms for optimizing gradient descent that will help you put them to use. We are first going to look at the different variants of gradient descent. We will then briefly summarize challenges during training. Subsequently, we will introduce the most common optimization algorithms by showing their motivation to resolve these challenges and how this leads to the derivation of their update rules. We will also take a short look at algorithms and architectures to optimize gradient descent in a parallel and distributed setting.

The determination of cluster centers generally depends on the scale that we use to analyze the data to be clustered. Inappropriate scale usually leads to unreasonable cluster centers and thus unreasonable results. In this study, we first consider the similarity of elements in the data as the connectivity of nodes in an undirected graph, then present the concept of a connection center and regard it as the cluster center of the data. Based on this definition, the determination of cluster centers and the assignment of class are very simple, natural and effective. One more crucial finding is that the cluster centers of different scales can be obtained easily by the different powers of a similarity matrix and the change of power from small to large leads to the dynamic evolution of cluster centers from local (microscopic) to global (microscopic). Further, in this process of evolution, the number of categories changes discontinuously, which means that the presented method can automatically skip the unreasonable number of clusters, suggest appropriate observation scales and provide corresponding cluster results.

We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function $f$, consistent estimators of the mean embedding of a random variable $X$ lead to consistent estimators of the mean embedding of $f(X)$. For Mat\'ern kernels and sufficiently smooth functions we also provide rates of convergence. Our results extend to functions of multiple random variables. If the variables are dependent, we require an estimator of the mean embedding of their joint distribution as a starting point; if they are independent, it is sufficient to have separate estimators of the mean embeddings of their marginal distributions. In either case, our results cover both mean embeddings based on i.i.d. samples as well as "reduced set" expansions in terms of dependent expansion points. The latter serves as a justification for using such expansions to limit memory resources when applying the approach as a basis for probabilistic programming.

While most of these algorithms have polynomial-time complexity, determinantal point processes are too slow in practice for large number N of items to choose a subset from. Simplest algorithms have cubic running-time complexity and do not scale well to more than N 1000. Some progress has been made recently to reach quadratic or linear time complexity in N when imposing low-rank constraints, for both learning and inference [Mariet and Sra, 2016, Gartrell et al., 2016]. This is not enough, in particular for applications in continuous DPPs where the base set is infinite, and for modelling documents as a subset of all possible sentences: the number of sentences, even taken with a bag-of-word assumption, scales exponentially with the vocabulary size. Our goal in this paper is to design a class of DPPs which can be manipulated (for inference and parameter learning) in potentially sublinear time in the number of items N. In order to circumvent even linear-time complexity, we consider a novel class of DPPs which relies on a particular low-rank decomposition of the associated positive definite matrices.

In the matrix completion problem, we seek to recover or estimate a matrix, when only a fraction of its entries are observed. While it is impossible to complete an arbitrary matrix using only partial observations of its entries, it may be possible to fully recover matrix entries when the matrix has an appropriate underlying structure. For example, most low-rank matrices can be completed accurately with high probability, by solving a convex optimization problem (Candés and Recht, 2009). Consequently, algorithms for lowrank matrix completion have enjoyed widespread use across many disciplines, including collaborative filtering and recommender systems (Koren et al., 2009), multi-task learning and classification (Amit et al., 2007; Argyriou et al., 2007; Wu and Lange, 2015), computer vision (Chen and Suter, 2004), statistical genetics (Chi et al., 2013), as well as remote sensing (Malek-Mohammadi et al., 2014). In this paper, we consider matrix completion under a structural assumption that is closely related to the low-rank assumption; i.e., we assume that the matrix entries vary "smoothly" with respect to a graphical organization of the rows and columns. For example, in the context of a movie recommendation system, we seek to complete a user-by-movies ratings matrix. We may have additional information about users, such as if pairs of users are friends on a social media application, as well as additional information from a movie database, such as the co-occurrence of certain film principles. We expect the entries of a movie ratings matrix to vary "smoothly" over a neighborhood of users, defined by a friendship graph, and over a neighborhood of movies, defined by a shared movie principles graph. When such local similarity structure exists, and is available, it behooves us to leverage this information to predict missing entries in a matrix.

In this paper we study the problem of exact recovery of the pure-strategy Nash equilibria (PSNE) set of a graphical game from noisy observations of joint actions of the players alone. We consider sparse linear influence games --- a parametric class of graphical games with linear payoffs, and represented by directed graphs of n nodes (players) and in-degree of at most k. We present an $\ell_1$-regularized logistic regression based algorithm for recovering the PSNE set exactly, that is both computationally efficient --- i.e. runs in polynomial time --- and statistically efficient --- i.e. has logarithmic sample complexity. Specifically, we show that the sufficient number of samples required for exact PSNE recovery scales as $\mathcal{O}(\mathrm{poly}(k) \log n)$. We also validate our theoretical results using synthetic experiments.

We develop a class of rules spanning the range between quadratic discriminant analysis and naive Bayes, through a path of sparse graphical models. A group lasso penalty is used to introduce shrinkage and encourage a similar pattern of sparsity across precision matrices. It gives sparse estimates of interactions and produces interpretable models. Inspired by the connected-components structure of the estimated precision matrices, we propose the community Bayes model, which partitions features into several conditional independent communities and splits the classification problem into separate smaller ones. The community Bayes idea is quite general and can be applied to non-Gaussian data and likelihood-based classifiers.

Python is one of the fastest-growing platforms for applied machine learning. In this mini-course, you will discover how you can get started, build accurate models and confidently complete predictive modeling machine learning projects using Python in 14 days. This is a big and important post. You might want to bookmark it. Python Machine Learning Mini-Course Photo by Dave Young, some rights reserved.

Let's go through an example of Cancer Tissue Observations: Logistic regression is a popular method to predict a binary response. It is a special case of Generalized Linear models that predicts the probability of the outcome. Logistic regression measures the relationship between the Y "Label" and the X "Features" by estimating probabilities using a logistic function. The model predicts a probability which is used to predict the label class. Our data is from the Wisconsin Diagnostic Breast Cancer (WDBC) Data Set which categorizes breast tumor cases as either benign or malignant based on 9 features to predict the diagnosis.