While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic bounds on the clustering accuracy and sample complexity of learning a mixture of two isotropic Gaussians in high dimensions under small mean separation. If there is a sparse subset of relevant dimensions that determine the mean separation, then the sample complexity only depends on the number of relevant dimensions and mean separation, and can be achieved by a simple computationally efficient procedure. Our results provide the first step of a theoretical basis for recent methods that combine feature selection and clustering.

While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic bounds on the clustering accuracy and sample complexity of learning a mixture of two isotropic Gaussians in high dimensions under small mean separation. If there is a sparse subset of relevant dimensions that determine the mean separation, then the sample complexity only depends on the number of relevant dimensions and mean separation, and can be achieved by a simple computationally efficient procedure. Our results provide the first step of a theoretical basis for recent methods that combine feature selection and clustering.

We describe a framework for learning an object classifier from a single example. This goal is achieved by emphasizing the relevant dimensions for classification using available examples of related classes. Learning to accurately classify objects from a single training example is often un- feasible due to overfitting effects. However, if the instance representa- tion provides that the distance between each two instances of the same class is smaller than the distance between any two instances from dif- ferent classes, then a nearest neighbor classifier could achieve perfect performance with a single training example. We therefore suggest a two stage strategy.

In several applications, besides getting a generative model of the data, we also want the model to be useful for specific downstream tasks. Mixture models are useful for identifying discrete components in the data, but may not identify components useful for downstream tasks if misspecified; further, current inference techniques often fail to overcome misspecification even when a supervisory signal is provided. We introduce the prediction-focused mixture model, which selects and models input features relevant to predicting the targets. We demonstrate that our approach identifies relevant signal from inputs even when the model is highly misspecified.

The design of efficient representations is well established as a fruitful way to explore and analyze complex or large data. In these representations, data are encoded with various visual attributes depending on the needs of the representation itself. To make coherent design choices about visual attributes, the visual search field proposes guidelines based on the human brain perception of features. However, information visualization representations frequently need to depict more data than the amount these guidelines have been validated on. Since, the information visualization community has extended these guidelines to a wider parameter space. This paper contributes to this theme by extending visual search theories to an information visualization context. We consider a visual search task where subjects are asked to find an unknown outlier in a grid of randomly laid out distractor. Stimuli are defined by color and shape features for the purpose of visually encoding categorical data. The experimental protocol is made of a parameters space reduction step (i.e., sub-sampling) based on a machine learning model, and a user evaluation to measure capacity limits and validate hypotheses. The results show that the major difficulty factor is the number of visual attributes that are used to encode the outlier. When redundantly encoded, the display heterogeneity has no effect on the task. When encoded with one attribute, the difficulty depends on that attribute heterogeneity until its capacity limit (7 for color, 5 for shape) is reached. Finally, when encoded with two attributes simultaneously, performances drop drastically even with minor heterogeneity.

For regular particle filter algorithm or Sequential Monte Carlo (SMC) methods, the initial weights are traditionally dependent on the proposed distribution, the posterior distribution at the current timestamp in the sampled sequence, and the target is the posterior distribution of the previous timestamp. This is technically correct, but leads to algorithms which usually have practical issues with degeneracy, where all particles eventually collapse onto a single particle. In this paper, we propose and evaluate using $k$ means clustering to attack and even take advantage of this degeneracy. Specifically, we propose a Stochastic SMC algorithm which initializes the set of $k$ means, providing the initial centers chosen from the collapsed particles. To fight against degeneracy, we adjust the regular SMC weights, mediated by cluster proportions, and then correct them to retain the same expectation as before. We experimentally demonstrate that our approach has better performance than vanilla algorithms.

Recommender systems, medical diagnosis, network security, etc., require on-going learning and decision-making in real time. These -- and many others -- represent perfect examples of the opportunities and difficulties presented by Big Data: the available information often arrives from a variety of sources and has diverse features so that learning from all the sources may be valuable but integrating what is learned is subject to the curse of dimensionality. This paper develops and analyzes algorithms that allow efficient learning and decision-making while avoiding the curse of dimensionality. We formalize the information available to the learner/decision-maker at a particular time as a context vector which the learner should consider when taking actions. In general the context vector is very high dimensional, but in many settings, the most relevant information is embedded into only a few relevant dimensions. If these relevant dimensions were known in advance, the problem would be simple -- but they are not. Moreover, the relevant dimensions may be different for different actions. Our algorithm learns the relevant dimensions for each action, and makes decisions based in what it has learned. Formally, we build on the structure of a contextual multi-armed bandit by adding and exploiting a relevance relation. We prove a general regret bound for our algorithm whose time order depends only on the maximum number of relevant dimensions among all the actions, which in the special case where the relevance relation is single-valued (a function), reduces to $\tilde{O}(T^{2(\sqrt{2}-1)})$; in the absence of a relevance relation, the best known contextual bandit algorithms achieve regret $\tilde{O}(T^{(D+1)/(D+2)})$, where $D$ is the full dimension of the context vector.

While several papers have investigated computationally and statistically efficient methods for learning Gaussian mixtures, precise minimax bounds for their statistical performance as well as fundamental limits in high-dimensional settings are not well-understood. In this paper, we provide precise information theoretic bounds on the clustering accuracy and sample complexity of learning a mixture of two isotropic Gaussians in high dimensions under small mean separation. If there is a sparse subset of relevant dimensions that determine the mean separation, then the sample complexity only depends on the number of relevant dimensions and mean separation, and can be achieved by a simple computationally efficient procedure. Our results provide the first step of a theoretical basis for recent methods that combine feature selection and clustering.

This paper compares a family of methods for characterizing neural feature selectivity with natural stimuli in the framework of the linear-nonlinear model. In this model, the neural firing rate is a nonlinear function of a small number of relevant stimulus components. The relevant stimulus dimensions can be found by maximizing one of the family of objective functions, Rényi divergences of different orders [1, 2]. We show that maximizing one of them, Rényi divergence of order 2, is equivalent to least-square fitting of the linear-nonlinear model to neural data. Next, we derive reconstruction errors in relevant dimensions found by maximizing Rényi divergences of arbitrary order in the asymptotic limit of large spike numbers. We find that the smallest errors are obtained with Rényi divergence of order 1, also known as Kullback-Leibler divergence.

This paper compares a family of methods for characterizing neural feature selectivity with natural stimuli in the framework of the linear-nonlinear model. In this model, the neural firing rate is a nonlinear function of a small number of relevant stimulus components. The relevant stimulus dimensions can be found by maximizing one of the family of objective functions, Rényi divergences of different orders [1, 2]. We show that maximizing one of them, Rényi divergence of order 2, is equivalent to least-square fitting of the linear-nonlinear model to neural data. Next, we derive reconstruction errors in relevant dimensions found by maximizing Rényi divergences of arbitrary order in the asymptotic limit of large spike numbers. We find that the smallest errors are obtained with Rényi divergence of order 1, also known as Kullback-Leibler divergence.