Simplex-valued data appear throughout statistics and machine learning, for example in the context of transfer learning and compression of deep networks. Existing models for this class of data rely on the Dirichlet distribution or other related loss functions; here we show these standard choices suffer systematically from a number of limitations, including bias and numerical issues that frustrate the use of flexible network models upstream of these distributions. We resolve these limitations by introducing a novel exponential family of distributions for modeling simplex-valued data - the continuous categorical, which arises as a nontrivial multivariate generalization of the recently discovered continuous Bernoulli. Unlike the Dirichlet and other typical choices, the continuous categorical results in a well-behaved probabilistic loss function that produces unbiased estimators, while preserving the mathematical simplicity of the Dirichlet. As well as exploring its theoretical properties, we introduce sampling methods for this distribution that are amenable to the reparameterization trick, and evaluate their performance. Lastly, we demonstrate that the continuous categorical outperforms standard choices empirically, across a simulation study, an applied example on multi-party elections, and a neural network compression task.
Bayesian Optimization (BO) methods are useful for optimizing functions that are expen- sive to evaluate, lack an analytical expression and whose evaluations can be contaminated by noise. These methods rely on a probabilistic model of the objective function, typically a Gaussian process (GP), upon which an acquisition function is built. The acquisition function guides the optimization process and measures the expected utility of performing an evaluation of the objective at a new point. GPs assume continous input variables. When this is not the case, for example when some of the input variables …
We revisit binary decision trees from the perspective of partitions of the data. We introduce the notion of partitioning function, and we relate it to the growth function and to the VC dimension. We consider three types of features: real-valued, categorical ordinal and categorical nominal, with different split rules for each. For each feature type, we upper bound the partitioning function of the class of decision stumps before extending the bounds to the class of general decision tree (of any fixed structure) using a recursive approach. Using these new results, we are able to find the exact VC…
Columbia University（哥伦比亚大学）；Layer6 AI（Layer6人工智能公司）；New York University（纽约大学）
Probability distributions supported on the simplex enjoy a wide range of applications across statistics and machine learning. Recently, a novel family of such distributions has been discovered: the continuous categorical. This family enjoys remarkable mathematical simplicity; its density function resembles that of the Dirichlet distribution, but with a normalizing constant that can be written in closed form using elementary functions only. In spite of this mathematical simplicity, our understanding of the normalizing constant remains far from complete. In this work, we characterize the numeric…
This paper presents a proposal of a faster Wasserstein $k$-means algorithm for histogram data by reducing Wasserstein distance computations and exploiting sparse simplex projection. We shrink data samples, centroids, and the ground cost matrix, which leads to considerable reduction of the computations used to solve optimal transport problems without loss of clustering quality. Furthermore, we dynamically reduced the computational complexity by removing lower-valued data samples and harnessing sparse simplex projection while keeping the degradation of clustering quality lower. We designate this…
Mining clusters from data is an important endeavor in many applications. The $k$-means method is a popular, efficient, and distribution-free approach for clustering numerical-valued data, but does not apply for categorical-valued observations. The $k$-modes method addresses this lacuna by replacing the Euclidean with the Hamming distance and the means with the modes in the $k$-means objective function. We provide a novel, computationally efficient implementation of $k$-modes, called OTQT. We prove that OTQT finds updates to improve the objective function that are undetectable to existing $k$-m…
Samsung AI Center Cambridge（三星人工智能中心剑桥）；Rutgers University（罗格斯大学）
In deep representational learning, it is often desired to isolate a particular factor (termed {\em content}) from other factors (referred to as {\em style}). What constitutes the content is typically specified by users through explicit labels in the data, while all unlabeled/unknown factors are regarded as style. Recently, it has been shown that such content-labeled data can be effectively exploited by modifying the deep latent factor models (e.g., VAE) such that the style and content are well separated in the latent representations. However, the approach assumes that the content factor is cat…
Data augmentation plays a key role in modern machine learning pipelines. While numerous augmentation strategies have been studied in the context of computer vision and natural language processing, less is known for other data modalities. Our work extends the success of data augmentation to compositional data, i.e., simplex-valued data, which is of particular interest in the context of the human microbiome. Drawing on key principles from compositional data analysis, such as the Aitchison geometry of the simplex and subcompositions, we define novel augmentation strategies for this data modality.…
With the scale of data growing every day, reducing the dimensionality (a.k.a. sketching) of high-dimensional data has emerged as a task of paramount importance. Relevant issues to address in this context include the sheer volume of data that may consist of categorical samples, the typically streaming format of acquisition, and the possibly missing entries. To cope with these challenges, the present paper develops a novel categorical subspace learning approach to unravel the latent structure for three prominent categorical (bilinear) models, namely, Probit, Tobit, and Logit. The deterministic P…
Industrial process control systems try to keep an output variable within a given tolerance around a target value. PID control systems have been widely used in industry to control input variables in order to reach this goal. However, this kind of Transfer Function based approach cannot be extended to complex processes where input data might be non-numeric, high dimensional, sparse, etc. In such cases, there is still a need for determining the subspace of input data that produces an output within a given range. This paper presents a non-stochastic heuristic to determine input values for a mathem…
Deriving insights from high-dimensional data is one of the core problems in data mining. The difficulty mainly stems from the fact that there are exponentially many variable combinations to potentially consider, and there are infinitely many if we consider weighted combinations, even for linear combinations. Hence, an obvious question is whether we can automate the search for interesting patterns and visualizations. In this paper, we consider the setting where a user wants to learn as efficiently as possible about real-valued attributes. For example, to understand the distribution of crime rat…
