# data points

• Trung Nguyen and Edwin Bonilla

### Fast Allocation of Gaussian Process Experts (pdf)

We propose a scalable nonparametric Bayesian regression model based on a mixture of Gaussian process (GP) experts and the inducing points formalism underpinning sparse GP approximations. Each expert is augmented with a set of inducing points, and the allocation of data points to experts is defined probabilistically based on their proximity to the experts. This allocation mechanism enables a fast variational inference procedure for learning of the inducing inputs and hyperparameters of the experts. When using $K$ experts, our method can run $K^2$ times faster and use $K^2$ times less memory than popular sparse methods such as the FITC approximation. Furthermore, it is easy to parallelize and handles non-stationarity straightforwardly. Our experiments show that on medium-sized datasets (of around $10^4$ training points) it trains up to 5 times faster than FITC while achieving comparable accuracy. On a large dataset of $10^5$ training points, our method significantly outperforms six competitive baselines while requiring only a few hours of training.

• Greg Ver Steeg and Aram Galstyan and Fei Sha and Simon DeDeo

### Demystifying Information-Theoretic Clustering (pdf)

We propose a novel method for clustering data which is grounded in information-theoretic principles and requires no parametric assumptions. Previous attempts to use information theory to define clusters in an assumption-free way are based on maximizing mutual information between data and cluster labels. We demonstrate that this intuition suffers from a fundamental conceptual flaw that causes clustering performance to deteriorate as the amount of data increases. Instead, we return to the axiomatic foundations of information theory to define a meaningful clustering measure based on the notion of consistency under coarse-graining for finite data.

• Binbin Lin and Ji Yang and Xiaofei He and Jieping Ye

### Geodesic Distance Function Learning via Heat Flow on Vector Fields (pdf)

Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if the original manifold is not Euclidean. In this paper, we propose to learn the distance function directly on the manifold without embedding. We first provide a theoretical characterization of the distance function by its gradient field. Based on our theoretical analysis, we propose to first learn the gradient field of the distance function and then learn the distance function itself. Specifically, we set the gradient field of a local distance function as an initial vector field. Then we transport it to the whole manifold via heat flow on vector fields. Finally, the geodesic distance function can be obtained by requiring its gradient field to be close to the normalized vector field. Experimental results on both synthetic and real data demonstrate the effectiveness of our proposed algorithm.

### Discrete Chebyshev Classifiers (pdf)

In large scale learning problems it is often easy to collect simple statistics of the data, but hard or impractical to store all the original data. A key question in this setting is how to construct classifiers based on such partial information. One traditional approach to the problem has been to use maximum entropy arguments to induce a complete distribution on variables from statistics. However, this approach essentially makes conditional independence assumptions about the distribution, and furthermore does not optimize prediction loss. Here we present a framework for discriminative learning given a set of statistics. Specifically, we address the case where all variables are discrete and we have access to various marginals. Our approach minimizes the worst case hinge loss in this case, which upper bounds the generalization error. We show that for certain sets of statistics the problem is tractable, and in the general case can be approximated using MAP LP relaxations. Empirical results show that the method is competitive with other approaches that use the same input.

• Si Si and Cho-Jui Hsieh and Inderjit Dhillon

### Memory Efficient Kernel Approximation (pdf)

The scalability of kernel machines is a big challenge when facing millions of samples due to storage and computation issues for large kernel matrices, that are usually dense. Recently, many papers have suggested tackling this problem by using a low rank approximation of the kernel matrix. In this paper, we first make the observation that the structure of shift-invariant kernels changes from low-rank to block-diagonal (without any low-rank structure) when varying the scale parameter. Based on this observation, we propose a new kernel approximation algorithm -- Memory Efficient Kernel Approximation (MEKA), which considers both low-rank and clustering structure of the kernel matrix. We show that the resulting algorithm outperforms state-of-the-art low-rank kernel approximation methods in terms of speed, approximation error, and memory usage. As an example, on the MNIST2M dataset with two-million samples, our method takes 550 seconds on a single machine using less than 500 MBytes memory to achieve 0.2313 test RMSE for kernel ridge regression, while standard Nystr\{o

• Hsiang-Fu Yu and Prateek Jain and Purushottam Kar and Inderjit Dhillon

### Large-scale Multi-label Learning with Missing Labels (pdf)

The multi-label classification problem has generated significant interest in recent years. However, existing approaches do not adequately address two key challenges: (a) scaling up to problems with a large number (say millions) of labels, and (b) handling data with missing labels. In this paper, we directly address both these problems by studying the multi-label problem in a generic empirical risk minimization (ERM) framework. Our framework, despite being simple, is surprisingly able to encompass several recent label-compression based methods which can be derived as special cases of our method. To optimize the ERM problem, we develop techniques that exploit the structure of specific loss functions - such as the squared loss function - to obtain efficient algorithms. We further show that our learning framework admits excess risk bounds even in the presence of missing labels. Our bounds are tight and demonstrate better generalization performance for low-rank promoting trace-norm regularization when compared to (rank insensitive) Frobenius norm regularization. Finally, we present extensive empirical results on a variety of benchmark datasets and show that our methods perform significantly better than existing label compression based methods and can scale up to very large datasets such as a Wikipedia dataset that has more than 200,000 labels.

• Sergey Bartunov and Dmitry Vetrov

### Variational Inference for Sequential Distance Dependent Chinese Restaurant Process (pdf)

Recently proposed distance dependent Chinese Restaurant Process (ddCRP) generalizes extensively used Chinese Restaurant Process (CRP) by accounting for dependencies between data points. Its posterior is intractable and so far only MCMC methods were used for inference. Because of very different nature of ddCRP no prior developments in variational methods for Bayesian nonparametrics are appliable. In this paper we propose novel variational inference for important sequential case of ddCRP (seqddCRP) by revealing its connection with Laplacian of random graph constructed by the process. We develop efficient algorithm for optimizing variational lower bound and demonstrate its efficiency comparing to Gibbs sampler. We also apply our variational approximation to CRP-equivalent seqddCRP-mixture model, where it could be considered as alternative to one based on truncated stick-breaking representation. This allowed us to achieve significantly better variational lower bound than variational approximation based on truncated stick breaking for Dirichlet process.

• Anshumali Shrivastava and Ping Li

### Densifying One Permutation Hashing via Rotation for Fast Near Neighbor Search (pdf)

The query complexity of {\em locality sensitive hashing (LSH)

• Cho-Jui Hsieh and Si Si and Inderjit Dhillon

### A Divide-and-Conquer Solver for Kernel Support Vector Machines (pdf)

The kernel support vector machine (SVM) is one of the most widely used classification methods; however, the amount of computation required becomes the bottleneck when facing millions of samples. In this paper, we propose and analyze a novel divide-and-conquer solver for kernel SVMs (DC-SVM). In the division step, we partition the kernel SVM problem into smaller subproblems by clustering the data, so that each subproblem can be solved independently and efficiently. We show theoretically that the support vectors identified by the subproblem solution are likely to be support vectors of the entire kernel SVM problem, provided that the problem is partitioned appropriately by kernel clustering. In the conquer step, the local solutions from the subproblems are used to initialize a global coordinate descent solver, which converges quickly as suggested by our analysis. By extending this idea, we develop a multilevel Divide-and-Conquer SVM algorithm with adaptive clustering and early prediction strategy, which outperforms state-of-the-art methods in terms of training speed, testing accuracy, and memory usage. As an example, on the covtype dataset with half-a-million samples, DC-SVM is 7 times faster than LIBSVM in obtaining the exact SVM solution (to within 10^{-6

• Jinfeng Yi and Lijun Zhang and Jun Wang and Rong Jin and Anil Jain

### A Single-Pass Algorithm for Efficiently Recovering Sparse Cluster Centers of High-dimensional Data (pdf)

Learning a statistical model for high-dimensional data is an important topic in machine learning. Although this problem has been well studied in the supervised setting, little is known about its unsupervised counterpart. In this work, we focus on the problem of clustering high-dimensional data with sparse centers. In particular, we address the following open question in unsupervised learning: is it possible to reliably cluster high-dimensional data when the number of samples is smaller than the data dimensionality?" We develop an efficient clustering algorithm that is able to estimate sparse cluster centers with a single pass over the data. Our theoretical analysis shows that the proposed algorithm is able to accurately recover cluster centers with only $O(s\log d)$ number of samples (data points), provided all the cluster centers are $s$-sparse vectors in a $d$ dimensional space. Experimental results verify both the effectiveness and efficiency of the proposed clustering algorithm compared to the state-of-the-art algorithms on several benchmark datasets.

• Brooks Paige and Frank Wood

### A Compilation Target for Probabilistic Programming Languages (pdf)

Forward inference techniques such as sequential Monte Carlo and particle Markov chain Monte Carlo for probabilistic programming can be implemented in any programming language by creative use of standardized operating system functionality including processes, forking, mutexes, and shared memory. Exploiting this we have defined, developed, and tested a probabilistic programming language intermediate representation language we call probabilistic C, which itself can be compiled to machine code by standard compilers and linked to operating system libraries yielding an efficient, scalable, portable probabilistic programming compilation target. This opens up a new hardware and systems research path for optimizing probabilistic programming systems.

2013-2014 ICML | International Conference on Machine Learning