We propose a specialized string kernel for small bio-molecules, peptides and pseudo-sequences of binding interfaces. The kernel incorporates physico-chemical properties of amino acids and elegantly generalize eight kernels, such as the Oligo, the Weighted Degree, the Blended Spectrum, and the Radial Basis Function. We provide a low complexity dynamic programming algorithm for the exact computation of the kernel and a linear time algorithm for it's approximation. Combined with kernel ridge regression and SupCK, a novel binding pocket kernel, the proposed kernel yields biologically relevant and good prediction accuracy on the PepX database. For the first time, a machine learning predictor is capable of accurately predicting the binding affinity of any peptide to any protein. The method was also applied to both single-target and pan-specific Major Histocompatibility Complex class II benchmark datasets and three Quantitative Structure Affinity Model benchmark datasets. On all benchmarks, our method significantly (p-value < 0.057) outperforms the current state-of-the-art methods at predicting peptide-protein binding affinities. The proposed approach is flexible and can be applied to predict any quantitative biological activity. The method should be of value to a large segment of the research community with the potential to accelerate peptide-based drug and vaccine development.

ARTINGALES are one of the fundamental tools in probability theory and statistics for modeling and studying sequences of random variables. Some of the most well-known and widely used concentration inequalities for individual martingales are Hoeffding-Azuma's and Bernstein's inequalities [1], [2], [3]. We present a comparison inequality that bounds the expectation of a convex function of a martingale difference sequence shifted to the [0, 1] interval by the expectation of the same function of independent Bernoulli variables. We apply this inequality in order to derive a tighter analog of Hoeffding-Azuma's inequality for martingales. More importantly, we present a set of inequalities that make it possible to control weighted averages of multiple simultaneously evolving and interdependent martingales (see Figure 1 for an illustration).

In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate latent positions for random dot product graphs provided the latent positions are i.i.d. from some distribution. If class labels are observed for a number of vertices tending to infinity, then we show that the remaining vertices can be classified with error converging to Bayes optimal using the $k$-nearest-neighbors classification rule. We evaluate the proposed methods on simulated data and a graph derived from Wikipedia.

Distributions over rankings are used to model data in a multitude of real world settings such as preference analysis and political elections. Modeling such distributions presents several computational challenges, however, due to the factorial size of the set of rankings over an item set. Some of these challenges are quite familiar to the artificial intelligence community, such as how to compactly represent a distribution over a combinatorially large space, and how to efficiently perform probabilistic inference with these representations. With respect to ranking, however, there is the additional challenge of what we refer to as human task complexity -- users are rarely willing to provide a full ranking over a long list of candidates, instead often preferring to provide partial ranking information. Simultaneously addressing all of these challenges -- i.e., designing a compactly representable model which is amenable to efficient inference and can be learned using partial ranking data -- is a difficult task, but is necessary if we would like to scale to problems with nontrivial size. In this paper, we show that the recently proposed riffled independence assumptions cleanly and efficiently address each of the above challenges. In particular, we establish a tight mathematical connection between the concepts of riffled independence and of partial rankings. This correspondence not only allows us to then develop efficient and exact algorithms for performing inference tasks using riffled independence based representations with partial rankings, but somewhat surprisingly, also shows that efficient inference is not possible for riffle independent models (in a certain sense) with observations which do not take the form of partial rankings. Finally, using our inference algorithm, we introduce the first method for learning riffled independence based models from partially ranked data.

In this paper, we propose a semiparametric approach, named nonparanormal skeptic, for efficiently and robustly estimating high dimensional undirected graphical models. To achieve modeling flexibility, we consider Gaussian Copula graphical models (or the nonparanormal) as proposed by Liu et al. (2009). To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This celebrating result suggests that the Gaussian copula graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare different estimators for their graph recovery performance under both ideal and noisy settings. The proposed methods are then applied on a large-scale genomic dataset to illustrate their empirical usefulness. The R language software package huge implementing the proposed methods is available on the Comprehensive R Archive Network: http://cran. r-project.org/.

In recent years, data streaming has gained prominence due to advances in technologies that enable many applications to generate continuous flows of data. This increases the need to develop algorithms that are able to efficiently process data streams. Additionally, real-time requirements and evolving nature of data streams make stream mining problems, including clustering, challenging research problems. In this paper, we propose a one-pass streaming soft clustering (membership of a point in a cluster is described by a distribution) algorithm which approximates the "soft" version of the k-means objective function. Soft clustering has applications in various aspects of databases and machine learning including density estimation and learning mixture models. We first achieve a simple pseudo-approximation in terms of the "hard" k-means algorithm, where the algorithm is allowed to output more than $k$ centers. We convert this batch algorithm to a streaming one (using an extension of the k-means++ algorithm recently proposed) in the "cash register" model. We also extend this algorithm when the clustering is done over a moving window in the data stream.

Many statistical $M$-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension $\pdim$ to grow with (and possibly exceed) the sample size $\numobs$. This high-dimensional structure precludes the usual global assumptions---namely, strong convexity and smoothness conditions---that underlie much of classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that projected gradient descent has a globally geometric rate of convergence up to the \emph{statistical precision} of the model, meaning the typical distance between the true unknown parameter $\theta^*$ and an optimal solution $\hat{\theta}$. This result is substantially sharper than previous convergence results, which yielded sublinear convergence, or linear convergence only up to the noise level. Our analysis applies to a wide range of $M$-estimators and statistical models, including sparse linear regression using Lasso ($\ell_1$-regularized regression); group Lasso for block sparsity; log-linear models with regularization; low-rank matrix recovery using nuclear norm regularization; and matrix decomposition. Overall, our analysis reveals interesting connections between statistical precision and computational efficiency in high-dimensional estimation.

The recent developments of basis pursuit and compressed sensing seek to extract information from as few samples as possible. In such applications, since the number of samples is restricted, one should deploy the sampling points wisely. We are motivated to study the optimal distribution of finite sampling points. Formulation under the framework of optimal reconstruction yields a minimization problem. In the discrete case, we estimate the distance between the optimal subspace resulting from a general Karhunen-Loeve transform and the kernel space to obtain another algorithm that is computationally favorable. Numerical experiments are then presented to illustrate the performance of the algorithms for the searching of optimal sampling points.

We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vector-valued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings minimise, providing an intuitive understanding of the embeddings and a justification for their use. Furthermore, the equivalence allows the application of vector-valued regression methods and results to the problem of learning conditional distributions. Using this link we derive a sparse version of the embedding by considering alternative formulations. Further, by applying convergence results for vector-valued regression to the embedding problem we derive minimax convergence rates which are O(\log(n)/n) -- compared to current state of the art rates of O(n^{-1/4}) -- and are valid under milder and more intuitive assumptions. These minimax upper rates coincide with lower rates up to a logarithmic factor, showing that the embedding method achieves nearly optimal rates. We study our sparse embedding algorithm in a reinforcement learning task where the algorithm shows significant improvement in sparsity over an incomplete Cholesky decomposition.

This paper presents an unsupervised algorithm for nonlinear unmixing of hyperspectral images. The proposed model assumes that the pixel reflectances result from a nonlinear function of the abundance vectors associated with the pure spectral components. We assume that the spectral signatures of the pure components and the nonlinear function are unknown. The first step of the proposed method consists of the Bayesian estimation of the abundance vectors for all the image pixels and the nonlinear function relating the abundance vectors to the observations. The endmembers are subsequently estimated using Gaussian process regression. The performance of the unmixing strategy is evaluated with simulations conducted on synthetic and real data.